Annals of Botany 114: 605– 617, 2014
doi:10.1093/aob/mcu078, available online at www.aob.oxfordjournals.org
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,*
1
Flanders 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 Zürich,
Switzerland and 3Laboratory for Building Science and Technology, Swiss Federal Laboratories for Materials Testing and Research
(Empa), Überlandstrasse 129, 8600 Dübendorf, 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 acknowledged
since the 19th century. Most of the available models developed to date are limited to symmetric cell division with
isotropic growth. Often, the actual growth of the cell wall is either not considered or is updated intermittently on a
separate time scale to the mechanics. This study presents a generic algorithm that accounts for both symmetrically
and asymmetrically dividing cells with isotropic and anisotropic growth. Actual growth of the cell wall is simulated
simultaneously with the mechanics.
† Methods The cell is considered as a closed, thin-walled structure, maintained in tension by turgor pressure. The cell
walls are represented as linear elastic elements that obey Hooke’s law. Cell expansion is induced by turgor pressure
acting on the yielding cell-wall material. A system of differential equations for the positions and velocities of the cell
vertices as well as for the actual growth of the cell wall is established. Readiness to divide is determined based on
cell size. An ellipse-fitting algorithm is used to determine the position and orientation of the dividing wall. The
cell vertices, walls and cell connectivity are then updated and cell expansion resumes. Comparisons are made
with experimental data from the literature.
† Key Results The generic plant cell division algorithm has been implemented successfully. It can handle both symmetrically and asymmetrically dividing cells coupled with isotropic and anisotropic growth modes. Development of
the algorithm highlighted the importance of ellipse-fitting to produce randomness (biological variability) even in
symmetrically dividing cells. Unlike previous models, a differential equation is formulated for the resting length
of the cell wall to simulate actual biological growth and is solved simultaneously with the position and velocity of
the vertices.
† Conclusions The algorithm presented can produce different tissues varying in topological and geometrical properties. This flexibility to produce different tissue types gives the model great potential for use in investigations of plant
cell 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.
IN T RO DU C T IO N
Cellular pattern studies and simulation of higher level processes
such as phyllotaxis or vascular patterning call for models of the
division and arrangement of cells into tissues (Smith et al., 2006;
Merks et al., 2007). Many biophysiological processes in plant
organs, such as gas transport, are strong functions of the microstructural geometry of the tissue (Ho et al., 2009, 2010, 2011,
2012), which is in turn dependent on cell division and the
arrangement of cells.
In contrast to animal cells, plant cells have relatively rigid cell
walls. The walls of the neighbouring cells are joined by the
middle lamellas (Romberger et al., 1993). Walls of neighbouring
cells do not slide with respect to each other. Therefore, cell topology is maintained except in the event of intercellular gas space
formation or cell division. These aspects should be taken into
account 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 for
modern thoughts on cell division. According to Hofmeister,
the dividing wall is inserted at right angles to the axis of
maximal growth, while Sachs suggested that the new wall intersects the side walls at right angles. Errera’s rule states that the dividing wall should be the shortest one that partitions the mother
cell into two equal daughter cells (reviewed by Prusinkiewicz
and Runions, 2012).
Computer models of tissues with cell division based on position and orientation have been developed by Sahlin and
Jönsson (2010). In their approach, the position of the new wall
is determined to be either the centre of mass of the mother cell
or a point inside the mother cell chosen randomly, while its orientation is chosen based on a number of criteria. Although this
model includes cell growth mechanics and includes actual
growth of the cell wall by changing the resting length of the
cell wall, it does not limit the actual growth of the cell-wall
# The Author 2014. Published by Oxford University Press on behalf of the Annals of Botany Company. All rights reserved.
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Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting
resting length, and how turgor force is implemented does not
allow for cell – cell interactions.
Besson and Dumais (2011) developed a rule for symmetric
division of plant cells based on probabilistic selection of division
planes. According to their work, Errera’s rule of cell division
failed to account for the variability observed in symmetric cell
divisions, in particular that cells of identical shape do not necessarily adopt the same division plane. The variability in symmetric cell division is accounted for by introducing the concept of
local minima rather than global minima. Robinson et al.
(2011) introduced an asymmetric cell division algorithm in
which the division wall is chosen as the shortest wall that
passes through the nucleus of the mother cell. In their model,
asymmetric cell division is achieved by displacing the nucleus
F I 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
F I G . 2. Illustration of the calculation of the interior angle. The procedure is
repeated for each cell at each vertex.
of the mother cell from the centroid of the cell in a random
direction.
The dynamic pattern of cell arrangement is not only a function
of the position and orientation of division walls but also of the
timing of cell division and growth of the tissue. The early work
of Korn (1969) reintroduced by Merks and Glazier (2005) represents cells as a set of points, and growth is achieved by the
addition of new points to a cell. Nakielski (2008) developed a
model for growth and cell division based on the postulate of
Hejnowicz (1984) that cells divide in relation to principal directions of growth (PDGs) and supported by the experimental
work of Lintilhac and Vesecky (1981) and Lynch and Lintilhac
(1997). PDGs are mutually perpendicular directions along
which extreme growth rates are observed, which lead to
unequal major and minor equivalent diameters for the cell.
This is true for anisotropic growth, associated with anisotropic
mechanical properties of the cell wall. They are in turn dependent
on a growth tensor field defined at the organ level according to the
work of Hejnowicz and Romberger (1984). Cell mechanicsbased models for 2-D cell growth have been developed by
several researchers (Dupuy et al., 2008, 2010; Sahlin and
Jönsson, 2010; Gibson et al., 2011; Merks et al., 2011; Abera
et al., 2013). The 2-D cell growth model developed by Abera
et al. (2013) has recently been extended to a 3-D cell growth
model (Abera et al., 2014).
Not all plant cells grow isotropically; anisotropic growth is
also a common phenomenon. This anisotropy, defined as
direction-dependent growth of a cell that leads to a mother cell
with unequal major and minor diameters, is manifested as a consequence of anisotropic molecular wall structure, determined by
differential spatial arrangement of the cellulose microfibrils that
are generally organized in layers of parallel fibres (reviewed by
Baskin, 2005; Schopfer, 2006). A different definition of anisotropy is the temporal and/or spatial difference in growth rate of
the cell. For the latter, a cell can still be isotropic in shape (see
Kwiatkowska and Dumais, 2003; Kwiatkowska, 2006). In the
present paper, the first definition is used. Nakielski (2008) used
this concept to incorporate overall tissue anisotropy in their
model, which was based on PDGs. To our knowledge, although
anisotropic growth of plant cells is a common phenomenon, the
available literature on cell division models that use anisotropic
cell growth is limited.
Based on the literature references detailed above, there is no
single generic model. Some models are intended either for symmetric or for asymmetric cell division. Some are focused simply
F I 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-fitting
A
D
B
E
C
607
F
F I 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 anisotropic
growth 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.
0·50
0·40
0·35
0·30
0·25
0·20
0·15
0·10
0·05
0
M AT E R I A L S A N D M E T H O D S
Cell growth algorithm
In our model, the cell is considered a closed thin-walled structure,
maintained in tension by turgor pressure. The cell walls of adjacent cells are modelled as parallel, linear elastic elements that
obey Hooke’s law, an approach similar to that used in other
plant tissue models (Prusinkiewicz and Lindenmayer, 1990;
Rudge and Haseloff, 2005; Dupuy et al., 2008, 2010; Gibson
Asymmetric, a = 0
Asymmetric, a = 0·5
Asymmetric, a = 1·0
Symmetric, a = 0
Symmetric, a = 0·5
Symmetric, a = 1·0
0·45
Frequency
on the division rules, without incorporating the timing of cell division or the actual expansive growth and others do not consider
cell mechanics when modelling cell growth. Most of the
models introduced so far are based on isotropic growth. Hence,
the objective of this paper is to develop a 2-D plant cell division
algorithm that is generic in that it includes symmetric cell division by taking account of the randomness without abandoning
Errera’s rule of cell division. Furthermore, it accounts for asymmetric cell division, isotropic cell growth and anisotropic cell
growth. It is based on cell growth mechanics, which accounts
for cell – cell interaction through turgor force calculation and
limits 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 position
and orientation of the new dividing wall.
3
4
5
6
7
8
9
10
Number of neighbours
11
12
F I G . 5. Distribution of the topology of cells. Values are means + s.d. for five different simulations runs. Asymmetric and symmetric cell divisions are indicated in
the key, with a ¼ anisotropic value.
et al., 2011; Merks et al., 2011). Cell expansion then results
from turgor pressure acting on the yielding cell-wall material.
Growth is modelled by considering Newton’s law. The following
system of equations is solved for the velocity v and position x of
608
Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting
TA B L E 1. Standard deviation and skewness values of topology distribution
Symmetric cell division
Anisotropy ¼ 0
Anisotropy ¼ 0.5
Anisotropy ¼ 1
Asymmetric cell division
s.d.
Skewness
s.d.
Skewness
0.99 + 0.05
1.10 + 0.06
0.95 + 0.05
0.17 + 0.23
0.32 + 0.19
0.23 + 0.52
1.42 + 0.07
1.45 + 0.08
1.14 + 0.14
0.80 + 0.27
0.70 + 0.34
0.28 + 0.18
Values are means for the five different runs for each case with + values indicating the standard deviation in each case.
Lewis' Law
Affine fit
the vertices i of the cell-wall network (only the main equation are
presented here; futher details are given by Abera et al., 2013,
2014):
(1)
2
B
Asym
1
e[E
where Fturgor (N) are turgor forces on the set of cell faces F sharing
the vertex, Fs (N) are tension forces from the set of edges
(springs) sharing the vertex and
Fd = − bv
Sym
0
(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 this
vertex. The resultant force on each vertex, the position of each
vertex and thus the shape of the cells is computed as follows.
The total force acting on a vertex is given by (Prusinkiewicz
and Lindenmayer, 1990):
FT =
Fturgor +
FS + Fd
(3)
f [F
A
1
(4)
is a damping force, expressed as the product of a damping factor
b (Ns m – 1) and the vertex velocity v. The damping force was
included not only to capture the viscous nature of the matrix
but also to give sufficient damping to avoid numerical oscillations in the solution. When the system is at equilibrium, the
total force in eqn (3) is equal to zero.
From the calculation of cell expansion, cell growth is modelled by increasing the natural length of the springs associated
with the growing cell, simulating biosynthesis of cell-wall material. At each time step the spring’s extension from its resting
length and the difference between the maximum attainable
resting length of the spring and its current resting length, ln, are
used to compute the natural lengths of the springs as:
dln 1
ln,max − ln
= (l − ln )
(5)
dt
ln,max
t
where ln,max (m) is the maximum resting length of the spring
above which the wall cannot expand irreversibly, and which is
determined as a fixed percentage of the initial resting length,
and t is a time constant (s). In this way, resting length is
coupled with force balance, allowing them to be solved together.
0
2
C
Sym
1
0
Normalized cell area
dVi
mi
= FT,i
dt
dXi
= Vi
dt
2
2
D
Asym
1
0
2
E
Sym
1
0
2
F
Asym
1
0
2
G
Symp
1
0
2
H
SAM
1
0
4
6
8
Number of neighbours
10
F I G . 6. The relationship between number of neighbours and cell area: (A) symmetric cell division with isotropic growth; (B) asymmetric cell division with
isotropic growth; (C) symmetric cell division with anisotropic growth value
of 0.5; (D) asymmetric cell division with anisotropic growth value of 0.5; (E)
symmetric cell division with anisotropic growth value of 1; (F) asymmetric
cell division with anisotropic growth value of 1; (G) symmetric cell division
with 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 normalized cell area and n is number of neighbours. The blue lines represent the
affine fit of the data. Cell area is normalized by dividing the individual cell
area by the mean of the cell areas in the tissue.
Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting
609
TA B L E 2. Norm of residuals (normalized by size of the data) of a linear fit to the relationship between topology and mean normalized
cell area for the different simulation cases
Sym, a ¼ 0
0.2121
Sym, a ¼ 0.5
Sym, a ¼ 1
Asym, a ¼ 0
Asym, a ¼ 0.5
Asym, a ¼ 1
Symp
SAM
0.1902
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.
k = kmin + kmin (C − 1)(1 − l)
(6)
ln,max = l(C − 1)ln,0 + ln,0
(7)
Asymmetric, a = 0
Asymmetric, a = 0·5
Asymmetric, a = 1·0
Symmetric, a = 0
Symmetric, a = 0·5
Symmetric, a = 1·0
0·6
0·5
Frequency
In contrast to previous models (Rudge and Haseloff, 2005), there
is thus no need to assume intermediate equilibrium to update the
resting length of the cell wall.
To allow anisotropic expansion, which is a common mode of
growth in plants (Baskin, 2005; Schopfer, 2006), the spring constant (k) and the maximum resting length (ln,max) can be made to
vary according to the orientation of the walls as follows:
0·4
0·3
0·2
where kmin is the spring constant of walls aligned along the
maximum growth direction, C is the ratio of the maximum
resting length of the edges and the initial resting length of
edges (ln,max/ln,0) and l is a parameter defined between 0 and 1
according to the orientation of the edges as follows (Rudge and
Haseloff, 2005):
l = 1 − asin2 u
(8)
where u is the angle between the edges and the major axis of the
cell and a is the degree of anisotropy defined on (0, 1). With a ¼ 0
we get isotropic growth and with a ¼ 1 we have anisotropic
growth in the direction of the major axis of the cell. These equations allow us to switch growth from totally isotropic to any
degree of anisotropy. All the parameters used in this model
were taken from Abera et al. (2013).
Cell division algorithm
The moment in time when the cell divides was determined
based on cell size. A dividing wall is inserted whenever a cell
doubles its area. To determine the position and orientation of
the new wall that divides the cell, an ellipse is fitted to the cell vertices (for details of the fitting algorithms, see Mebatsion et al.,
2006, 2008). The outputs of the ellipse-fitting algorithm are the
major and minor diameters (which are orthogonal to each
other) and orientation (the direction of the major diameter) of
the fitted ellipse. The new wall is then inserted along the shortest
diameter of the fitted ellipse perpendicular to the longest diameter of the fitted ellipse (Fig. 1). Eventually, the mother cell,
along with its entities (vertices, walls), is replaced by the two
daughter cells. The cell vertices, walls and cell connectivity
are then updated and cell expansion resumes. The division
rules of the algorithm are based on the position and orientation
of the dividing wall. The orientation the dividing wall is made
to be along the minor diameter of the fitted ellipse normal to
the major diameter (in accordance with both Errera’s rule and
Hofmeister’s rule) whereas the position of the dividing wall
can be made to vary to produce daughter cells with different
0·1
0
<0·5
0·5–1·0
1·0–1·5
1·5–2·0
Mean normalized cell area
>2·0
F I G . 7. Cell area distribution. Cell area is normalized by the mean area of the
cells in the tissue. Values are means + s.d. for five different simulations runs.
Asymmetric, asymmetric cell division; Symmetric, symmetric cell division;
a, anisotropic value.
sizes. For example, if the position is made to be at the centroid
of the cell, the division will result in two similar daughter cells
(geometrically symmetric cell division, which is in accordance
with Errera’s rule); otherwise it will result in two different daughter cells (geometrically asymmetric cell division). The position
of the dividing wall in this case is moved to either direction
along the major axis based on a random factor chosen from a
uniform distribution between – 0.8 and 0.8 (the range is chosen
to avoid non-viable cells, which are very small if the range is
between – 1 and 1). There is also a good chance of symmetric
cell division when the random factor is 0.
Experimental data
The simulation data were compared with experimental data for
shoot apical meristem (SAM) of Arabidopsis thaliana and leaf
tissue of A. thaliana obtained from De Reuille et al. (2005) and
De Veylder et al. (2001), respectively. We digitized the images
to get individual cell coordinates using a Matlab program. The
topological and geometrical property distributions are then calculated from the digitized coordinates. In De Reuille et al.
(2005), their figure 5 shows how cell division in the SAM takes
place by maintaining the same colour for the mother cell and
the two daughter cells. The observation suggests the division is
mostly symmetric. In De Veylder et al. (2001), their figure 7
provides a typical example of asymmetric cell division of
meristematic leaf cells.
610
Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting
TA B L E 3. Standard deviation and skewness values of cell area distribution
Symmetric cell division
Anisotropy ¼ 0
Anisotropy ¼ 0.5
Anisotropy ¼ 1
Asymmetric cell division
s.d.
Skewness
s.d.
Skewness
0.36 + 0.03
0.33 + 0.05
0.30 + 0.03
1.00 + 0.20
1.18 + 0.41
1.30 + 0.69
0.76 + 0.07
0.67 + 0.08
0.55 + 0.05
1.70 + 0.67
0.64 + 0.32
0.22 + 0.22
Values are means for the five different runs for each case with + values indicating the standard deviation in each case.
Geometric and topological properties
A
Asymmetric, a = 0, after expansion
Asymmetric, a = 0, before expansion
Asymmetric, a = 0·5, after expansion
Asymmetric, a = 0·5, before expansion
Asymmetric, a = 1·0, after expansion
Asymmetric, a = 1·0, before expansion
0·6
Frequency
0·5
0·4
0·3
0·2
0·1
0
B
Symmetric, a = 0, after expansion
Symmetric, a = 0, before expansion
Symmetric, a = 0·5, after expansion
Symmetric, a = 0·5, before expansion
Symmetric, a = 1·0, after expansion
Symmetric, a = 1·0, before expansion
0·6
Frequency
0·5
Topology. The topology of the cell aggregate (cell collection in a
tissue) is defined in terms of the number neighbour cells that are
in contact with a given cell. For both the simulated tissues and
SAM from the experimental data, the algorithm calculates the
number of cells that are in contact (neighbour cells) with a
given cell and a frequency distribution of the number of cells
was determined. The topology distributions of the tissues
obtained from the model, using different cell division rules (geometrically similar daughter cells and geometrically different
daughter cells together with the different anisotropic values),
were compared statistically with each other and with that of the
experimental tissue.
Cell shape. The cell shape distribution is characterized in this
0·3
study by two distinct geometrical properties, namely aspect
ratio and interior angle of the polygons representing the cell
boundary.
The aspect ratio (ar) of the cells was defined as:
0·2
ar = 1 − (l1 /l2 )
0·4
0·1
0
C
Symmetric, a = 0, after expansion
Symmetric, a = 0, before expansion
Asymmetric, a = 0, after expansion
Asymmetric, a = 0, before expansion
0·6
0·5
Frequency
We have developed a program that calculates the geometric
and topological properties of the cells for both the simulated
tissues and the tissue from the confocal microscopy experimental
data. The properties that we have considered were topology, cell
shape (aspect ratio and interior angles) and cell size.
0·4
0·3
0·2
(9)
where l1 and l2 are the minor and major diameter of the fitted
ellipse, respectively. With this definition, circular cells will
have an aspect ratio of 0, whereas cells which have shapes far
from circular will approach an aspect ratio of 1. The diameters
of the equivalent ellipses were calculated according to the procedure outlined by Mebatsion et al. (2006).
The interior angles of the polygons, which represent cells in
the virtual tissues generated with the different rules of cell division (symmetric and asymmetric) and growth modes (isotropic
and anisotropic), were calculated. The distributions were then
compared with each other. A comparison was also made with a
distribution obtained by assuming that the cell polygons are
regular (an ideal situation in which all interior angles of a
polygon are assumed to be equal). The interior angles (i.e. at
the vertices where two cell walls meet) were calculated using
0·1
0
0–0·2
0·2–0·4
0·4–0·6
Aspect ratio
0·6–0·8
0·8–1·0
F I G . 8. Aspect ratio distributions of cells: (A) asymmetric cell division with different degrees of anisotropy; (B) symmetric cell division with different degrees of
anisotropy; and (C) a comparison between symmetric and asymmetric cell divisions. Values are means + s.d. for five different simulations runs. Asymmetric,
asymmetric cell division; Symmetric, symmetric cell division; a, anisotropic value.
Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting
the inverse cosine function by defining two vectors starting from
the common vertex heading away from it along the two walls of
the cell sharing that vertex (Fig. 2). The interior angles for the
regular polygons were calculated as:
u=
n−2
× 180 8
n
(10)
where u is interior angle and n is the number of sides of the
polygon.
Cell size. The size distribution of cell areas (2-D) was calculated.
The areas of the cells were calculated by applying Green’s
theorem (Kreyszig, 2005).
Statistical comparison
Topological and geometrical (shape and size) properties of
both microscopic cellular images and virtual cells were calculated and compared statistically. A two-sample Kolmogorov –
Smirnov test was used to compare the distributions of these
values. The null hypothesis was that both are from the same continuous distribution. The alternative hypothesis was that they
were from different continuous distributions. The test statistic
is the maximum height difference of the two data distributions
on a cumulative distribution function. If the test statistic is
greater than the critical value the null hypothesis is rejected.
The P-value, which is dependent on the test statistics and the
threshold with reference to the test significance, is normally
used as criterion for whether to reject the null hypothesis. If
the P-value is less than the test significance, the null hypothesis
is rejected meaning that the two distributions are different. The
result of the test was 1 if the test rejects the null hypothesis at a
specified significance level, and 0 otherwise. We have used a
5 % significance level (Justel et al., 1997). The distributions
were also compared by using the mean, standard deviation and
skewness. The statistical comparision was done in Matlab (The
Mathworks, Natick, MA, USA).
R E S U LT S
A demonstration of the cell division algorithm at different stages
in time for symmetric cell division with isotropic growth
is shown in Fig. 3. Several examples of tissues generated by
the 2-D cell division algorithm are presented in Fig. 4. The
tissues shown were generated using: (A) symmetric cell division
611
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 anisotropic
growth value of 0.5 and (F) symmetric cell division with anisotropic growth value of 1. As can be seen from the figure, the
algorithm can generate different kinds of tissues by employing
different cell division rules and different cell growth modes.
The visual differences between the tissues are evident from the
figure. The topological and geometrical properties are analysed
and discussed below. Five different simulation runs are used
for each case. The number of cells used was 781, 350, 246,
474, 286 and 310 for symmetric cell division– isotropic
growth, symmetric cell division – anisotropic growth (0.5), symmetric cell division – anisotropic growth (1), asymmetric cell
division– isotropic growth, asymmetric cell division– anisotropic growth (0.5) and asymmetric cell division – anisotropic growth
(1), respectively.
Topology
The topological distribution, defined as the number of neighbour cells, showed a clear difference between the different
tissues presented above. As can be seen from Fig. 5, there is a
clear difference between tissues using the different cell division
rules and growth modes, where the virtual tissues using symmetric cell division have a narrower distribution than those using
asymmetric cell division rules (Table 1). Another interesting
feature of these distributions is the skewness. Although all the
cases presented here display skewness in the distribution of
number of neighbours, the degree of skewness is lower for symmetric cell division (Table 1).
The different cell divisions are also tested on how well they fit
to Lewis’ law, which states that a linear relationship exists
between the number of neighbours and the area of cells
(Lewis, 1928). The results are presented in Fig. 6. The error
bars indicate the standard deviation of the mean normalized
cell area. The linear relationship between the mean normalized
cell area and the number of neighbours is respected more in the
symmetric 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 data
used in the fitting to get a fair comparison. The symmetric cell
division rules have a lower norm of residuals than their asymmetric counterparts, suggesting a better linear fit. The asymmetric
cell division rules demonstrated higher standard deviations
TA B L E 4. Mean, standard deviation and skewness values of aspect ratio distribution
Anisotropy ¼ 0
Symmetric cell division
Asymmetric cell division
Mean
s.d.
Skewness
Mean
s.d.
Skewness
Anisotropy ¼ 0.5
Anisotropy ¼ 1
Expanded
Not expanded
Expanded
Not expanded
Expanded
Not expanded
0.2 + 0.001
0.09 + 0.002
0.31 + 0.11
0.24 + 0.01
0.13 + 0.01
0.56 + 0.19
0.36 + 0.001
0.13 + 0.01
20.04 + 0.10
0.36 + 0.02
0.20 + 0.02
0.40 + 0.14
0.23 + 0.01
0.10 + 0.01
0.10 + 0.30
0.27 + 0.01
0.16 + 0.01
0.80 + 0.11
0.32 + 0.01
0.13 + 0.01
20.05 + 0.13
0.33 + 0.03
0.20 + 0.02
0.71 + 0.20
0.26 + 0.01
0.12 + 0.02
0.09 + 0.27
0.27 + 0.01
0.15 + 0.004
0.69 + 0.15
0.32 + 0.02
0.14 + 0.02
0.13 + 0.38
0.34 + 0.01
0.16 + 0.004
0.33 + 0.13
Values are means for the five different runs for each case with + values indicating the standard deviation in each case.
612
Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting
whereas the symmetric cell division rules showed a lower slope
than that given by Lewis’ law. These results are in agreement
with those of Sahlin and Jönsson (2010). The symmetric cell
division algorithm with isotropic growth, in which only the peripheral cells were allowed to divide, matched perfectly with
Lewis’ law.
Cell size distribution
0·6
Frequency
0·5
Asymmetric, a = 0, after expansion
Asymmetric, a = 0, before expansion
Asymmetric, a = 0, regular polygon
Symmetric, a = 0, after expansion
Symmetric, a = 0, before expansion
Symmetric, a = 0, regular polygon
The cell size distribution, presented here as the distribution of
mean normalized cell area, is an important geometrical property
used to compare the performance of different cell division rules.
As can be seen from Fig. 7, the cell size distribution shows a
clear distinction between the various division rules investigated.
In particular, the difference between the symmetric cell division
and asymmetric cell division is remarkable. Besides the difference
between the spread of the distribution, which is evident from Fig. 7,
the degree of skewness is another important difference between the
different division rules employed (Table 3). The spread of the distributions as well as the degree of skewness are more pronounced in
the asymmetric cell division than the corresponding symmetric cell
division, similar to what was found for the cell topology.
0·4
0·3
0·2
0·1
0
<50
50–80
80–110 110–140
Internal angle (°)
140–170
Cell shape distribution
F I G . 9. Interior angle distribution comparison of symmetric and asymmetric
cell divisions. Values are means + s.d. for five different simulations runs.
Asymmetric, asymmetric cell division; Symmetric, symmetric cell division; a,
anisotropic value.
Cell shape distribution is analysed here by means of two geometrical properties, namely aspect ratio and the internal angle of
the polygons representing the cell boundary (see Methods).
TA B L E 5. Mean, standard deviation and skewness values of internal angle distribution
Symmetric cell division
Expanded
Not expanded
Regular polygon
Asymmetric cell division
Mean
s.d.
Skewness
Mean
s.d.
Skewness
116.8 + 0.26
116.8 + 0.26
116.8 + 0.26
23.4 + 0.47
35.6 + 0.07
11.2 + 0.47
0.12 + 0.03
0.56 + 0.018
–0.83 + 0.30
115.8 + 0.20
115.8 + 0.20
115.8 + 0.20
27.0 + 0.72
38.0 + 0.54
16.1 + 0.87
–0.211 + 0.06
0.28 + 0.08
– 0.74 + 0.17
Values are means for the five different runs for each case with + values indicating the standard deviation in each case.
A
B
F I 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 meristematic leaf cells taken from De Veylder et al. (2001) where cell division is mainly asymmetric (B).
Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting
A
0·8
0·6
0·4
0·2
0
0
0·40
Interior angle. The interior angles of the polygons representing
0·5
1·0
4
5
2·0
2·5
3·0
1·5
Mean normalized cell area
3·5
4·0
9
10
11
0·7
0·8
B
0·35
0·30
0·25
Density
the cell are calculated just after cell division and after expansive
growth, when the cells have achieved equilibrium. These distributions are compared with each other and with the interior angle
distribution obtained assuming all the polygons are regular (an
ideal situation in which all interior angles of a polygon representing a cell are assumed to be equal). As most of the cells have six
sides, symmetric cell division converges to 1208, better than its
asymmetric counterparts (Fig. 9). Although their mean values
are more or less equal, there is a clear difference in the width
of the distribution as represented by its standard deviation
(Table 5). The ideal internal angle distributions of the different
cell division rules employed are narrower than their respective
actual distributions. The interior angle distributions of the
tissues obtained after cell expansion better approximate the respective ideal interior angle distributions in terms of both the
standard deviation and the skewness of the distributions
(Table 5). In both the aspect ratio distribution and the interior
angle distribution, the distributions after cell expansion approximate the ideal distributions better than those just after cell
division.
Asym, a = 0
Asym, a = 0·5
Asym, a = 1·0
SAM
Sym, a = 0
Sym, a = 0·5
Sym, a = 1·0
1·0
Density
Aspect ratio distribution. The cell aspect ratio distributions of the
tissues obtained using different cell division rules and growth
modes are presented in Fig. 8 and Table 4. As can be seen from
the figure and table, the mean aspect ratio values of the corresponding symmetric and asymmetric cell division rules immediately after cell division (before expansion) are almost equal.
Their difference is reflected in the spread of the distribution, as
shown by the corresponding standard deviation values. The symmetric cell division generally produces a narrower distribution
than the asymmetric cell division. The difference in the skewness
of the distribution is apparent: the asymmetric cell division rules
result in a higher positive degree of skewness as opposed to little
or no skewness for their symmetric counterparts. Among the cell
growth types investigated, isotropic cell growth has lower aspect
ratio values than anisotropic cell growth, although this difference
is reduced after cell division. The effect of the growth type used is
mainly manifested in the overall tissue shape (see Fig. 4).
613
0·20
0·15
0·10
0·05
3
6
7
8
Number of neighbours
C
3·0
2·5
As demonstrated by Besson and Dumais (2011) and
Prusinkiewicz (2011), seeking an exact prediction of a cell
division pattern is as futile as attempting to predict the exact sequence of numbers produced by repetitively throwing a die. Only
the statistical characteristics of these processes, as opposed to
individual outcomes, can be meaningfully anticipated. In this
regard, the visual, topological and geometrical comparisons of
the different cell division rules with experimental data are presented here. The image with 347 cells obtained from a confocal
microscopy was taken from De Reuille et al. (2005) and digitized
by a Matlab code (Fig. 10A). Comparison of the cell area distribution, topology distribution and aspect ratio distribution
(Fig. 11) suggests that symmetric cell division with isotropic
growth has superior visual, topological and geometrical similarity to the experimental data than the other division rules investigated (Table 6). The experimental data as well as symmetric cell
division, where cell division is restricted to the peripheral cells,
Density
2·0
Comparison of real and virtual topological and geometrical
properties
1·5
1·0
0·5
0
0
0·1
0·2
0·3
0·4
0·5
0·6
0·9
Aspect ratio
F I G . 11. Geometrical and topological property comparisons of virtual tissues
generated by the algorithm and a tissue obtained experimentally: (A) cell area distribution; (B) topology distribution; and (C) aspect ratio distribution. Asym,
asymmetric cell division; Sym, symmetric cell division; SAM, shoot apical meristem; a, degree of anisotropy. Density in these plots represents frequency density.
were well fitted to Lewis’ law (see Fig. 6G, H). Two-sample
Kolmogorov – Smirnov tests were made between the best performing division rule (symmetric cell division with isotropic
614
Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting
TA B L E 6. P-values (expressed as a percentage) of the comparisons of the experimental data with the data obtained by the different
simulation cases
Symmetric cell division
Cell area
SAM
Meristematic leaf
SAM
Meristematic leaf
SAM
Meristematic leaf
Topology
Aspect ratio
Asymmetric cell division
a¼0
a ¼ 0.5
a¼1
a¼0
a ¼ 0.5
a¼1
21
8 × 10 – 7
94
5 × 10 – 26
40
2 × 10 – 2
7
8 × 10 – 6
24
9 × 10 – 14
38
13
4.5
1 × 10 – 6
2.9 × 10 – 2
1 × 10 – 7
46
19
4×10 – 12
23
0.27
70
33
32
7 × 10 – 6
11.6
9 × 10 – 4
28
4
51
9 × 10 – 6
32
0.3
2 × 10 – 4
93
89
The comparison is done using a two-sample Kolmogorov –Smirnov test. SAM, shoot apical meristem; a, anisotropic value.
1·0
Cumulative probability
0·9
A
B
C
0·8
0·7
P = 0·94
P = 0·21
0·6
P = 0·40
0·5
0·4
0·3
0·2
Experiment
Simulation
0·1
0
4
5
6
7
8
9
Number of neighbours
10
0·5
1·0
1·5
2·0
Mean normalized cell area
0
0·1 0·2 0·3 0·4 0·5 0·6 0·7 0·8
Aspect ratio
F I G . 12. Statistical comparison of geometrical and topological properties of virtual tissues generated by the algorithm using symmetric cell division with isotropic
growth and a shoot apical meristem obtained from experiment using a two-sample Kolmogorov– Smirnov test.
growth) and the experimental data (Fig. 12 and Table 6). The
comparisons are made at 5 % significance level (see Methods).
The results suggest the distributions are not significantly different. The different simulation cases were also compared with a
meristematic leaf tissue image obtained from De Veylder et al.
(2001) and digitized with a Matlab code (see Fig. 10B). The
image contains 253 cells. Comparison of the cell area distribution, topology distribution and aspect ratio distribution (Fig. 13
and Table 6) suggest that asymmetric cell division generally
shows superior topological and geometrical similarity to the experimental data than their symmetric counterparts. Two-sample
Kolmogorov – Smirnov tests made between the data obtained
using asymmetric cell division with isotropic growth and the experimental data obtained from De Veylder et al. (2001) showed
the results are not significantly different. The results of the comparison are presented in Fig. 14.
DISCUSSION
We have developed a generic cell division algorithm based on
cell-wall mechanics that is capable of producing different
tissues varying both in geometrical and in topological property
distributions of the cells as well as in the overall shape of the
tissues. The algorithm provides a robust cell division model
based on ellipse-fitting to find the position and orientation of
the dividing wall and incorporates cell-wall mechanics for
growth. Using the developed algorithm, we have generated different tissues. We have studied tissues which were generated
using cell division rules of equally sized daughter cells and
unequally sized daughter cells. Isotropic or anisotropic growth
modes were used for each case. We have analysed the topologies
and geometries of simulated tissues and compared them with
experimental data to test the performance of different division
rules (symmetric and asymmetric in a geometric sense). A comparison is also made among the virtual tissues generated using
the different cell division and growth types (isotropic, anisotropic). We have also investigated how well tissues fitted to Lewis’
law, which states that a linear relationship exists between the
number of neighbours and the area of cells (Lewis, 1928).
Alim et al. (2012) investigated the effect cell division orientation on tissue growth heterogeneity. Cell growth in their approach
was implemented by minimizing the difference between the
second area moment of a target ellipse (which grows with
time) and that of the cell calculated from its vertices. They
found that tissue heterogeneity is more pronounced where cell
division with randomly orientated division walls is used. Our
result also allows the same conclusion.
The role of mechanics on cell expansion was detailed by Abera
et al. (2013, 2014, and references therein), where a cell is represented as a thin-walled structure maintained in tension by turgor
pressure. The cells will respond by increasing the resting length
of the cell walls (simulating actual growth) to alleviate the
Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting
A
Asym, a = 0
Asym, a = 0·5
Asym, a = 1·0
Meristematic leaf
Sym, a = 0
Sym, a = 0·5
Sym, a = 1·0
1·0
Density
0·8
0·6
0·4
0·2
0
0
0·5
1·0
1·5
2·0
2·5
3·0
Mean normalized cell area
3·5
4·0
9
10
11
0·7
0·8
0·9
B
0·40
0·35
Density
0·30
0·25
0·20
0·15
0·10
0·05
3
4
5
0·1
0·2
6
7
8
Number of neighbours
C
3·0
2·5
Density
2·0
1·5
1·0
0·5
0
0
0·3
0·4 0·5 0·6
Aspect ratio
F I G . 13. Geometrical and topological property comparisons of virtual tissues
generated by the algorithm using asymmetric cell division with isotropic growth
and a real meristematic leaf tissue obtained experimentally (from De Veylder
et al., 2001): (A) cell area distribution; (B) topology distribution; and (C) aspect
ratio distribution. Asym, asymmetric cell division; Sym, symmetric cell division;
a, degree of anisotropy. Density in these plots represents frequency density.
mechanical stress induced by the tension. The role of mechanical
stress on the orientation of the dividing wall has been explained
by different authors (e.g. Lynch and Lintilhac, 1997; Lintilhac
615
and Vesecky, 1981; Hejnowicz and Romberger, 1984).
Nakielski (2008) developed a model based on these studies for
root growth. These are studies based on the idea of growth
tensors (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 to
define the velocities of the vertices of the cell mesh and the
cell area increases as the vertices move governed by the GT
field defined on the organ level. When a cell is ready to divide,
the PDGs that are predefined by the GT based on the location
of the cell with regard to the GT will be used to determine the
orientation of the dividing wall. The diameters of the mother
cell along the PDGs will be calculated and the dividing wall is
inserted along the shorter of these diameters.
In our approach, the growth biomechanics govern the shape of
the mother cell on which an ellipse is fitted to decide how the cell
should divide. PDGs are mutually perpendicular diameters that
can be obtained if you fit an equivalent ellipse to the vertices
of the cell (minor and major diameters of the ellipse).
Therefore, regarding the way the cell division wall orientation
is determined, the two approaches should lead to a similar
result (see Alim et al., 2012). The most important difference
between their approach and ours is in their approach the PDGs
are predefined based on location (because the GTs are predefined
at organ level) whereas in our approach it is the mechanics that
leads to the shape of the mother cell based on which it divides.
Note that in cases where growth is started from an already elongated cell and maximum growth rate is along the shorter diameter, the two approaches will lead to different orientation for
the dividing wall. In this case, when using the ellipse-fitting algorithm, the orientation of the dividing wall should be along the
orientation of the major diameter of the fitted ellipse.
The algorithm can produce the variability observed in symmetric cell division. Besson and Dumais (2011) achieved the
variability observed in symmetric cell division by introducing
the concept of local minima instead of the global minima of
Errera’s rule for selection of the dividing wall. In our model,
this was achieved without abandoning the founding rule of cell
division suggested by Errera (1886). An ellipse-fitting procedure
was used to determine the position and orientation of the dividing
wall. The minor diameter of the ellipse through the centroid was
used as the position and orientation of the dividing wall. If the
fitted ellipse is a circle, we have infinitely many possible orientations for the candidate dividing wall. The random selection of
one of them made it possible to produce the variability observed
in symmetric cell division (see Fig. 4A). Ellipse-fitting has been
used in previous studies (e.g. Gibson et al., 2011; Merks et al.,
2011) to determine the position of the new dividing wall, but
its importance to produce randomness even in symmetric cell
division has not been put forward. This randomness is an important property that represents biological variability observed in
nature.
Robinson et al. (2011) demonstrated asymmetric cell division
in the leaves of arabidopsis, where the nucleus is displaced from
the geometrical centre of the cell by cell polarity switching.
In our model, asymmetric cell division is achieved by random
displacement of the dividing wall along the major diameter of
the fitted ellipse. This random displacement of the dividing
wall is the equivalent of displacement of the nucleus from
the geometrical centre of the cell, which could lead to two
616
Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting
1·0
Cumulative probability
0·9
A
B
C
0·8
P = 0·23
0·7
P = 0·70
P = 0·32
0·6
0·5
0·4
0·3
Experiment
Simulation
0·2
0·1
0
2
4
6
8
Number of neighbours
10
12 0
0·5 1·0 1·5 2·0 2·5 3·0 3·5 4·0
Mean normalized cell area
0·1
0·2
0·3
0·4
0·5
0·6
0·7
Aspect ratio
F I G . 14. Statistical comparison of geometrical and topological properties of virtual tissues generated by the algorithm using asymmetric cell division and a meristematic leaf tissue obtained experimentally using a two-sample Kolmogorov –Smirnov test.
daughter cells that are different in size as well as in topology (see
Fig. 4D– F). Our model has the added advantage that it incorporates cell mechanics in the growth model. By making the mechanical properties and maximum resting length of the walls
dependent on direction, the model allows both isotropic and anisotropic cell growth, which leads to different simulated tissues
(see Fig. 4). Although most of the meristematic cells that are undifferentiated obey symmetric cell division with isotropic
growth, asymmetric cell division and anisotropic growth also
have great potential to account for differentiation of meristem
tissue into different specialized tissues.
Gibson et al. (2011) used the same cell representation as we do
here. They developed differential equations and Newton’s law
was used to solve force balance on the vertices. In contrast to
their model, ours assigns spring constant values, which are
inversely proportional to wall length, whereas in their model
the same spring constant value, independent of length, is
assumed. Moreover, their model did not simulate actual growth
of the cell walls; rather, they used a constant resting length
throughout the simulation. They studied only isotropic mechanical properties of the cell wall. Therefore, our model provides a
better representation of the cell mechanics and actual growth
of the cell wall, which actually mimics the biosynthesis of cellwall materials.
Merks et al. (2011) introduced a cell-based computer modelling 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-related
turgor pressure force and the tension force of the walls. They
used the same cell representation as we do here. In contrast to
their probabilistic determination of position of vertices, ours
uses a deterministic approach based on Newton’s law of force
balances. In addition, they insert a new node to simulate cell-wall
yielding after length exceeded four times its initial value while
ours actually simulates growth of the resting length of the walls
using differential equations simultaneously with mechanics.
While their model presents a simplified software platform, ours
offers better representation of the actual growth of the cell wall.
The cell division algorithms can be coupled to 2-D expansive
plant cell growth models (Abera et al., 2013), where the initial
topology was, for example, obtained from a 2-D Voronoi
tessellation. The cell division algorithm has more biological justification than the Voronoi tessellations, which were initiated
from random generating points, which is simply a partition of a
given area cell topology into a number of regions, representing
the individual cells.
In particular cases where the dividing wall is not a function of
the shape of the mother cell, such as those observed in periclinal
and anticlinal cell divisions (Howell, 1998; Evert, 2006), special
care should be taken in using the ellipse-fitting algorithm.
Periclinal (division walls parallel to the surface): this usually
leads to relatively large new wall as the mother cells are elongated
cells along the periphery. In this case, the ellipse-fitting can still
be used to determine the position and orientation of the new wall
but the major diameter instead of the minor diameter of the
ellipse has to be used. Anticlinal (division walls at right angles
to the surface): this usually leads to short new wall and the
ellipse-fitting can still be used although caution has to be taken
when the fitted ellipse is a circle, where there are infinitely
many candidate short walls, to ensure the chosen short wall is
the one that is normal to the surface.
CON C L U S IO NS
The cell division algorithm developed here can produce tissues
that have different topological and geometrical properties. This
flexibility to produce different tissue types gives the model
great potential for use in in silico investigations of plant cell division and growth. The cell division algorithms take account of
both cell shape and topology. The model is based on cell
mechanics, where the cell wall mechanical properties, fluid
matrix inside the cell and cell turgor pressure are taken into
account. The equations for actual growth of the cell walls
(change in resting length of the walls) and cell division are
solved continuously. It is generic in that a switch between isotropic growth and anisotropic growth as well as between symmetric cell division and asymmetric cell division is automatic and
easy, which makes the model convenient to adapt to a specific
case study. Finally, the model is robust as there is no need for
an iterative procedure to find the shortest wall for cell division.
In our algorithm, the division wall is inserted along the orientation of the minor diameter of the fitted ellipse. The geometrical
Abera et al. — Plant cell division algorithm based on biomechanics and ellipse-fitting
properties of the simulated tissues were compared with experimental data for SAM. Symmetric cell division with isotropic
growth best fits the experimental data.
ACK N OW L E DG E M E N T S
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 Institute
for the Promotion of Innovation by Science and Technology in
Flanders (IWT scholarship SB/0991469) is gratefully acknowledged. T.D. is a postdoctoral fellow of the Flanders Fund for
Scientific Research (FWO Vlaanderen).
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