simple geometric rules such as Thom's division rule (Thom, 1972). ... organism at each stage of its life, embryogenesis during the development, neogenesis .... Cambium is represented by a cellular automaton Î(t) of N(t) fusiform cells, where.
Morphogenetic processes. Application to cambial growth dynamics. Loïc Forest+, Jaime San Martín* , Fernando Padilla* , Fabrice Chassat* , Françoise Giroud +, Jacques Demongeot+ o +
Laboratoire Techniques de l’Imagerie, de la Modélisation et de la Cognit ion (TIMC UMR CNRS 5525). Faculté de Médecine – Domaine de la Merci - 38706 La Tronche cedex – France * Centro de Modelamiento Matemático (CMM UMR CNRS 2071). Av. Blanco Encalada 2120 piso 7 Santiago de Chile - Chile o Institut Universitaire de France
Abstract: Both physiologic and pathologic morphogenetic processes we can meet in embryogenesis, neogenesis and degenerative dysgenesis present common features: they are ruled by three different kinds of mechanisms, one related to cell migration, the second to cell differentiation and the third to cell proliferation. We deal here with an application to the cambial growth which involves essentially the third type of mechanism. Woody plants produce secondary tissue (secondary xylem and phloem) from a meristematic tissue called vascular cambium, responsible for the radial growth of a tree. The paper focuses on the formation of secondary xylem, considered in two dimensions in a cross-section framework. A new discrete modelling approach is used, based on the cellular scale, in order to attain a more accurate understanding of how the elementary microscopic behaviour of each cell takes part in the macroscopic morphogenesis. The mathematical modelling uses essentially to account for the cell proliferation an occurrence method simulating the main features of radial growth with simple geometric rules such as Thom’s division rule (Thom, 1972). The study applies to concrete instances in which the changes made in the geometrical cellular patterns of the vascular cambium clearly affect the shape of the tree, as in Pinus radiata . Keywords : morphogenesis, proliferation, cambium, vegetal cell, radial growth, xylem
1. Introduction The morphogenetic processes in biology and pathology involve three main mechanisms: the cell migration, the cell differentiation and the cell proliferation. These processes concern a living organism at each stage of its life, embryogenesis during the development, neogenesis during the adult stage and dysgenesis at the early stage, when degenerative processes are perturbating the ageing. For example the invagination of salivary acini in Drosophila embryo is a morphogenetic process which requires together the three mechanisms differentiation, migration and proliferation (Myat and Andrew 2000). During Drosophila development, the salivary primordia are internalized to form the salivary gland tubes. By studying histological sections and electron micrographs done at multiple stages of salivary gland development, the authors have shown that internalization involved coordinated cell shape changes and migrations with the dorsal-posterior cells of the primordia. Many genes are sequentially expressed as hkb starting the sequence of cell movements. The genetic control is exerted directly on differentiation and indirectly on migration and proliferation processes by modifying the cell geometry and the morphogenes concentration. The epigenetic control concerns essentially the migration process with mechanisms like diffusion, reaction, convection, haptotaxis, chemotaxis,… (Benkherourou et al. 2000) and the proliferation process in which the rate of mitosis depends on the nutritive surface/inner volume cell ratio (Thom 1972).
1
Morphogene concentration
Genetic control bifurcation Epigenetic control trajectory
A T G G C A T G
T
Gene Fig. 1 The morphogenetic landscape by C.H. Waddington (Waddington 1966).
The run of a living organism during the morphogenetic dynamics has been symbolized by C.H. Waddington (Waddington 1966) in a morphogenetic landscape by trajectories on a surface whose shape is fixed by the physico-chemical laws (diffusion, reaction, convection,…) for the epigenetic control and is modified by the gene expression for the genetic control (Fig. 1). This genetic control is described with interaction graphs (Fig. 2) which express the relationships (repression or induction) between genes. The presence of positive loops (i.e. paths from a gene to itself passing through a even number of repressions) in these graphs (Aracena, 2001; Cinquin and Demongeot 2002; Demongeot et al. 2003; Aracena et al. 2003) is a necessary condition for the differentiation and the presence of negative loops (i.e. paths from a gene to itself passing through a odd number of repressions) ensures the stability of the morphogenetic trajectories. A classical example of such an interaction graph in gastrulation process in Drosophila has been studied by M. Leptin (Leptin 1999) and J. Aracena (Aracena 2001) has proved that four positive loops were necessary to account for the experimental evidence of four differentiated kinds of cells involved in the process of gastrulation (Fig. 2). The initial graph proposed in (Leptin 1999) had only two positive loops ensuring the existence of only two kinds of differentiated cells.
2
+ +
+ +
+
-
+
+
-
+ + +
-
+
+ +
+
-
CyT
+
NDK
ADK
+
-
+
+
+ +
Fig. 2 The genetic regulation network of the gastrulation in Drosophila [(Leptin 1999), (Aracena 2001)].
2. The cambial growth. 2.1. Generalities Radial growth of conifer tree provides a good example of a morphogenetic process mainly ruled by cell proliferations. Indeed, radial growth in conifer is under control of the dynamic of a tissue called vascular cambium or cambium. On a cross-section of a tree such as that presented in Fig. 3a taken from an experimental study conducted by Padilla and San Martín (Cominetti et al., 2002; Padilla, 2001) or Fig. 3b, a microscopic view (Zimmermann et al., 1971), several zones can be observed by function [(Harris, 1991), (Zimmermann et Brown, 1971)]. Bark Phloem Vascular cambium Cambial initials Xylem Pith
a
b
3 cm Fig. 3 Real cross-sections of a Pinus radiata taken from (Padilla, 2001; Zimmermann and Brown, 1971).
The central zone is called the pith. Next to it is the xylem (the wood), mainly made of dead cells involved in the transit of water and minerals. The bark makes up the outer zone, the zone inside the
3
bark is the phloem, where the elaborated sap is transported. Using Butterfield 's well-accepted terminology, between the xylem and the phloem there is the vascular cambium or cambium (typically 5 cells in width) (Butterfield, 1975). The cambium is entirely made of living proliferating cells. These constitutive cells are of two types: the elongated fusiform cell and the cuboid ray cells (Larson, 1994; Barlow et al., 2002). The first type of cell dominates in Pinus radiata (D. Don.). Cellular divisions take place mainly in the cambium. Inside the cambium, a central uniseriate tissue made of cambial initials can be distinguished. Divisions taking place in the cambial initial tissue can be classified into two categories according to the side on which they occur (Harris, 1991; Larson, 1994). The most common division is periclinal: it produces a new cellular wall orthogonal to the radial direction. Following this process, either the most outer cell remains in the cambial initial tissue and the other becomes a xylem mother cell (this is the most common phenomenon), or it progressively differentiates into a phloem mother cell and in this case the most inner cell remains in the cambium initial tissue (Barlow et al., 2002; Harris, 1991). Xylem and phloem mother cells that are part of the cambium may divide at other times but almost always by periclinal division; then they join the differentiating xylem or phloem cells, which are cells in transition to mature xylem and mature phloem. Cells in these differentiating tissues have a limited capacity to divide. These cells enlarge and undergo secondary wall thickening. Those periclinal divisions explain the radial growth of the tree, which is carried out by a progressive outward movement of the cambium. To phloem
r n
To xylem
r Fig. 4 Examples of a fusiform initial cell dividing in the two types of periclinal division. n is the normal unit vector related to the external perimeter of the cell pointing outward: from the xylem to the phloem.
While the tree grows in the radial direction, the girth also increases. As the size of a cell is limited (Harris, 1991), this growth implies that the number of cells inside the cambium increases. Anticlinal divisions make this increase possible. Indeed, a cambium cell can divide itself radially, resulting in two daughter cells that then stay in the cambium initial tissue. Moreover, the cambium is a highly competitive zone where the cells compete to develop. High competition degree can produce fusiform cells elimination from the cambium. Cells resulting from anticlinal divisions are thinner and more fragile with a higher elimination rate (Larson, 1994).
r n
Fig. 5 Example of a fusiform initial cell dividing anticlinally.
2.2. Physiological component These mechanisms of radial growth are controlled by hormonal systems. We will focus on the influence of a growth hormone called indol-3-acetic acid (IAA) (belonging to a family of hormones called auxin) on radial growth. Indeed, even if the hormonal regulating mechanisms of xylem production are still not completely understood, IAA has been designated, notably in Uggla's work of (Uggla, 1998, 2001), as the main growth factor in the tree. This hormone is mainly produced by the principal apex of the tree and then moves to the roots (Estelle 1998; Kramer, 2002). IAA is for the most part transported through the cambial zone, where it stimulates the proliferation of cells (Uggla, 1998). This occurs at a very low speed: a molecule of IAA covers the average distance of 1 cm in 1 hour (Kramer, 2002). Transportation of IAA looks like a flowing fluid submitted to diffusion and
4
transport phenomena in a pipe: the cambium (Kramer, 2002). Kramer proposed an experimental law that measures the effect of this physiological component (Kramer, 2001). It stipulates that the rate r of xylem production is proportional to m, the mass of IAA per unit area on the cambial surface according to the relation: r = λm
(1)
where λ is called the coefficient of sensitivity of cambium to IAA. Uggla gives a scale for λ of 1 µm of formed xylem by day for a concentration of 1 ng/cm of IAA in a tree such as Pinus sylvestris. λ is a complex parameter that depends on several variables. It notably evolves as a function of the seasons throughout the year. During the tree's dormant period, this coefficient is very low because there are inhibitory factors against growth (an increase in the production of inhibitory hormones such as ethylene, less favourable outdoor conditions, for example, temperature and a decrease in the photoperiod). This explains why the quantity of formed xylem is very low in this period. Moreover, cells produced at this time are smaller, in particular because of the diminution of nutriments. These two phenomena combined lead to the formation of substantial differences in cellular density: the xylem formed in dormant periods is much denser than the xylem formed during the active period of growth. These changes explain that rings of growth are discernible macroscopically and microscopically in trees growing in temperate zones (Fig. 6) (Uggla, 2001; Roberts, 2001).
to the bark
Fig. 6 A real microscopic view showing the transition between two periods of growth and the difference in cellular density between these periods (binocular microscope image; see “Comparison of results with observation”).
3. Modelling 3.1. Cellular modelling The model developed here does not claim to take into account all the mechanisms of tree growth. On the contrary, the aim of this paper is to explain the main features of radial growth with simple rules, a first attempt at cellular modelling of secondary radial growth. It is based on a number of simplifying assumptions. We list here important hypotheses of the model concerning wood structure and development: (H1) Radial growth is considered as reduced to xylem production (i.e. xylogenesis). In the case of radiata pine trees, the phloem layer of the trees is especially thin. It is assumed in Fig. 3a that a 6-yearold tree has a xylem layer approximately 30 times thicker than t he layer of phloem and bark. (H2) Cambium will be considered as reduced to a uniseriate layer of initial cells able to grow and to divide anticlinally and periclinally. Daughter cells produced by periclinal divisions on the xylem side are assumed to integrate the mature xylem directly. This simplification consists in neglecting the progressive cellular differentiation to mature xylem. The study is concentrated on the dynamics of
5
cambial initials because it is the most important in pattern formation in trees. Cells are competing for available space but without elimination. More primitive systems such as Coleochaete orbicularis would satisfy (H1) and (H2) (Barlow, 2002). (H3) We will only take into account the initial fusiform-type cambium cells, which are more numerous (Harris, 1991), ignoring the role of the ray initial cells. (H4) IAA is assumed to be the only hormone involved in the control of the growth of the cambium. Other hormones such as gibberellins and cytokinins, whose roles in secondary growth are less clear, are not involved in the modelling process undertaken here. (H5) We have retained some of Kramer’s hypotheses (Kramer, 2001): possible formation of reaction wood due to elastic stress is ignored, water and minerals are supposed to be sufficiently available to respond to the needs of the tree. (H6) The third dimension is never considered. Cambiums in conifer are non-storied but the model does not take into account the influence of axial gaps between cells and also possible cell migration between horizontal cell layers. The purpose and the originality of modelling these phenomena are to simulate the behaviour of each cell as the morphogenetic entity in a two-dimensional framework and to study the coherent arrangement of neighbouring cells. It is worth mentioning that in forest science authors generally study the global three-dimensional architecture of morphogenesis based on the L-system (Prusinkiewicz and Lindenmayer, 1990; Lück and Lück, 1985) and that in medicine cellular models have been used in tissue renewal (Potten, 1979; Clem et al., 1997). Morphogenesis will be considered as the consecutive transformations of the cambium due to cell proliferation. To do this, we used an occurrence method. Cambium is represented by a cellular automaton Γ(t) of N(t) fusiform cells, where t is a discrete time. Each cell Xi (t) is represented by a quadrilateral defined by its 4 vertices Ai (t), Bi(t), Ci (t), D i (t). Figure 7 shows how the elements of the model were described. ∀ t ≥ 0, Γ(t) = {Xi (t)}0 ≤ i ≤ N(t) ∀ 0 ≤ i ≤ N(t), Xi(t)={Ai (t),Bi (t),Ci (t),Di (t)}
(2) (3)
Phloem
Xylem Fig. 7 Representation of a section of cambium and of three constitutive cells.
3.2. Law of evolution and occurrences The evolution of the system between t and t+1 depends on a law of evolution linked to cellular growth and on possible occurrences of events in the life of cells. The growth of the cambium tissue is uuur uuur determined by the growth of each constitutive cell. Growth occurs in the direction of the AC and BD vectors (applying the natural trend observed on microscopic sections ((Harris, 1991; Zimmermann and Brown, 1971), see “Comparison of results with observation”). We focus on the longitudinal transport of IAA, which determines the global circumferential distribution of IAA around the cambial surface. A more advanced study should model cambium made of various rows of cambial cells and the importance of the radial component of IAA transport and concentration, which control the number of cambial cell rows (Uggla, 1998). Rate of cambium cell growth is directly determined by the IAA concentration inside the cambial zone (Kramer, 2001). Because of its geometry (vertical length of approximately 5 mm in conifers (Larson, 1994)), a cambium cell exchanges IAA molecules essentially through its outer surface. To adapt this rule to the cellular model, we state that the growth in area of a cell Xi (t) will then be considered as proportional to IAA concentration on its useful perimeter for external exchanges
6
pui (t)=Ci Di(t). The IAA concentration along the cambium will be simulated by a function of distribution f(s). s is the curvilinear abscissa relative to the cambium cell's external perimeter, whose origin is C0 (t) and which goes through the cambium external perimeter L (t ) in the trigonometric direction. f as a distribution function will always be chosen so that its average is 1:
∫ U[ N (t ) i =1
Ci (t) ,Di (t )]
=1
L( t )
L (t ) =
with
f (s ) ds
(4)
N (t)
∑ pu ( t )
(5)
i
i =1
Let ∆A be the total growth in area of the cambium between t and t+1. The growth of a cambial cell is calculated as a fraction of total growth ∆A, proportionally to the distribution of auxin on its useful perimeter. So for all cells of area ∆i , the discrete dynamics of growth in area is: ∆ i ( t + 1) = ∆ i ( t ) +
(∫
C (t ),D ( t ) i i
f (s ) ds
) ∆A
(6) L( t ) The total growth in area of cambium between t and t+1 is the sum of the growth of all the cambial cell
∑ ( ∆i (t + 1) − ∆i (t ) ) . We calculate with equation (6) that: N (t )
between t and t+1, i.e.
i =1
N (t )
N (t )
N ( t)
∑ ∆ i ( t + 1) = ∑ ∆i (t ) + ∑ i =1 i =1 i =1
(∫
C ( t ),D ( t ) i i
f ( s ) ds
L (t )
As expected, we check that ∆A is equal to
) ∆A = ∑ ∆ ( t ) + ∆A . N (t )
i =1
i
∑ ( ∆i (t + 1) − ∆i (t ) ) . N (t )
i =1
∆A retains a constant value during the iterations because IAA is assumed to be synthesized at a constant rate and because IAA flux is considered conservative during transport (Kramer, 2001). From this cell dynamics of growth in area, we describe the consequently dynamics of cambial cell’s external perimeter (i.e. dynamics of Ci ( t ) and Di ( t ) ) by equation (7) to (10). Area ( Ai ( t + 1) , Bi ( t + 1), C i (t +1) , Di ( t + 1) ) = ∆i ( t + 1) (7) uuuuuuuuuuuuuuuur C (t ) C ( t + 1) f s ( Ci (t ) ) + f s (CDi ( t ) ) i i uuuuuuuuuuuuuuuu (8) ru = D (t ) D ( t + 1) f s D t + f s CD t ( ) ( ) ( ) ( ) i i i i uuuuuuuuuuuuuuuur uuuuuuuuuuuur (9) det Ci (t ) Ci ( t + 1), Ai ( t ) Ci ( t ) = 0 uuuuuuuuuuuuuuuuru uuuuuuuuuuuuru det D (t ) D ( t + 1), B ( t ) D ( t) = 0 (10) i i i i C (t ) + Di (t ) Let CDi (t ) = i . 2 Equation (7) states that the transition of Ci (t) and Di (t) to Ci (t+1) and Di (t+1) is such that the area of the cell Xi grows from ∆i (t ) to ∆ i (t + 1) . Both of the last equations of the system respect the uuuuuuuuuuur uuuuuuuuuuur hypothesis of growth along lateral vectors Ai (t ) Ci ( t) and Bi (t ) Di (t ) . Equation (8) states that the growth along each vector is proportional to the IAA mean distribution on the corresponding half useful perimeter (i.e. Ci ( t ) , CDi ( t ) for Ci (t + 1) and CDi ( t ) ,D i (t ) for Di ( t + 1) ). Appendix 1 uuuuuuuuuuuuuuuur explains the practical exploitation of these equations (calculation of vectors Ci (t ) Ci ( t + 1) and uuuuuuuuuuuuuuuuru Di ( t ) Di ( t + 1) ) for the algorithm with more mathematical details.
( (
( (
) ( ) (
) )
) )
7
In addition to this growth rule, directly related to the flux of IAA, we will take into account two occurrences in the life of cells: cellular division and additional growth in width. Occurrences of events are tested for each cell after each growth of cambium from t to t+1. (i) Cellular division: We assume that cambium cells divide according to the geometric criterion stated by Thom (Thom, 1972). This criterion states that a cell divides when its useful perimeter pui (t) = Ci Di (t), which feeds the cell through hormonal control, becomes too small compared to the area ∆ i (t) of the cell, which is the surface to feed. This condition states that once a cell grows to the point that its perimeter is no longer sufficient to meet its metabolic needs, it is triggered to begin preparation for cell division (Steven, 1988; Edelstein -Keshet, 1988). More accurately, it is triggered to proceed from the G1 to the S phases of the cycle. Different studies can be found on the surface-to-volume ratio (perimeter-to-area ratio in 2D) or similar concepts (Cottet and Maday, 1981; Kondorsi et al., 2000; Darzynkiewicz et al., 1979). When a cell divides, it creates two daughter cells of area ∆i (t)/2. The division condition is the following: A cell divides itself when: pu (t ) τ i (t ) = i < τ C (11) ∆ i (t ) where τC is the critical division threshold, which depends most notably on the availability of nutriments, in particular those dedicated to cytoskeleton building (tubulin and GTP) and those devoted to cell shape deformation (actin, myosin and ATP), whose critical concentration causes the start of the cdk kinase cycle (Shibaoka and Nagai, 1994; Robinson et al., 2001; Goldbeter, 1991). When the critical threshold τC is crossed, the type of the division is determined according to a law of probability favourable to periclinal division determined at each iteration. More accurately, when a division can occur (see (iii)), a number p is randomly chosen in the interval [0,1] and compared to p c, a critical decision criterion. When (p ≥ p c) a periclinal division can occur; otherwise an anticlinal division occurs. p c is chosen so as to respect the desired number of occurrences of anticlinal divisions to produce the right number of new cells in the cambium at each iteration. p c was determined by a theoretical approach. Let us consider a circular cambium of internal radius R0 made of N identical rectangular (of radial length l and circular width L) synchronous cells. Just after the synchronous divisions, each cell is characterized by τi (t) = 2τC. After an increase in the radial length of each cell of l, τi (t) increases to the value τi (t+∆t)=τC. So N divisions occur. During this growth, the perimeter of the cambium has increased by 2πl (2π(R0 +2l)-2π(R0 +l)). This means that in order to retain the average circular width of the cells equal to L, 2πl/L cells have to be created. In other words, among N divisions, 2πl/L has to be anticlinal, which determines the theoretical value of p c to 2πl/(NL). Barlow used a similar approach to determine this theoretical value (Barlow et al., 2002). This random approach was chosen because the localization of the occurrence of anticlinal divisions is not clear. Fusiform initial cells seem to be induced to divide in a specific way by stimuli that could be related to hormones, but the precise mechanism of control is poorly understood. Studies have been conducted on a possible local determination of these occurrences (Smith, 2001). The geometry of cells and their neighbouring cells could also trigger the occurrence of the anticlinal division. Anticlinal divisions also take part when a cell has reached its upper limit in width (Harris, 1991). This limit is called τL. The complete conditions for a cell Xi (t) of area ∆i (t) and useful perimeter pui (t) to divide at time t are: - for a periclinal division: pui (t ) < τ C and ( p ≥ pC ) τ i (t ) = ∆ i (t )
- for an anticlinal division: pu (t ) τ i (t ) = i < τ C and ( p < pC ) or ( pui (t ) > τ L ) ∆i (t )
(12)
(13)
8
(ii) Additional growth in width: Cambium tissue is ruled by intense cell dynamics. It is thought that initials compete intensely for available space (Kramer, 2002). To take into account the competitive effect between cells in the uuuuuuuuuuur uuuuuuuuuuur process of spreading, a characteristic angle α i (t ) = Ai (t ) Ci (t ),Bi (t )Di (t ) is defined for each cell. If
(
)
this angle is negative, the cell will be said to be dominated (i.e its external perimeter for exchange pui uuuuuuuuuuur uuuuuuuuuuur tend to decrease with time since growth occurs along Ai (t ) Ci ( t) and Bi (t ) Di (t ) vectors). Otherwise it will be dominating. For each iteration we determine the set M(t) of the n most dominated cells (typically we take n=3). M ( t) =
{( X
i1
(t ) ,..., X i ( t ) )/ ∀ j ∈ { i1 ,..., in }, ∀k ∈ {1,..., N ( t )} / {i 1 ,...,i n } , α i ≤ α j }
(14)
n
Configuration of cells according to the α sign.
Reaction and mechanical waves.
Fig. 8 Cellular reaction.
Admitting that cambium cells behave as elastic bodies, we assume that the cells of M(t) react by applying a local low-amplitude mechanic relaxation wave (for example, the points C’i and D’i of Fig. 8 uuuuur uuuuuuur uuuuur Ci Di are chosen so that C 'i D 'i = Ci Di + ). This wave increases the value of the characteristic angle 100 of these cells, which goes from the value α to the value α + dα. Then, if the size of the side of the cell transmitting the wave is bigger than the size of the destination cell, the wave spreads with attenuation to the maximal distance (generally 3-4 cells). This makes it possible to stimulate the trend of the cells to grow more in width when they cannot proliferate enough in length. This means that cells try to maintain the length of their external perimeter for external exchange. Figure 9 shows an initial situation and then the result of the reaction of one of the cells. This figure shows that this reaction enables a very local redistribution of the growth vectors of the concerned cells.
a
b Fig. 9 Reaction wave in a concave zone of cells. (Figure 9a represents a situation before the wave and Fig. 9b the situation after.).
These first two events make it possible to simulate natural tendencies in plant development as stated by Heberle-Bors (Heberle-Bors, 2001): “Plants are multicellular organisms with cells in fixed positions. Thus, not only the regulation of the timing and the number of divisions, but also the orientation of these divisions and the orientation of cell enlargement are critical to elaborate the plant body shape”.
9
4. Appl ications to Pinus radiata The first purpose of our model is to study the evolution over time of a pine tree which has been attacked by a parasite (Rhyacionia buoliana). This parasite is capable of attacking the apex of a young tree by feeding on pine cells: this creates a local concavity inside the cambium and a global deformation of the tree. Figure 10 shows the instance of pine trees affected by substantial deformation (compared with the direction of gravity). These deformations in the trunk cause important impacts on log values, affecting the log conversion due to manufacturing waste (Tsoumis, 1991). These deviations from the typical shape of a tree may be associated with an abnormal tissue called compression wood in conifers, which presents serious problems in lumber production and chemical wood use. These deformations also cause problems related to the transport of the logs extracted from these Fig. 10 Example of deformed tree. Right figure deformed trees. shows deviations from gravity direction (from The tree is able in most cases to recover (Padilla, 2001)). (i.e. regain a more cylindrical global shape) progressively from these deformations according to the scheme in Fig. 11. This figure comes from (Cominetti et al., 2002) and represents the growth of a real tree that has a deformation (and a symmetry plane). This figure was obtained by an experimental study. Different cross-section were collected from the same deformed tree and an interpolation of the different growth rings was made to reconstruct the growth history of the tree. This reconstruction enables to distinguish differential growth, in this deformed case the xylem formation is not uniform along the tree. In such a case, deformations are very important and necessitate several years to be recovered.
Study plane IAA accumulation
a
b
Fig. 11 The recovery process year after year observed in a radial section (arrows in Fig. 11a represent the flux of IAA) (Adapted from (Cominetti et al., 2002)).
The deformation phenomenon can be divided into two parts: recovery of the concavity (i.e. the concavity presented by the cambium becomes less and less discernible and then the cambium regains a circular shape) and global recovery. Contrary to concavity recovery, global recovery requires several
10
years and can be observed on the growth rings. The evolution can be qualitatively explained by IAA distribution. The hormone accumulates in the side of the attack following the effect of gravity (Fig. 11a). This accumulation seems essentially due to relocation of some auxin efflux carriers with gravity, which enable lateral movement of auxin from the upper sides to lower sides (Friml et al., 2002) which produce an accumulation of IAA on the lower sides and on contrary a depletion on upper sides (measured in (Funada, 1990) for inclined stem of Cryptomeria japonica). Some of this accumulation could also be due to a rapid synthesis of IAA by gravitropic stimulation if the speculation that cambial tissue can produce IAA is true (Funada, 1990). This dissymmetry in auxin gravitational drift with respect to the cambium surface induces uncentred growth. The outer perimeters of the growth rings remain circular but their centres shift (Fig. 12) (Cominetti et al., 2002). The difference between recoveries is located in the distribution of IAA along the cambium, and therefore in the choice of the f function. - In the case of concavity, we suppose a slight perturbation of the flux still due to gravity can be simulated by a local low-amplitude sinusoid (for example, typically the amplitude of the perturbation is 0.1, which then decreases with the recovery of concavity) so as to have a more concentrated area of IAA in the centre of the concavity and a slightly less concentrated area on its border.
Fig. 12 Real cross-section showing the pathologic differential growth of the rings.
- The f function for the phenomenon of global recovery considered, as in Fig. 11, will be taken according to the following expression: s π f (s ) = 1 + h (θ ( t )) cos 2π - (15) L(t ) 2 to calculate the heterogeneous symmetric distribution of IAA. h is a function that give the degree of heterogeneity of IAA repartition and that stands for the influence of the angle θ(t), which tends π towards 0 when θ tends towards . The more simple expression of h we can choose is 2 π h (θ (t ) ) = − θ (t ) . 2 We also study a control case of a tree in normal growth: the xylem is formed by circular concentric layers around the medullar zone (Fig. 13): in this case, the distribution of IAA is uniform (f = 1). In particular, we note that two sections of trees can have a similar external aspect and a very different internal evolution (Figs. 12 and 13).
11
Fig. 13 Real cross-section showing the normal concentric growth.
5. Examples of results The current model has been implemented in a Matlab® 6.5 environment, operating with real scales for parameters. However, the diameter of the simulated tree will often be reduced to be able to do more iterations. Typically, a cell will be represented by a rectangle of 50 µm of basic width for a radial length of 25 µm. p c has been chosen equal to 1/N (p c=π/N ≈ 1/N, with ≈ for “of the same order of magnitude”). The initial situation Γ(0) will be given by a class C1 cambium (two exterior frontiers are C1 , i.e. continually differentiable) made of identical cells Xi (0). By convention, the cells of cambium will appear in grey and the zoomed areas will be bordered in black. Normal concentric growth. Here a tree with no deformation is treated: the distribution of IAA is taken as homogeneous (f = 1). We model the formation of xylem during the succession of two different periods of growth. Therefore, we alternate two seasons of identical length: one high-growth season and the other low-growth. For this application, τC and λ are not chosen as constant but as functions of time constant by interval. We use a τC value 2.5 times larger and a value of the sensitivity-to-IAA coefficient 15 times smaller for the period of low growth compared to the high-growth period. Indeed, the increase in the production of ethylene during the low-growth period provokes an increase in τC and therefore the formation of smaller cells (Kondorski et al., 2000). The decrease in λ is due to inhibitory factors against growth as stated above. Fig. 14 shows the results.
a
a
b
1 mm
b
Fig. 14 Simulation of the case of a tree 2 mm in initial internal radius, in normal circular growth.
It is noteworthy that at the start of development the cells are equally distributed and of same dimension, the consequence of having chosen an initial situation where the cells were perfectly in
12
phase. But after a period of development, differences in position and dimension can be noted between cells, which are biologically satisfactory. The global evolution remains concentric around the initial situation, despite the proper development of each cell. The distinction between growth rings was obtained naturally by a change in cellular density, as in the real case. Figures 14a and 14b clearly distinguish three periods of dense xylem formation. Double recovery phenomenon. To simulate this process, the initial situation chosen here presents a sinusoidal function making up a local concavity. The function f is defined according to the discussion in the preceding section. The low concavity implied is recovered quickly by the faster growth of its constitutive cells (Fig. 15b). The tree grows more on the side of the attack, retaining a circular exterior shape (apart for the concave zone) (Fig. 15b, 15d).
b
a c
1 mm d
c
a
d
b Fig. 15 Simulation of the evolution of a tree 2 mm in initial int ernal radius reacting to the parasite attack.
6. Comparison of results with observation The first step in the validation of the model is on the macroscopic level. The results of the model were compared to the growth rings. The simulation of cambium evolution is coherent qualitatively with the succession of the shapes of the rings. The outer perimeter of each growth ring delimits cells that were in the cambium at the same time. This observable synchronization is due to a change in the growth period. Precedent microscopic views (Figs. 3, 6, 10, 12 and 13) and those of this part were obtained directly from the results of Padilla and Martín's experimental study (Cominetti, 2002; Padilla, 2001). Figure 16 shows an example of a part of the xylem in normal wood.
13
to the bark ray cells 0.25 mm
Fig. 16 Microscopic observation of the xylem.
Figures 16 and 17 show ordinaries distributions of xylem cells. Xylem cells naturally tend to be disposed so as to form straight radial files. These uuur dispositions show that cambial cells tend to grow in the direction of the AC uur and BD vectors, because this process produces the new cells with a periclinal division, in the continuity of the existing files. This was assumed in the model rules (equations (9) and (10)). Figure 17 also prove that in first approximation a xylem cell can be modeled by quadrilateral. The cells located in radial lines on figure 16 are xylem ray cells . Figure 16 also shows the proportion between the two species of cells. As in our initial hypothesis (H3), we observed that the fusiform cell is the dominant species. The homogeneity of the dimensions of the cells belonging to the same files and the heterogeneity of those belonging to different files is also striking. This observation confirms that Thom’s criterion for division is well adapted. Because cells that belong to the same file have approximately the same useful perimeter, Thom’s rule implies that the areas of these cells must be similar. The heterogeneity of the different radial files also respects Thom’s rule because a file that has a perimeter of little use is made of small area cells and those of a large perimeter is made of large area cells. Experimental observations also permit to shows the variability of xylem radials files disposition. Figure 17 is an example of ordinary dispositions in straight files and figure 18a shows a deviation in the xylem cells alignment. Our modelling of competition effects can generate these deviations as seen in figure 16b.
a
to the bark
D C
B A
D
C
B A
D
C
B A to the bark
Fig. 17 Example of an ordinary distribution of xylem cells in radial file.
b
Fig. 18 Observations of deviations. Figure 18a shows an experimental deviation and figure 18b is a deviation observed in the result of the modelling. Figure 18a is a zoom of figure 16.
14
7. Discussion and Conclusion The developed model sets up a simulation of the morphogenetic process of the radial growth of conifer trees via the production of xylem. The occurrence method used has an evolution law based on the concentration of a growth hormone, IAA, inside the cambium and two occurrences. The first one is the cellular division that enables production of xylem. The second one, called additional growth in width, models the competition between cambium cells. Although there are some simplifications, our model allows us to simulate the evolution of cambium and the production of xylem by capturing the main characteristics of these processes, the occurrence method being particularly suitable to modelling cell proliferation. Modelling makes it possible to see the microscopic relevant arrangements of the cells to obtain the macroscopic evolution observed and obtain a great number of possible shapes for the cells, which tend to produce the microscopic sections we encountered (Zimmermann and Brown, 1971; Harris, 1991). It provides a first mathematical cellular approach to the phenomena, essential for a better understanding of the radial growth of a tree. An interesting application case for our model could be this of the radial growth of a tree from which a branch has been lopped. In this case, the cut creates a discontinuity in the cambium tissue. After the cut, the cambium is able to recover its continuity forming a concavity. This case is similar to the local attack by the parasite. In general, the created concavity is much greater and can be directly observed in the growth rings (Padilla, 2001; Sommerville, 1980). In this case, a further model have to include a new event “elimination of the cambium” for the most dominated cells. Indeed, in an important concavity this phenomena can not be neglected. We can suggest improvements that would increase the realism of the cellular configurations obtained. It is possible to generalize the model by considering the radial growth in a more realistic way, also taking into account phloem production. This can be done by distinguishing between two subtypes of periclinal division. During a periclinal division, a new law of probability should be implemented, respecting the relative proportion of xylem and phloem for the considered family of trees. As mentioned above, it would also be very important to deal with cambium made of various rows. Furthermore, it could be interesting to be able to treat reaction wood formation and thus to insert a new type of round cell (Bamber and Burley, 1983). Moreover, our work also consists in first modelling a qualitative forecast for more complex situations than the one dealing only with the simple theoretical case of a cylindrical tree. We have generalized the application of the IAA control to asymmetric cases. Other factors that are much less well understood influence the growth of a tree, such as the effect of elastic stresses, the availability of nutriments, and the influence of other hormones (gibberellins, cytokinins, etc.). As Kramer stated, taking into account these other criteria as growth factors would be mathematically performable but premature from a scientific point of view (Kramer, 2001). Indeed, modelling secondary growth of the tree is a new field, still unexp lored. The modelling and analysis explored here takes into account scales and not real quantitative data. A quantitative forecast model of radial growth of the tree still seems difficult to set up currently, because of the great number of influence parameters and above all the lack of understanding of many mechanisms. At the same time, we have done a theoretical study on the three-dimensional transportation of IAA inside cambium subjected to diffusion and transport phenomena. This would greatly improve the accuracy of the model since then we would have an effective numerical expression of f and the possibility to test the hypothesis of the IAA control in a quantitative way. Nevertheless, to approach reality more closely, a reaction–diffusion-transport model should be introduced to calculate the concentration of IAA since there is some evidence that the cambium or adjacent cells can synthesize IAA (Uggla, 1998). This local production of IAA, even if it is minor, might allow adding, beyond the gravitational perturbation, other factors explaining the increase in the growth speed of cells located in a concavity. Indeed, if IAA is produced at a precise moment of the cellular cycle and/or if it is located in the outer part of cells, an increase in IAA concentration inside the concavity would occur, due to a quicker succession of cellular cycles and/or to a smaller distance between the outer parts of the cells favouring diffusion.
15
8. Conclusions To conclude, we have shown by using the spatial repartition of the diffusive morphogen auxin and the cell proliferation how a pure epigenetic process could account for the corrective morphogenesis after a growth abnormality due to a parasitic attack. Other processes like cambial cell migration (along the vertical axis of the trunk under the bark) as well as a possible differentiation to apoptotic cells probably participate also to the cambial growth correcting the initial dysmorphy due to the temporary presence of the parasite. In the future we intend to model complex morphogenetic events like gastrulation, by not using only proliferation as in this paper but also by introducing both migration and differentiation processes in the equations of the dynamical systems involved in the growth.
Acknowledgements We acknowledge the support provided by FONDEF D01I1021. We extend our sincere thanks to Thibaud Hisler for his help in translating the article into English. Special thanks are extended to R. Thom, who died in October 2002. He presented the first version of Thom’s proliferation law ruling cell divisions at the Institut Henri Poincaré (Paris) in October 1972, inspired by L. Bertalanffy (Bertalanffy, 1968). The law appears in his famous book “Structural Stability and Morphogenesis”.
References Aracena, J. (2001). Modèles mathématiques discrets associés à des systèmes biologiques. Ph.D. Dissertation. Univ. Chile. Santiago de Chile. Aracena, J., S. Ben Lamine, M.A. Mermet, O. Cohen and J. Demongeot (2003). Mathematical modelling in genetic networks : relationships between the genetic expression and both chromosomic breakage and positive circuits. IEEE Trans. Systems Man Cyber. 33: 825-834. Bamber, R.K. and J. Burley (1983). The wood properties of Radiata Pine. Commonwealth Agricultural Bureau. Wallingford UK, pp. 84. Barlow, P.W., P. Brain and S.J. Powers (2002). Estimation of directional division frequencies in vascular cambium and in marginal meristematic cells of plants. Cell Proliferation 35: 49-68. Benkherourou, M, P.Y. Gumery, L. Tranqui and P. Tracqui (2000). Quantification and macroscopic modeling of the nonlinear viscoelastic behavior of strained gels with varying fibrin concentrations. IEEE Trans Biomed Eng. 47:1465-1475. Bertalanffy, L.V. (1968). General Systems Theory. Braziller. New York. Butterfield, B.G. (1975). Terminology used for describing the cambium. IAWA Bulletin 1:13-14. Cinquin, O. and J. Demongeot (2002). Positive and negative feedback: mending the ways of sloppy systems. C. R. Acad. Sci. Biologies 325:1085-1095. Clem, C.J, D. Konig and J.P. Rigaut (1997). A three-dimensional dynamic simulation model of epithelial tissue renewal. Anal Quant Cytol Histol 19: 174-84. Cominetti, R., F. Padilla and J. San Martín (2002). Field methodology for reconstruction of a Pinus radiata log. New Zealand Journal of Forestry Science 32: 309-321. Cottet, G.H. and Y. Maday (1981). Hydrodynamic and cell division. Lecture notes in Biomathematics. 49: 227-235. Demongeot, J. , J. Aracena, F. Thuderoz, T.P. Baum and O. Cohen (2003). Genetic regulation networks: circuits, regulons and attractors. C. R. Acad. Sci. Biologies 326: 171-188. Demongeot, J., F. Thuderoz, T.P. Baum, F. Berger and O. Cohen (2003). Bio-array images processing and genetic networks modelling. C. R. Acad. Sci. Biologies 326: 487-500.
16
Demongeot, J., J. Bezy-Wendling, J. Mattes, P. Haigron, N. Glade and J.L. Coatrieux (2003). Multiscale Modeling and Imaging: The Challenges of Biocomplexity. Proc. IEEE 91: 1723-1737. Darzynkiewicz, Z., D.P. Evenson, L. Stainano-Coico, T.K. Sharpless and M.L. Melamed (1979). Correlation between Cell Cycle Duration and RNA Content. Journal of Cell Physiology 100: 425-438. Edelstein-Keshet, L. (1988). Mathematical Models in Biology, McGraw Hill. New York. Estelle, M. (1998). Polar Auxin Transport: New Support for an Old Model. The Plant Cell 10: 17751778. Friml, J., J. Wisniewska, E. Benkova, K. Mendgen and K. Palme (2002). Lateral relocation of auxin efflux regulator PIN3 mediates tropism in Arabidopsis. Nature 415: 806-809. Funada, R. (1990). Distribution of indole -3-acetic acid and compression wood formation. Holzforschung 44: 331-334. Goldbeter, A. (1991). A minimal cascade model for the mitotic oscillator involving cyclin and cdc2 kinase. Proceedings of the National Academy of Sciences of USA 88: 9107-9111. Han, K. (2001). Molecular Biology of Secondary Growth. Journal of Plant Biotechnology 3(2): 45-57. Harris, J.M. (1991). Structure of wood and bark. In: Properties and uses of New Zealand Radiata Pine. Vol. 1. Wood properties. Chapter 2 and 3. (Kininmonth, J. and Whiteside, I. Eds.) Published by New Zealand Ministry of Forestry. Forest Research Institute. Wellington. Heberle -Bors, E. (2001). Cyclin-dependent protein kinases, mitogen-activated protein kinases and the plant cell cycle. Current scie nce 80-2: 225-230. Kondorsi, E., F. Roudier and E. Gendreau (2000). Plant cell-size control: growing by ploidy ? Current Opinion in Plant Biology 3, 488-492. Kramer, E.M. (2001). A Mathematical Model of Auxin-mediated Radial Growth in Trees. Journal of Theoretical Biology 208: 387-397. Kramer, E.M. (2002). A Mathematical Model of Pattern Formation in the Vascular Cambium of Trees. Journal of Theoretical Biology 216: 147-158. Larson, P.R. (1994). The vascular cambium. Development and Structure. Springer-Verlag. Berlin, pp. 725. Leptin, M. (1999). Gastrulation in Drosophila: the logic and the cellular mechanisms. EMBO J. 18: 3187-3192. Lück, J. and H.B. Lück (1985). Comparative Plant Morphogenesis Founded on Map and Stereomap Generating System. In: Dynamical Systems and cellular automata. (Demongeot, J., Goles, E. and Tchuente, M. , eds.). Academic Press. London, pp. 111-121. Myat, M.M. and D.J. Andrew (2000). Organ shape in the Drosophila salivary gland is controlled by regulated sequential internalization of the primordia. Development 127: 679-691. Padilla, F. (2001). Estudio de la deformación del fuste causada por polilla del brote rhyacionia buoliana en Pinus radiata en la décima región. Memoria de Ingenerio forestal. Facultad de Ciencias Forestales. Univ. Chile. Santiago de Chile. Potten, C.S. (1979). Proliferative Cell Populations in Surface Epithelia: Biological Model for Cell Replacement. Lecture notes in Biomathematics 38: 22-35. Prusinkiewicz, P. and A. Lindenmayer (1990). The Algorithmic Beauty of Plants. Springer-Verlag. New York. Roberts, K. (2001). How the Cell Wall Acquired a Cellular Context. Plant Physiology 125: 127-130. Robinson, D.N., G. Cavet, H.M. Warrick and J.A. Spudich (2001). Quantitation of the distribution and flux of myosin-II during cytokinesis. BMC Cell Biology 3(1):4. Shibaoka, H. and R. Nagai (1994). The plant cytoskeleton. Current Opinin in Cell Biology 6: 10-15.
17
Smith, G.S. (2001). Plant cell division: building walls in the right place. Nature Review. Molecular Cell Biology 2: 33-39. Somerville, A. (1980). Resin pockets and related defects of Pinus radiata grown in New Zealand. New Zealand Journal of Forestry Science 10-2: 439-444. Tabony, J., N. Glade, J. Demongeot and C. Papaseit (2002). Biological self-organisation by way of microtubule reaction-diffusion processes. Langmuir 18: 7196-7207. Thom, R. (1972). Stabilité structurelle et morphogenèse. Essai d'une théorie générale des modèles. W. A. Benjamin Inc. New York. Tsoumis, G. (1991). Science and technology of wood. Structure, properties, utilisation. Van Nostrand Reinhold. New York, pp. 494. Uggla C., E. Magel, T. Moritz and B. Sunderg (2001). Function and Dynamics of Auxin and Carbohydrates during Earlywood / Latewood Transition in Scots Pine. Plant Physiology 125: 2029-2039.
18
Appendix 1: Dynamics of the external perimeters of cells Here we detail the discrete dynamics ruling the cells' external perimeter. For each cell Xi , Ci (t+1) and Di(t+1) are determined from the system of equations {(7),(8),(9),(10)}. In this part, index i is not written but results are valid for each cell. The time dependencies are not always written. When they do not appear, the value is evaluated at time t. uuur uuur f s ( Ci ( t )) + f s ( CDi (t )) uuur AC uuur BD Let u AC = uuuuur , uBD = uuuuur , and r = AC BD f s ( Di ( t ) ) + f s ( CDi ( t ) ) uuuuuuuuuuuuuuur uuuuuuuuuuuuuuuru and xC = Ci (t ) Ci (t + 1) , xD = Di (t ) Di (t + 1)
( (
) ( ) (
) )
xC = rxD uuur Equation (8), (9) and (10) could be rewritten: C(t + 1) = C (t ) + xC u AC uuur D(t + 1) = D(t ) + xD u BD Equation (7) is rewritten: Area (C (t ), D(t ), C(t + 1), D (t + 1) ) = ∆(t +1) − ∆(t ) uuuuuuuuuuuuuru uuuuuuuuuuuuuru det C (t )D (t + 1), D(t )C (t + 1) = 2 (∆ (t + 1) − ∆ (t )) uuur uuur uuur uuur which can be written: det CD + x D u BD , DC + xC u AC = 2 ( ∆(t +1) − ∆(t ) )
(
so,
)
(
)
uuur uuur uuur uuur uuur uuur xD det uBD , DC + x D xC det u BD , u AC + xC det CD , u AC = 2 (∆ (t + 1) − ∆ (t ))
(
)
(
)
(
)
Solutions are determined by an approximation of this equation using: uuur uuur uuur uuur uuur uuur uuur uuur xC det uBD ,u AC = det u BD , DC and xD det u BD , u AC = det CD , uAC .
(
)
(
)
(
)
(
)
So, we use: uuur uuur uuur uuur xD det uBD , DC + xC det CD ,u AC ; 2 ( ∆ (t +1) − ∆ (t ) ) uuur uuur uuur uuur As xC , xD > 0 and det uBD , DC ,det CD, uAC < 0 , the final formulae used in the algorithm are:
(
)
(
(
)
) (
2 ( ∆ (t + 1) − ∆ (t ) ) uuur uuur uuur uuur xD ; det u BD , CD + r det DC , u AC xC = rxD
(
)
(
)
)
The value of ∆ (t + 1) − ∆ (t ) is obtained by equation (6).
19
Appendix 2: The dynamic representation
200 µm Fig. 19 Different steps in the concavity recuperation process. The observation steps are regularly chosen.
20
Appendix 3: Spatial location of the event (ii) The location of the occurrence of event (ii) for the local redistribution of the cells' direction of growth is also important. To study these locations, we have represented the state of each cell that has emitted a wave of relaxation in black in Fig. 20. In general, these states are transitory states of the cambium.
b
a 1 mm
Fig. 20 Location of event (ii). (a) is an example for concave growth and (b) for normal growth. (a) and (b) were simulated from the same initial internal ratio and for the same growth time.
Figure 20 shows a case of concave growth and a case of normal growth. For concave growth, the global distribution of the emitting cells is very localized since the emissions take place principally in the concavity. After a while, when concavity begins to be recuperated, the emission of waves progressively takes place in the edges of the concavity (the centre of the concavity is less and less concerned by wave emission) until it leaves this concavity. This important concentration in concavity permits the redistribution of the vector of cell growth, which is necessary to recuperate the concavity. This representation shows that a cell could be so dominated that it has to emit waves during many successive iterations. In normal growth (Fig. 20a), the distribution of the cells emitting waves is more random. However, it must be noted that when a cell emits a wave that stimulates its neighbours , they also emit a wave. Indeed, the total sum of the characteristic angles is a constant equal to 2π. The conservation of this constant implies that the reaction of a cell will often involve the domination of its neighbours. This is why groups of cells emitting waves appear.
21
