A plant cell division algorithm based on cell biomechanics and ...

PART OF A SPECIAL ISSUE ON FUNCTIONAL–STRUCTURAL PLANT MODELLING

A plant cell division algorithm based on cell biomechanics and ellipse-fitting

Metadel K. Abera1, Pieter Verboven1, Thijs Defraeye1, Solomon Workneh Fanta1, Maarten L. A. T. M. Hertog1,Jan Carmeliet2,3 and Bart M. Nicolai1,*

1Flanders Centre of Postharvest Technology/BIOSYST-MeBios, University of Leuven, Willem de Croylaan 42, B-3001, Leuven,Belgium, 2Building Physics, Swiss Federal Institute of Technology Zurich (ETHZ), Wolfgang-Pauli-Strasse 15, 8093 Zurich,

Switzerland and 3Laboratory for Building Science and Technology, Swiss Federal Laboratories for Materials Testing and Research(Empa), Uberlandstrasse 129, 8600 Dubendorf, Switzerland* For correspondence. E-mail [email protected]

Received: 16 September 2013 Returned for revision: 23 October 2013 Accepted: 24 March 2014 Published electronically: 25 May 2014

† Background and Aims The importance of cell division models in cellular pattern studies has been acknowledgedsince the 19th century. Most of the available models developed to date are limited to symmetric cell division withisotropic growth. Often, the actual growth of the cell wall is either not considered or is updated intermittently on aseparate time scale to the mechanics. This study presents a generic algorithm that accounts for both symmetricallyand asymmetrically dividing cells with isotropic and anisotropic growth. Actual growth of the cell wall is simulatedsimultaneously with the mechanics.† Methods The cell is considered as a closed, thin-walled structure, maintained in tension by turgor pressure. The cellwalls are represented as linear elastic elements that obey Hooke’s law. Cell expansion is induced by turgor pressureacting on the yielding cell-wall material. A system of differential equations for the positions and velocities of the cellvertices as well as for the actual growth of the cell wall is established. Readiness to divide is determined based oncell size. An ellipse-fitting algorithm is used to determine the position and orientation of the dividing wall. Thecell vertices, walls and cell connectivity are then updated and cell expansion resumes. Comparisons are madewith experimental data from the literature.† Key Results The generic plant cell division algorithm has been implemented successfully. It can handle both sym-metrically and asymmetrically dividing cells coupled with isotropic and anisotropic growth modes. Development ofthe algorithm highlighted the importance of ellipse-fitting to produce randomness (biological variability) even insymmetrically dividing cells. Unlike previous models, a differential equation is formulated for the resting lengthof the cell wall to simulate actual biological growth and is solved simultaneously with the position and velocity ofthe vertices.† Conclusions The algorithm presented can produce different tissues varying in topological and geometrical prop-erties. This flexibility to produce different tissue types gives the model great potential for use in investigations of plantcell division and growth in silico.

Key words: Cell division, biomechanics, turgor pressure, thin-walled structure, ellipse-fitting, geometric symmetry,geometric asymmetry, functional–structural plant modelling.

INTRODUCTION

Cellular pattern studies and simulation of higher level processessuch as phyllotaxis or vascular patterning call for models of thedivision and arrangement of cells into tissues (Smith et al., 2006;Merks et al., 2007). Many biophysiological processes in plantorgans, such as gas transport, are strong functions of the micro-structural geometry of the tissue (Ho et al., 2009, 2010, 2011,2012), which is in turn dependent on cell division and thearrangement of cells.

In contrast to animal cells, plant cells have relatively rigid cellwalls. The walls of the neighbouring cells are joined by themiddle lamellas (Romberger et al., 1993). Walls of neighbouringcells do not slide with respect to each other. Therefore, cell top-ology is maintained except in the event of intercellular gas spaceformation or cell division. These aspects should be taken intoaccount when considering cell division and expansive growth.

The cell division rules proposed by Hofmeister (1863), Sachs(1878) and Errera (1886) remain the most prominent basis formodern thoughts on cell division. According to Hofmeister,the dividing wall is inserted at right angles to the axis ofmaximal growth, while Sachs suggested that the new wall inter-sects the sidewalls at right angles. Errera’s rule states that the div-iding wall should be the shortest one that partitions the mothercell into two equal daughter cells (reviewed by Prusinkiewiczand Runions, 2012).

Computer models of tissues with cell division based on pos-ition and orientation have been developed by Sahlin andJonsson (2010). In their approach, the position of the new wallis determined to be either the centre of mass of the mother cellora point inside the mothercell chosen randomly, while its orien-tation is chosen based on a number of criteria. Although thismodel includes cell growth mechanics and includes actualgrowth of the cell wall by changing the resting length of thecell wall, it does not limit the actual growth of the cell-wall

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Annals of Botany 114: 605–617, 2014

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resting length, and how turgor force is implemented does notallow for cell–cell interactions.

Besson and Dumais (2011) developed a rule for symmetricdivision of plant cells based on probabilistic selection of divisionplanes. According to their work, Errera’s rule of cell divisionfailed to account for the variability observed in symmetric celldivisions, in particular that cells of identical shape do not neces-sarily adopt the same division plane. The variability in symmet-ric cell division is accounted for by introducing the concept oflocal minima rather than global minima. Robinson et al.(2011) introduced an asymmetric cell division algorithm inwhich the division wall is chosen as the shortest wall thatpasses through the nucleus of the mother cell. In their model,asymmetric cell division is achieved by displacing the nucleus

of the mother cell from the centroid of the cell in a randomdirection.

The dynamic pattern of cell arrangement is not only a functionof the position and orientation of division walls but also of thetiming of cell division and growth of the tissue. The early workof Korn (1969) reintroduced by Merks and Glazier (2005) repre-sents cells as a set of points, and growth is achieved by theaddition of new points to a cell. Nakielski (2008) developed amodel for growth and cell division based on the postulate ofHejnowicz (1984) that cells divide in relation to principal direc-tions of growth (PDGs) and supported by the experimentalwork of Lintilhac and Vesecky (1981) and Lynch and Lintilhac(1997). PDGs are mutually perpendicular directions alongwhich extreme growth rates are observed, which lead tounequal major and minor equivalent diameters for the cell.This is true for anisotropic growth, associated with anisotropicmechanical properties of the cell wall. Theyare in turn dependenton a growth tensor field defined at the organ level according to thework of Hejnowicz and Romberger (1984). Cell mechanics-based models for 2-D cell growth have been developed byseveral researchers (Dupuy et al., 2008, 2010; Sahlin andJonsson, 2010; Gibson et al., 2011; Merks et al., 2011; Aberaet al., 2013). The 2-D cell growth model developed by Aberaet al. (2013) has recently been extended to a 3-D cell growthmodel (Abera et al., 2014).

Not all plant cells grow isotropically; anisotropic growth isalso a common phenomenon. This anisotropy, defined asdirection-dependent growth of a cell that leads to a mother cellwith unequal major and minor diameters, is manifested as a con-sequence of anisotropic molecular wall structure, determined bydifferential spatial arrangement of the cellulose microfibrils thatare generally organized in layers of parallel fibres (reviewed byBaskin, 2005; Schopfer, 2006). A different definition of anisot-ropy is the temporal and/or spatial difference in growth rate ofthe cell. For the latter, a cell can still be isotropic in shape (seeKwiatkowska and Dumais, 2003; Kwiatkowska, 2006). In thepresent paper, the first definition is used. Nakielski (2008) usedthis concept to incorporate overall tissue anisotropy in theirmodel, which was based on PDGs. To our knowledge, althoughanisotropic growth of plant cells is a common phenomenon, theavailable literature on cell division models that use anisotropiccell growth is limited.

Based on the literature references detailed above, there is nosingle generic model. Some models are intended either for sym-metric or for asymmetric cell division. Some are focused simply

FI G. 1. Illustration of the cell division algorithm. 2-D cell division is shown,where the blue lines are the boundaries of the mother cell, the green line is the

fitted ellipse and the red line is the new wall dividing the two daughter cells.

q

FI G. 2. Illustration of the calculation of the interior angle. The procedure isrepeated for each cell at each vertex.

FI G. 3. Illustration of the cell division algorithm at different stages in time.

Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting606


on the division rules, without incorporating the timing of cell div-ision or the actual expansive growth and others do not considercell mechanics when modelling cell growth. Most of themodels introduced so far are based on isotropic growth. Hence,the objective of this paper is to develop a 2-D plant cell divisionalgorithm that is generic in that it includes symmetric cell div-ision by taking account of the randomness without abandoningErrera’s rule of cell division. Furthermore, it accounts for asym-metric cell division, isotropic cell growth and anisotropic cellgrowth. It is based on cell growth mechanics, which accountsfor cell–cell interaction through turgor force calculation andlimits the resting length of the cell walls. In our model,because the cell division rules are based on ellipse-fitting,there is no need for an iterative procedure to find the positionand orientation of the new dividing wall.

MATERIALS AND METHODS

Cell growth algorithm

In our model, the cell is considered aclosed thin-walled structure,maintained in tension by turgor pressure. The cell walls of adja-cent cells are modelled as parallel, linear elastic elements thatobey Hooke’s law, an approach similar to that used in otherplant tissue models (Prusinkiewicz and Lindenmayer, 1990;Rudge and Haseloff, 2005; Dupuy et al., 2008, 2010; Gibson

et al., 2011; Merks et al., 2011). Cell expansion then resultsfrom turgor pressure acting on the yielding cell-wall material.Growth is modelled by considering Newton’s law. The followingsystem of equations is solved for the velocity v and position x of

0·50

0·45

0·40

Asymmetric, a = 0Asymmetric, a = 0·5Asymmetric, a = 1·0

Symmetric, a = 0·5Symmetric, a = 1·0

Symmetric, a = 00·35

0·30

0·25

Freq

uenc

y

0·20

0·15

0·10

0·05

03 4 5 6 7 8

Number of neighbours9 10 11 12

FI G. 5. Distributionof the topologyof cells. Valuesare means+ s.d. forfive dif-ferent simulations runs. Asymmetric and symmetriccell divisions are indicated in

the key, with a ¼ anisotropic value.

A D

B E

C F

FI G. 4. Different virtual tissues obtained using: (A) symmetric cell division with isotropic growth; (B) symmetric cell division with anisotropic growth value of 0.5;(C) symmetric cell division with anisotropic growth value of 1; (D) asymmetric cell division with isotropic growth; (E) asymmetric cell division with anisotropicgrowth value of 0.5; and (F) asymmetric cell division with anisotropic growth value of 1. The red lines are new dividing walls separating the two daughter cells

and the blue lines are old walls inherited from the mother cells.

Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting 607


the vertices i of the cell-wall network (only the main equation arepresented here; futher details are given by Abera et al., 2013,2014):

mi

dVi

dt= FT,i (1)

dXi

dt= Vi (2)

where mi is the mass of the vertex, which is assumed to be unity,xi (m) and vi(m s– 1) are the position and velocity of vertex i,respectively, and FT,i (N) is the total force acting upon thisvertex. The resultant force on each vertex, the position of eachvertex and thus the shape of the cells is computed as follows.The total force acting on a vertex is given by (Prusinkiewiczand Lindenmayer, 1990):

FT =∑f [F

Fturgor +∑e[E

FS + Fd (3)

where Fturgor (N) are turgor forces on the set of cell faces F sharingthe vertex, Fs (N) are tension forces from the set of edges(springs) sharing the vertex and

Fd = − bv (4)

is a damping force, expressed as the product of a damping factorb (Ns m– 1) and the vertex velocity v. The damping force wasincluded not only to capture the viscous nature of the matrixbut also to give sufficient damping to avoid numerical oscilla-tions in the solution. When the system is at equilibrium, thetotal force in eqn (3) is equal to zero.

From the calculation of cell expansion, cell growth is mod-elled by increasing the natural length of the springs associatedwith the growing cell, simulating biosynthesis of cell-wall mater-ial. At each time step the spring’s extension from its restinglength and the difference between the maximum attainableresting length of the spring and its current resting length, ln, areused to compute the natural lengths of the springs as:

dln

dt= 1

t(l − ln)

ln,max − ln

ln,max

( )(5)

where ln,max (m) is the maximum resting length of the springabove which the wall cannot expand irreversibly, and which isdetermined as a fixed percentage of the initial resting length,and t is a time constant (s). In this way, resting length iscoupled with force balance, allowing them to be solved together.

TABLE 1. Standard deviation and skewness values of topology distribution

Symmetric cell division Asymmetric cell division

s.d. Skewness s.d. Skewness

Anisotropy ¼ 0 0.99+0.05 0.17+0.23 1.42+0.07 0.80+0.27Anisotropy ¼ 0.5 1.10+0.06 0.32+0.19 1.45+0.08 0.70+0.34Anisotropy ¼ 1 0.95+0.05 0.23+0.52 1.14+0.14 0.28+0.18

Values are means for the five different runs for each case with+values indicating the standard deviation in each case.

2

1

0

Sym

Lewis' LawAffine fit

A

B

C

D

E

F

G

H

2

1

0

Nor

mal

ized

cel

l are

aSym

2

1

0

Sym

2

1

0

Symp

2

1

0

Asym

2

1

0

Asym

2

1

0

Asym

2

1

04 6

Number of neighbours

SAM

8 10

FI G. 6. The relationship between number of neighbours and cell area: (A) sym-metric cell division with isotropic growth; (B) asymmetric cell division withisotropic growth; (C) symmetric cell division with anisotropic growth valueof 0.5; (D) asymmetric cell division with anisotropic growth value of 0.5; (E)symmetric cell division with anisotropic growth value of 1; (F) asymmetriccell division with anisotropic growth value of 1; (G) symmetric cell divisionwith isotropic growth where cell division is restricted to the boundary cells;and (H) experimental data of Arabidopsis shoot apical meristem (SAM).Sym, symmetric; Asym, asymmetric; Symp, symmetric peripheral cells.Lewis’ law equation, An ¼ (n – 2)/4, is shown in red, where An is the mean nor-malized cell area and n is number of neighbours. The blue lines represent theaffine fit of the data. Cell area is normalized by dividing the individual cell

area by the mean of the cell areas in the tissue.



In contrast to previous models (Rudge and Haseloff, 2005), thereis thus no need to assume intermediate equilibrium to update theresting length of the cell wall.

To allow anisotropic expansion, which is a common mode ofgrowth in plants (Baskin, 2005; Schopfer, 2006), the spring con-stant (k) and the maximum resting length (ln,max) can be made tovary according to the orientation of the walls as follows:

k = kmin + kmin(C − 1)(1 − l) (6)

ln,max = l(C − 1)ln,0 + ln,0 (7)

where kmin is the spring constant of walls aligned along themaximum growth direction, C is the ratio of the maximumresting length of the edges and the initial resting length ofedges (ln,max/ln,0) and l is a parameter defined between 0 and 1according to the orientation of the edges as follows (Rudge andHaseloff, 2005):

l = 1 − asin2u (8)

where u is the angle between the edges and the major axis of thecell and a is the degree of anisotropy defined on (0, 1). With a ¼ 0we get isotropic growth and with a ¼ 1 we have anisotropicgrowth in the direction of the major axis of the cell. These equa-tions allow us to switch growth from totally isotropic to anydegree of anisotropy. All the parameters used in this modelwere taken from Abera et al. (2013).

Cell division algorithm

The moment in time when the cell divides was determinedbased on cell size. A dividing wall is inserted whenever a celldoubles its area. To determine the position and orientation ofthe new wall that divides the cell, an ellipse is fitted to the cell ver-tices (for details of the fitting algorithms, see Mebatsion et al.,2006, 2008). The outputs of the ellipse-fitting algorithm are themajor and minor diameters (which are orthogonal to eachother) and orientation (the direction of the major diameter) ofthe fitted ellipse. The new wall is then inserted along the shortestdiameter of the fitted ellipse perpendicular to the longest dia-meter of the fitted ellipse (Fig. 1). Eventually, the mother cell,along with its entities (vertices, walls), is replaced by the twodaughter cells. The cell vertices, walls and cell connectivityare then updated and cell expansion resumes. The divisionrules of the algorithm are based on the position and orientationof the dividing wall. The orientation the dividing wall is madeto be along the minor diameter of the fitted ellipse normal tothe major diameter (in accordance with both Errera’s rule andHofmeister’s rule) whereas the position of the dividing wallcan be made to vary to produce daughter cells with different

sizes. For example, if the position is made to be at the centroidof the cell, the division will result in two similar daughter cells(geometrically symmetric cell division, which is in accordancewith Errera’s rule); otherwise it will result in two different daugh-ter cells (geometrically asymmetric cell division). The positionof the dividing wall in this case is moved to either directionalong the major axis based on a random factor chosen from auniform distribution between –0.8 and 0.8 (the range is chosento avoid non-viable cells, which are very small if the range isbetween –1 and 1). There is also a good chance of symmetriccell division when the random factor is 0.

Experimental data

The simulation datawere compared with experimental data forshoot apical meristem (SAM) of Arabidopsis thaliana and leaftissue of A. thaliana obtained from De Reuille et al. (2005) andDe Veylder et al. (2001), respectively. We digitized the imagesto get individual cell coordinates using a Matlab program. Thetopological and geometrical property distributions are then cal-culated from the digitized coordinates. In De Reuille et al.(2005), their figure 5 shows how cell division in the SAM takesplace by maintaining the same colour for the mother cell andthe two daughter cells. The observation suggests the division ismostly symmetric. In De Veylder et al. (2001), their figure 7provides a typical example of asymmetric cell division ofmeristematic leaf cells.

TABLE 2. Norm of residuals (normalized by size of the data) of a linear fit to the relationship between topology and mean normalizedcell area for the different simulation cases

Sym, a ¼ 0 Sym, a ¼ 0.5 Sym, a ¼ 1 Asym, a ¼ 0 Asym, a ¼ 0.5 Asym, a ¼ 1 Symp SAM

0.2121 0.1902 0.1896 0.5209 0.4762 0.4665 0.3232 0.2626

Sym, symmetric; Asym, asymmetric; Symp, symmetric peripheral cells; SAM, shoot apical meristem; a, anisotropic value.

0·6

Asymmetric, a = 0Asymmetric, a = 0·5Asymmetric, a = 1·0Symmetric, a = 0Symmetric, a = 0·5Symmetric, a = 1·00·5

0·4

Freq

uenc

y

0·3

0·2

0·1

0<0·5 0·5–1·0 1·0–1·5

Mean normalized cell area

1·5–2·0 >2·0

FI G. 7. Cell area distribution. Cell area is normalized by the mean area of thecells in the tissue. Values are means+ s.d. for five different simulations runs.Asymmetric, asymmetric cell division; Symmetric, symmetric cell division;

a, anisotropic value.



Geometric and topological properties

We have developed a program that calculates the geometricand topological properties of the cells for both the simulatedtissues and the tissue from the confocal microscopyexperimentaldata. The properties that we have considered were topology, cellshape (aspect ratio and interior angles) and cell size.

Topology. The topology of the cell aggregate (cell collection in atissue) is defined in terms of the number neighbour cells that arein contact with a given cell. For both the simulated tissues andSAM from the experimental data, the algorithm calculates thenumber of cells that are in contact (neighbour cells) with agiven cell and a frequency distribution of the number of cellswas determined. The topology distributions of the tissuesobtained from the model, using different cell division rules (geo-metrically similar daughter cells and geometrically differentdaughter cells together with the different anisotropic values),were compared statistically with each other and with that of theexperimental tissue.

Cell shape. The cell shape distribution is characterized in thisstudy by two distinct geometrical properties, namely aspectratio and interior angle of the polygons representing the cellboundary.

The aspect ratio (ar) of the cells was defined as:

ar = 1 − (l1/l2) (9)

where l1 and l2 are the minor and major diameter of the fittedellipse, respectively. With this definition, circular cells willhave an aspect ratio of 0, whereas cells which have shapes farfrom circular will approach an aspect ratio of 1. The diametersof the equivalent ellipses were calculated according to the pro-cedure outlined by Mebatsion et al. (2006).

The interior angles of the polygons, which represent cells inthe virtual tissues generated with the different rules of cell div-ision (symmetric and asymmetric) and growth modes (isotropicand anisotropic), were calculated. The distributions were thencompared with each other. A comparison was also made with adistribution obtained by assuming that the cell polygons areregular (an ideal situation in which all interior angles of apolygon are assumed to be equal). The interior angles (i.e. atthe vertices where two cell walls meet) were calculated using

TABLE 3. Standard deviation and skewness values of cell area distribution


s.d. Skewness s.d. Skewness

Anisotropy ¼ 0 0.36+0.03 1.00+0.20 0.76+0.07 1.70+0.67Anisotropy ¼ 0.5 0.33+0.05 1.18+0.41 0.67+0.08 0.64+0.32Anisotropy ¼ 1 0.30+0.03 1.30+0.69 0.55+0.05 0.22+0.22


FI G. 8. Aspect ratio distributions of cells: (A) asymmetric cell division with dif-ferent degrees of anisotropy; (B) symmetric cell division with different degrees ofanisotropy; and (C) a comparison between symmetric and asymmetric cell divi-sions. Values are means+ s.d. for five different simulations runs. Asymmetric,asymmetric cell division; Symmetric, symmetriccelldivision; a, anisotropicvalue.

0·6

A

B

C

0·5

0·4

Freq

uenc

y

0·3

0·2

0·1

0

0·6

0·5

0·4

Freq

uenc

y

0·3

0·2

0·1

0

0·6

0·5

0·4

Freq

uenc

y

0·3

0·2

0·1

00–0·2 0·2–0·4 0·4–0·6

Aspect ratio0·6–0·8 0·8–1·0

Asymmetric, a = 0, after expansionAsymmetric, a = 0, before expansionAsymmetric, a = 0·5, after expansionAsymmetric, a = 0·5, before expansionAsymmetric, a = 1·0, after expansionAsymmetric, a = 1·0, before expansion

Symmetric, a = 0, after expansionSymmetric, a = 0, before expansionSymmetric, a = 0·5, after expansionSymmetric, a = 0·5, before expansionSymmetric, a = 1·0, after expansionSymmetric, a = 1·0, before expansion

Symmetric, a = 0, after expansionSymmetric, a = 0, before expansion

Asymmetric, a = 0, before expansionAsymmetric, a = 0, after expansion



the inverse cosine function by defining two vectors starting fromthe common vertex heading away from it along the two walls ofthe cell sharing that vertex (Fig. 2). The interior angles for theregular polygons were calculated as:

u = n − 2

n× 1808 (10)

where u is interior angle and n is the number of sides of thepolygon.

Cell size. The size distribution of cell areas (2-D) was calculated.The areas of the cells were calculated by applying Green’stheorem (Kreyszig, 2005).

Statistical comparison

Topological and geometrical (shape and size) properties ofboth microscopic cellular images and virtual cells were calcu-lated and compared statistically. A two-sample Kolmogorov–Smirnov test was used to compare the distributions of thesevalues. The null hypothesis was that both are from the same con-tinuous distribution. The alternative hypothesis was that theywere from different continuous distributions. The test statisticis the maximum height difference of the two data distributionson a cumulative distribution function. If the test statistic isgreater than the critical value the null hypothesis is rejected.The P-value, which is dependent on the test statistics and thethreshold with reference to the test significance, is normallyused as criterion for whether to reject the null hypothesis. Ifthe P-value is less than the test significance, the null hypothesisis rejected meaning that the two distributions are different. Theresult of the test was 1 if the test rejects the null hypothesis at aspecified significance level, and 0 otherwise. We have used a5 % significance level (Justel et al., 1997). The distributionswere also compared by using the mean, standard deviation andskewness. The statistical comparision was done in Matlab (TheMathworks, Natick, MA, USA).

RESULTS

A demonstration of the cell division algorithm at different stagesin time for symmetric cell division with isotropic growthis shown in Fig. 3. Several examples of tissues generated bythe 2-D cell division algorithm are presented in Fig. 4. Thetissues shown were generated using: (A) symmetric cell division

with isotropic growth, (B) symmetric cell division with aniso-tropic growth value of 0.5, (C) symmetric cell division with an-isotropic growth value of 1, (D) asymmetric cell division withisotropic growth, (E) asymmetric cell division with anisotropicgrowth value of 0.5 and (F) symmetric cell division with aniso-tropic growth value of 1. As can be seen from the figure, thealgorithm can generate different kinds of tissues by employingdifferent cell division rules and different cell growth modes.The visual differences between the tissues are evident from thefigure. The topological and geometrical properties are analysedand discussed below. Five different simulation runs are usedfor each case. The number of cells used was 781, 350, 246,474, 286 and 310 for symmetric cell division–isotropicgrowth, symmetric cell division–anisotropic growth (0.5), sym-metric cell division–anisotropic growth (1), asymmetric celldivision–isotropic growth, asymmetric cell division–anisotrop-ic growth (0.5) and asymmetric cell division–anisotropic growth(1), respectively.

Topology

The topological distribution, defined as the number of neigh-bour cells, showed a clear difference between the differenttissues presented above. As can be seen from Fig. 5, there is aclear difference between tissues using the different cell divisionrules and growth modes, where the virtual tissues using symmet-ric cell division have a narrower distribution than those usingasymmetric cell division rules (Table 1). Another interestingfeature of these distributions is the skewness. Although all thecases presented here display skewness in the distribution ofnumber of neighbours, the degree of skewness is lower for sym-metric cell division (Table 1).

The different cell divisions are also tested on how well they fitto Lewis’ law, which states that a linear relationship existsbetween the number of neighbours and the area of cells(Lewis, 1928). The results are presented in Fig. 6. The errorbars indicate the standard deviation of the mean normalizedcell area. The linear relationship between the mean normalizedcell area and the number of neighbours is respected more in thesymmetric cell division rules than their asymmetric counterparts(see the affine fit of the data drawn in blue in Fig. 6 and Table 2).The norm of the residuals is normalized by the size of the dataused in the fitting to get a fair comparison. The symmetric celldivision rules have a lower norm of residuals than their asymmet-ric counterparts, suggesting a better linear fit. The asymmetriccell division rules demonstrated higher standard deviations

TABLE 4. Mean, standard deviation and skewness values of aspect ratio distribution

Anisotropy ¼ 0 Anisotropy ¼ 0.5 Anisotropy ¼ 1

Expanded Not expanded Expanded Not expanded Expanded Not expanded

Symmetric cell division Mean 0.2+0.001 0.36+0.001 0.23+0.01 0.32+0.01 0.26+0.01 0.32+0.02s.d. 0.09+0.002 0.13+0.01 0.10+0.01 0.13+0.01 0.12+0.02 0.14+0.02Skewness 0.31+0.11 20.04+0.10 0.10+0.30 20.05+0.13 0.09+0.27 0.13+0.38

Asymmetric cell division Mean 0.24+0.01 0.36+0.02 0.27+0.01 0.33+0.03 0.27+0.01 0.34+0.01s.d. 0.13+0.01 0.20+0.02 0.16+0.01 0.20+0.02 0.15+0.004 0.16+0.004Skewness 0.56+0.19 0.40+0.14 0.80+0.11 0.71+0.20 0.69+0.15 0.33+0.13




whereas the symmetric cell division rules showed a lower slopethan that given by Lewis’ law. These results are in agreementwith those of Sahlin and Jonsson (2010). The symmetric cell

division algorithm with isotropic growth, in which only the per-ipheral cells were allowed to divide, matched perfectly withLewis’ law.

Cell size distribution

The cell size distribution, presented here as the distribution ofmean normalized cell area, is an important geometrical propertyused to compare the performance of different cell division rules.As can be seen from Fig. 7, the cell size distribution shows aclear distinction between the various division rules investigated.In particular, the difference between the symmetric cell divisionand asymmetric cell division is remarkable. Besides the differencebetween the spread of the distribution, which is evident from Fig. 7,the degree of skewness is another important difference between thedifferent division rules employed (Table 3). The spread of the dis-tributions as well as the degree of skewnessare more pronounced intheasymmetric celldivision than thecorrespondingsymmetriccelldivision, similar to what was found for the cell topology.

Cell shape distribution

Cell shape distribution is analysed here by means of two geo-metrical properties, namely aspect ratio and the internal angle ofthe polygons representing the cell boundary (see Methods).

TABLE 5. Mean, standard deviation and skewness values of internal angle distribution


Mean s.d. Skewness Mean s.d. Skewness

Expanded 116.8+0.26 23.4+0.47 0.12+0.03 115.8+0.20 27.0+0.72 –0.211+0.06Not expanded 116.8+0.26 35.6+0.07 0.56+0.018 115.8+0.20 38.0+0.54 0.28+0.08Regular polygon 116.8+0.26 11.2+0.47 –0.83+0.30 115.8+0.20 16.1+0.87 –0.74+0.17


A

B

FI G. 10. Digitized shoot apical meristem of Arabidopsis thaliana taken from De Reuille et al. (2005) where cell division is mainly symmetric (A) and digitized meri-stematic leaf cells taken from De Veylder et al. (2001) where cell division is mainly asymmetric (B).

0·6

Asymmetric, a = 0, after expansionAsymmetric, a = 0, before expansionAsymmetric, a = 0, regular polygonSymmetric, a = 0, after expansionSymmetric, a = 0, before expansionSymmetric, a = 0, regular polygon

0·5

0·4

Freq

uenc

y

0·3

0·2

0·1

0<50 50–80 80–110

Internal angle (°)110–140 140–170

FI G. 9. Interior angle distribution comparison of symmetric and asymmetriccell divisions. Values are means+ s.d. for five different simulations runs.Asymmetric, asymmetric cell division; Symmetric, symmetric cell division; a,

anisotropic value.



Aspect ratio distribution. The cell aspect ratio distributions of thetissues obtained using different cell division rules and growthmodes are presented in Fig. 8 and Table 4. As can be seen fromthe figure and table, the mean aspect ratio values of the corre-sponding symmetric and asymmetric cell division rules immedi-ately after cell division (before expansion) are almost equal.Their difference is reflected in the spread of the distribution, asshown by the corresponding standard deviation values. The sym-metric cell division generally produces a narrower distributionthan the asymmetric cell division. The difference in the skewnessof the distribution is apparent: the asymmetric cell division rulesresult in a higher positive degree of skewness as opposed to littleor no skewness for their symmetric counterparts. Among the cellgrowth types investigated, isotropic cell growth has lower aspectratiovalues than anisotropic cell growth, although this differenceis reduced aftercell division. The effect of the growth type used ismainly manifested in the overall tissue shape (see Fig. 4).

Interior angle. The interior angles of the polygons representingthe cell are calculated just after cell division and after expansivegrowth, when the cells have achieved equilibrium. These distri-butions are compared with each other and with the interior angledistribution obtained assuming all the polygons are regular (anideal situation in which all interior angles of a polygon represent-ing a cell are assumed to be equal). As most of the cells have sixsides, symmetric cell division converges to 1208, better than itsasymmetric counterparts (Fig. 9). Although their mean valuesare more or less equal, there is a clear difference in the widthof the distribution as represented by its standard deviation(Table 5). The ideal internal angle distributions of the differentcell division rules employed are narrower than their respectiveactual distributions. The interior angle distributions of thetissues obtained after cell expansion better approximate the re-spective ideal interior angle distributions in terms of both thestandard deviation and the skewness of the distributions(Table 5). In both the aspect ratio distribution and the interiorangle distribution, the distributions after cell expansion approxi-mate the ideal distributions better than those just after celldivision.

Comparison of real and virtual topological and geometricalproperties

As demonstrated by Besson and Dumais (2011) andPrusinkiewicz (2011), seeking an exact prediction of a celldivision pattern is as futile as attempting to predict the exact se-quence of numbers produced by repetitively throwing a die. Onlythe statistical characteristics of these processes, as opposed toindividual outcomes, can be meaningfully anticipated. In thisregard, the visual, topological and geometrical comparisons ofthe different cell division rules with experimental data are pre-sented here. The image with 347 cells obtained from a confocalmicroscopy was taken from De Reuille et al. (2005) and digitizedby a Matlab code (Fig. 10A). Comparison of the cell area distri-bution, topology distribution and aspect ratio distribution(Fig. 11) suggests that symmetric cell division with isotropicgrowth has superior visual, topological and geometrical similar-ity to the experimental data than the other division rules investi-gated (Table 6). The experimental data as well as symmetric celldivision, where cell division is restricted to the peripheral cells,

were well fitted to Lewis’ law (see Fig. 6G, H). Two-sampleKolmogorov–Smirnov tests were made between the best per-forming division rule (symmetric cell division with isotropic

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FI G. 11. Geometrical and topological property comparisons of virtual tissuesgenerated by the algorithm and a tissue obtained experimentally: (A) cell area dis-tribution; (B) topology distribution; and (C) aspect ratio distribution. Asym,asymmetric cell division; Sym, symmetric cell division; SAM, shoot apical meri-stem; a, degree of anisotropy. Density in these plots represents frequency density.



growth) and the experimental data (Fig. 12 and Table 6). Thecomparisons are made at 5 % significance level (see Methods).The results suggest the distributions are not significantly differ-ent. The different simulation cases were also compared with ameristematic leaf tissue image obtained from De Veylder et al.(2001) and digitized with a Matlab code (see Fig. 10B). Theimage contains 253 cells. Comparison of the cell area distribu-tion, topology distribution and aspect ratio distribution (Fig. 13and Table 6) suggest that asymmetric cell division generallyshows superior topological and geometrical similarity to the ex-perimental data than their symmetric counterparts. Two-sampleKolmogorov–Smirnov tests made between the data obtainedusing asymmetric cell division with isotropic growth and the ex-perimental data obtained from De Veylder et al. (2001) showedthe results are not significantly different. The results of the com-parison are presented in Fig. 14.

DISCUSSION

We have developed a generic cell division algorithm based oncell-wall mechanics that is capable of producing differenttissues varying both in geometrical and in topological propertydistributions of the cells as well as in the overall shape of thetissues. The algorithm provides a robust cell division modelbased on ellipse-fitting to find the position and orientation ofthe dividing wall and incorporates cell-wall mechanics for

growth. Using the developed algorithm, we have generated dif-ferent tissues. We have studied tissues which were generatedusing cell division rules of equally sized daughter cells andunequally sized daughter cells. Isotropic or anisotropic growthmodes were used for each case. We have analysed the topologiesand geometries of simulated tissues and compared them withexperimental data to test the performance of different divisionrules (symmetric and asymmetric in a geometric sense). A com-parison is also made among the virtual tissues generated usingthe different cell division and growth types (isotropic, anisotrop-ic). We have also investigated how well tissues fitted to Lewis’law, which states that a linear relationship exists between thenumber of neighbours and the area of cells (Lewis, 1928).

Alim et al. (2012) investigated the effect cell division orienta-tion on tissue growth heterogeneity. Cell growth in theirapproachwas implemented by minimizing the difference between thesecond area moment of a target ellipse (which grows withtime) and that of the cell calculated from its vertices. Theyfound that tissue heterogeneity is more pronounced where celldivision with randomly orientated division walls is used. Ourresult also allows the same conclusion.

The role of mechanics on cell expansion was detailed by Aberaet al. (2013, 2014, and references therein), where a cell is repre-sented as a thin-walled structure maintained in tension by turgorpressure. The cells will respond by increasing the resting lengthof the cell walls (simulating actual growth) to alleviate the

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FI G. 12. Statistical comparison of geometrical and topological properties of virtual tissues generated by the algorithm using symmetric cell division with isotropicgrowth and a shoot apical meristem obtained from experiment using a two-sample Kolmogorov–Smirnov test.

TABLE 6. P-values (expressed as a percentage) of the comparisons of the experimental data with the data obtained by the differentsimulation cases


a ¼ 0 a ¼ 0.5 a ¼ 1 a ¼ 0 a ¼ 0.5 a ¼ 1

Cell area SAM 21 7 4.5 4×10– 12 7 × 10–6 9 × 10– 6

Meristematic leaf 8 × 10– 7 8 × 10– 6 1 × 10– 6 23 11.6 32Topology SAM 94 24 2.9 × 10– 2 0.27 9 × 10–4 0.3

Meristematic leaf 5 × 10– 26 9 × 10– 14 1 × 10– 7 70 28 2 × 10– 4

Aspect ratio SAM 40 38 46 33 4 93Meristematic leaf 2 × 10– 2 13 19 32 51 89

The comparison is done using a two-sample Kolmogorov–Smirnov test. SAM, shoot apical meristem; a, anisotropic value.



mechanical stress induced by the tension. The role of mechanicalstress on the orientation of the dividing wall has been explainedby different authors (e.g. Lynch and Lintilhac, 1997; Lintilhac

and Vesecky, 1981; Hejnowicz and Romberger, 1984).Nakielski (2008) developed a model based on these studies forroot growth. These are studies based on the idea of growthtensors (GTs), which lead to mutually orthogonal PDGs.In their work (Hejnowicz and Romberger, 1984; Nakielski,2008), they first define organ shape-based GTs, which help todefine the velocities of the vertices of the cell mesh and thecell area increases as the vertices move governed by the GTfield defined on the organ level. When a cell is ready to divide,the PDGs that are predefined by the GT based on the locationof the cell with regard to the GT will be used to determine theorientation of the dividing wall. The diameters of the mothercell along the PDGs will be calculated and the dividing wall isinserted along the shorter of these diameters.

In our approach, the growth biomechanics govern the shape ofthe mother cell on which an ellipse is fitted to decide how the cellshould divide. PDGs are mutually perpendicular diameters thatcan be obtained if you fit an equivalent ellipse to the verticesof the cell (minor and major diameters of the ellipse).Therefore, regarding the way the cell division wall orientationis determined, the two approaches should lead to a similarresult (see Alim et al., 2012). The most important differencebetween their approach and ours is in their approach the PDGsare predefined based on location (because the GTs are predefinedat organ level) whereas in our approach it is the mechanics thatleads to the shape of the mother cell based on which it divides.Note that in cases where growth is started from an already elon-gated cell and maximum growth rate is along the shorter diam-eter, the two approaches will lead to different orientation forthe dividing wall. In this case, when using the ellipse-fitting algo-rithm, the orientation of the dividing wall should be along theorientation of the major diameter of the fitted ellipse.

The algorithm can produce the variability observed in sym-metric cell division. Besson and Dumais (2011) achieved thevariability observed in symmetric cell division by introducingthe concept of local minima instead of the global minima ofErrera’s rule for selection of the dividing wall. In our model,this was achieved without abandoning the founding rule of celldivision suggested by Errera (1886). An ellipse-fitting procedurewas used to determine the position and orientation of the dividingwall. The minor diameter of the ellipse through the centroid wasused as the position and orientation of the dividing wall. If thefitted ellipse is a circle, we have infinitely many possible orienta-tions for the candidate dividing wall. The random selection ofone of them made it possible to produce the variability observedin symmetric cell division (see Fig. 4A). Ellipse-fitting has beenused in previous studies (e.g. Gibson et al., 2011; Merks et al.,2011) to determine the position of the new dividing wall, butits importance to produce randomness even in symmetric celldivision has not been put forward. This randomness is an import-ant property that represents biological variability observed innature.

Robinson et al. (2011) demonstrated asymmetric cell divisionin the leaves of arabidopsis, where the nucleus is displaced fromthe geometrical centre of the cell by cell polarity switching.In our model, asymmetric cell division is achieved by randomdisplacement of the dividing wall along the major diameter ofthe fitted ellipse. This random displacement of the dividingwall is the equivalent of displacement of the nucleus fromthe geometrical centre of the cell, which could lead to two

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FI G. 13. Geometrical and topological property comparisons of virtual tissuesgenerated by the algorithm using asymmetric cell division with isotropic growthand a real meristematic leaf tissue obtained experimentally (from De Veylderet al., 2001): (A) cell area distribution; (B) topology distribution; and (C) aspectratio distribution. Asym, asymmetric cell division; Sym, symmetric cell division;

a, degree of anisotropy. Density in these plots represents frequency density.



daughter cells that are different in size as well as in topology (seeFig. 4D–F). Our model has the added advantage that it incorpo-rates cell mechanics in the growth model. By making the mech-anical properties and maximum resting length of the wallsdependent on direction, the model allows both isotropic and an-isotropic cell growth, which leads to different simulated tissues(see Fig. 4). Although most of the meristematic cells that are un-differentiated obey symmetric cell division with isotropicgrowth, asymmetric cell division and anisotropic growth alsohave great potential to account for differentiation of meristemtissue into different specialized tissues.

Gibson et al. (2011) used the same cell representation as we dohere. They developed differential equations and Newton’s lawwas used to solve force balance on the vertices. In contrast totheir model, ours assigns spring constant values, which areinversely proportional to wall length, whereas in their modelthe same spring constant value, independent of length, isassumed. Moreover, their model did not simulate actual growthof the cell walls; rather, they used a constant resting lengththroughout the simulation. They studied only isotropic mechan-ical properties of the cell wall. Therefore, our model provides abetter representation of the cell mechanics and actual growthof the cell wall, which actually mimics the biosynthesis of cell-wall materials.

Merks et al. (2011) introduced a cell-based computer model-ling framework, ‘VirtualLeaf’, for plant tissue morphogenesis.They used a Monte Carlo-based energy minimization algorithm.The energy of the system was calculated from cell area-relatedturgor pressure force and the tension force of the walls. Theyused the same cell representation as we do here. In contrast totheir probabilistic determination of position of vertices, oursuses a deterministic approach based on Newton’s law of forcebalances. In addition, they insert a new node to simulate cell-wallyielding after length exceeded four times its initial value whileours actually simulates growth of the resting length of the wallsusing differential equations simultaneously with mechanics.While their model presents a simplified software platform, oursoffers better representation of the actual growth of the cell wall.

The cell division algorithms can be coupled to 2-D expansiveplant cell growth models (Abera et al., 2013), where the initialtopology was, for example, obtained from a 2-D Voronoi

tessellation. The cell division algorithm has more biological jus-tification than the Voronoi tessellations, which were initiatedfrom random generating points, which is simply a partition of agiven area cell topology into a number of regions, representingthe individual cells.

In particular cases where the dividing wall is not a function ofthe shape of the mother cell, such as those observed in periclinaland anticlinal cell divisions (Howell, 1998; Evert, 2006), specialcare should be taken in using the ellipse-fitting algorithm.Periclinal (division walls parallel to the surface): this usuallyleads to relatively large new wall as the mothercells are elongatedcells along the periphery. In this case, the ellipse-fitting can stillbe used to determine the position and orientation of the new wallbut the major diameter instead of the minor diameter of theellipse has to be used. Anticlinal (division walls at right anglesto the surface): this usually leads to short new wall and theellipse-fitting can still be used although caution has to be takenwhen the fitted ellipse is a circle, where there are infinitelymany candidate short walls, to ensure the chosen short wall isthe one that is normal to the surface.

CONCLUSIONS

The cell division algorithm developed here can produce tissuesthat have different topological and geometrical properties. Thisflexibility to produce different tissue types gives the modelgreat potential for use in in silico investigations of plant cell div-ision and growth. The cell division algorithms take account ofboth cell shape and topology. The model is based on cellmechanics, where the cell wall mechanical properties, fluidmatrix inside the cell and cell turgor pressure are taken intoaccount. The equations for actual growth of the cell walls(change in resting length of the walls) and cell division aresolved continuously. It is generic in that a switch between iso-tropic growth and anisotropic growth as well as between symmet-ric cell division and asymmetric cell division is automatic andeasy, which makes the model convenient to adapt to a specificcase study. Finally, the model is robust as there is no need foran iterative procedure to find the shortest wall for cell division.In our algorithm, the division wall is inserted along the orienta-tion of the minor diameter of the fitted ellipse. The geometrical

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FI G. 14. Statistical comparison of geometrical and topological properties of virtual tissues generated by the algorithm using asymmetric cell division and a meri-stematic leaf tissue obtained experimentally using a two-sample Kolmogorov–Smirnov test.



properties of the simulated tissues were compared with experi-mental data for SAM. Symmetric cell division with isotropicgrowth best fits the experimental data.

ACKNOWLEDGEMENTS

Financial support by the Flanders Fund for Scientific Research(project FWO G.0645.13), K.U.Leuven (project OT 12/055)and the EC (project InsideFood FP7-226783) and the Institutefor the Promotion of Innovation by Science and Technology inFlanders (IWT scholarship SB/0991469) is gratefully acknowl-edged. T.D. is a postdoctoral fellow of the Flanders Fund forScientific Research (FWO Vlaanderen).

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