Modeling of Branching Patterns in Plants

Transcription

1 Bulletin of Mathematical Biology (2008) 70: DOI /s ORIGINAL ARTICLE Modeling of Branching Patterns in Plants N. Bessonov a, N. Morozova b, V. Volpert c, a Institute of Mechanical Engineering Problems, Saint Petersburg, Russia b Department of Biological Sciences, University of Illinois, Chicago, IL 60607, USA c Camille Jordan Institute of Mathematics, UMR 5208 CNRS, University Lyon 1, Villeurbanne, France Received: 17 February 2007 / Accepted: 3 October 2007 / Published online: 12 February 2008 Society for Mathematical Biology 2008 Abstract A major determinant of plant architecture is the arrangement of branches around the stem, known as phyllotaxis. However, the specific form of branching conditions is not known. Here we discuss this question and suggest a branching model which seems to be in agreement with biological observations. Recently, a number of models connected with the genetic network or molecular biology regulation of the processes of pattern formation appeared. Most of these models consider the plant hormone, auxin, transport and distribution in the apical meristem as the main factors for pattern formation and phyllotaxis. However, all these models do not take into consideration the whole plant morphogenesis, concentrating on the events in the shoot or root apex. On the other hand, other approaches for modeling phyllotaxis, where the whole plant is considered, usually are mostly phenomenological, and due to it, do not describe the details of plant growth and branching mechanism. In this work, we develop a mathematical model and study pattern formation of the whole, though simplified, plant organism where the main physiological factors of plant growth and development are taken into consideration. We model a growing plant as a system of intervals, which we will consider as branches. We assume that the number and location of the branches are not given a priori, but appear and grow according to certain rules, elucidated by the application of mathematical modeling. Four variables are included in our model: concentrations of the plant hormones auxin and cytokinin, proliferation and growth factor, and nutrients we observe a wide variety of plant forms and study more specifically the involvement of each variable in the branching process. Analysis of the numerical simulations shows that the process of pattern formation in plants depends on the interaction of all these variables. While concentrations of auxin and cytokinin determine the appearance of a new bud, its growth is determined by the concentrations of nutrients and proliferation factors. Possible mechanisms of apical domination in the frame of our model are discussed. Keywords Plant growth Branching Numerical simulations Corresponding author. address: volpert@math.univ-lyon1.fr (V. Volpert).

2 Modeling of Branching Patterns in Plants Introduction Theories of phyllotaxis, which can be defined as a construction determined by organs, parts of organs, or primordia of plants (Jean, 1994) appeared already in the seventeenth century. There are several approaches in which plants are considered as dynamic objects, changing their size and form over time based on different growth mechanisms. The best-known mechanism of pattern formation in mathematical biology is related to reaction-diffusion systems and Turing structures (Murray, 2001; Turing, 1952; Meinhardt, 1984). However, the biological evidences that this mechanism is really involved in biological pattern formation are limited (Meinhardt, 2003; Wolpert, 2002; Jean, 1994). Another approach to plant modeling is based on attempts to describe the empirical kinetics of plant growth. Such kinetic equations have been proposed since the early twentieth century (Thompson, 1992) with no significant progress since then (reviewed in Mazliak, 1998). Such models where the final plant size and form are considered as given, are overly simplified and cannot describe details of plant growth and branching mechanism. Among other more specific models, we could mention models using the optimization mechanism; for example, the branching pattern in plants can be related to maximization of light interception (Niklas, 1986); biophysical models, where the main factors for modeling phyllotaxis are tension, compression and shear (Green, 1999); and models based on plant topology and design (Godin and Caraglio, 1998; Boudon et al., 2003). Plant growth and architecture are studied in (Yan et al., 2004) under the assumption that a priori given plant units appear when there are enough nutrients. Some other aspects of plant modeling can be found in the Proceedings of the Workshop on Plant Models (Godin et al., 2004). Recently, a number of models connected with the genetic network and/or molecular biology regulation of the processes of pattern formation appeared. Most of these models consider the plant hormone, auxin, transport and distribution in the apical meristem as the main factor for pattern formation and phyllotaxis (Treml et al., 2005; Reinhardt, 2005; Smith et al., 2006; Jonssonetal.,2006; Fleming, 2005; Blilou et al., 2005; Reinhardtetal.,1998). Auxin flux also regulates the radial growth of branches (Kramer, 2001; Forest and Demongeot, 2006). In this work, we develop a mathematical model and study pattern formation of the whole, though simplified, plant organism based on the main factors of plant growth and development: nutrients supply, plant hormones auxin and cytokinin signaling, and the regulation of cell cycle progression. It is now established that plant hormone auxin is the main factor for the positioning and developing of new buds. The highest auxin level corresponds to the place of initiation of bud formation. Recently, much work has been done to elucidate the details of auxin regulatory activity (Okada et al., 1991; Heisler et al., 2005; Treml et al., 2005; Reinhardt, 2005; Smith et al., 2006; Jonsson et al., 2006; Fleming, 2005; ZhuandDavies,1997; Thingnaes et al., 2003; Sitbon et al., 2000; Delisle, 1937; Went,1944; Benkova et al., 2003; Casimiro et al., 2001; Reinhardtetal.,2000, 2003; Blilou et al., 2005; Vernoux et al., 2000; Galweiler et al., 1998;Aidaetal.,2002; Stieger et al., 2002). Some of the recent studies show that bud development, after its initiation, is related to depletion of auxin in the growing bud, which starts to work like sink of auxin, transporting it to the lower layers or to the phloem (Heisler et al., 2005; Reinhardtetal.,2003; Stieger et al., 2002). The most recent investigations reveal that auxin efflux carrier PIN1 protein plays the central role in this regulation. The expression of this protein is considered as a key factor

3 870 Bessonov et al. for formation of plant organs (Heisler et al., 2005; Treml et al., 2005; Reinhardt, 2005; Smith et al., 2006; Jonssonetal.,2006; Fleming, 2005; Reinhardtetal.,2000, 2003; Blilou et al., 2005; Vernoux et al., 2000; Galweiler et al., 1998; Aidaetal.,2002; Stieger et al., 2002). On the other hand, regulation of the PIN genes is itself under the auxin control (via PLETHORA gene feedback loop Blilou et al., 2005). Therefore, we have an auxin induced auxin efflux which can be modeled with or without the intermediate PIN protein. The fact that auxin works as a trigger of bud formation seems to be the same for leaf and flower bud development. No difference between auxin influence on leaf or flower bud has been reported, except for Zhu and Davies (1997) where it is affirmed that auxin content is lower in the flower bud than in the leaf bud. There are numerous works showing that different genes are expressed in flower and in vegetative bud. Some of these genes are regulated by auxin but this is true for both types of buds (Okada et al., 1991; Heisler et al., 2005; Reinhardt, 2005; Smith et al., 2006; Jonssonetal.,2006; Fleming, 2005; Stirnberg et al., 1999; Rajeevan and Lang, 1993). Therefore, the model developed in this work will be equally applicable for vegetative branches or inflorescences. However, auxin by itself cannot initiate cell proliferation. It can happen only in the presence of cytokinin, another plant hormone (actually a group of hormones), the main role of which is the regulation of cell proliferation. On the other hand, initiation of cell proliferation alone does not lead to leaf initiation (Wyrzykowska et al., 2002). Therefore, it can be an interplay of these two hormones that causes formation of a new bud. Experiments on plant cell culture show that the main factor for the regulation of cell growth, cell differentiation and for the initiation of specific processes of organogenesis is the auxin/cytokinin ratio. Namely, it is shown that the increase of the auxin/cytokinin ratio in callus causes formation of roots and the decrease of this ratio results in shoot formation (Skoog and Miller, 1957; Yamaguchi et al., 2003). It is also known that bud formation can be initiated by auxin in the place of a sufficient level of metabolic activity (Pien et al., 2001; Reinhardt et al., 1998; Cosgrove, 2000). This observation shows that nutrient supply can be important for plant organ formation. This correlates with the observations that the sugar content (mostly sucrose) is elevated in the cells undergoing organogenesis (Borisjuk et al., 2002, 2003; Rosche et al., 2002; Leon and Sheen, 2003; Morozova, 1993; Lorenzetal.,2003). It can be an independent factor but there are a number of works showing that auxin causes the efflux of nutrients to the place with its high content. This concerns sugars (Zhu and Davies, 1997;Went, 1944), phosphates (Lopez-Bucio et al., 2002) and nitrates (Himanen et al., 2002). Summarizing the experimental observations, we can say that phyllotaxis is mainly determined by concentrations of nutrients and metabolites together with auxin as the main regulatory factor and cytokinin as the hormone necessary for proliferation. Cell cycle is regulated by auxin (in cyclind- E2F-RB part) (Traas and Bohn-Courseau, 2005; Magyar et al., 2005; Boucheron et al., 2002; Himanen et al., 2002) and by cytokinin (in cdc2-cyclinb part and in cyclind-e2f-rb part) (Boucheron et al., 2002; Soni et al., 1995). In particular, it was reported that a possible mechanism of auxin-mediated cell cycle activation could be realized by the splitting of the E2F-RB complex by auxin which positively regulates the passage of the G1-to-S restriction point (Traas and Bohn-Courseau, 2005; Magyar et al., 2005). Cytokinin influences the cyclinb-cdc2 complex and positively regulates G2-M transition (Boucheron et al., 2002; Soni et al., 1995).

4 Modeling of Branching Patterns in Plants 871 In this work, we continue the modeling of plants started in Bessonov and Volpert (2003, 2006). We can now specify more precisely the biological meaning of growth and mitosis factors introduced in the previous works. The role of cell cycle and its interaction with plant hormones are better understood. This allows us to refine the model. 2. From biological observations to a mathematical model We will use the biological observations and experimental results to develop a mathematical model of plant growth. It will take into account the interaction of cell cycle and fluxes of nutrients and metabolites. We consider the entire plant though with many simplifications, in particular without taking into account root growth, photosynthesis and plant leaves, but trying to describe the most essential features of growth mechanisms. We suppose that the apical meristem, which consists only of several cell layers, is much smaller than the whole plant. Therefore, we can consider it as a mathematical surface and describe plant growth as a free boundary problem. The speed of the free boundary corresponds to proliferation rate. We begin with the basic model of plant growth considered in this work. It is a 1D model without branching where the growing plant is an interval whose length depends on time. The growth rate depends on the concentration of nutrients and on the concentration of a special substance, growth and mitosis factor. We have the following model for the axial shoot growth: C + u C t x = d 2 C x, (1) 2 h dr = F(A,K)g(R)C σr, (2) dt where 0 x L(t), L(t) is the shoot length, C is the concentration of nutrients coming from the root, u is the convective speed of nutrients determined from the continuity equation for the incompressible fluid and related to the speed of growth of the interval, u = dl dt. (3) The concentration of growth and mitosis factor (GMF or GM-factor) R is defined at the growing end x = L(t) which corresponds to the apical meristem. Equation (1) describes diffusive and convective transport of nutrients through the plant, Eq. (2) describes the production and consumption or destruction of the GMF in the apical meristem. Here h is a parameter (the width of the meristem), the function F(A,K) describes the dependence of the rate of the production of the GMF on plant hormones auxin A and cytokinin K. The plant growth rate depends on R, dl dt = f(r). (4) The specific forms of the functions f(r)and g(r) are described below.

5 872 Bessonov et al. System (1)and(2) should be completed by initial conditions L t=0 = L 0, R t=0 = R 0, C t=0 = C 0 (5) and boundary conditions for C at x = 0 (supply of nutrients) and at x = L(t) (flux of nutrients to the meristem): x = 0 : C = C 0, x = L(t) : d C x = g(r)c. (6) The second boundary condition shows that the flux of nutrients from the main body of the plant to the meristem is proportional to the concentration C(L,t). This is a conventional relation for mass exchange at the boundary, Robin boundary conditions. The factor g(r) shows that this flux can be regulated by proliferating cells. We discuss this assumption in more detail in Section 3.1. In the 1D model without branching, the concentrations A and K are constant. A more complete model, where we take into account the production of the hormones and their distribution will be considered in Section 4. Growth and mitosis factor is a generic name for a number of biochemical products related to cell cycle. Its production is self-accelerating, which determines the specific form of the function g(r). At the first part of the cycle (G1-S), R can be related to the concentration of transcription factor E2F, which determines (together with RB protein) the transition from the G 1 phase to the S phase, enabling the consequent cell proliferation. At the second part of the cycle (G2-M), R is related to the complex cyclinb-cdc2, controlling the process of mitosis. Since the production of cyclinb-cdc2 complex is self-accelerating through the activation by cdc25 (Hoffmann et al., 1993), we can consider that production of R is self-accelerating. The cell cycle can be influenced by auxin in its first part and by cytokinin in the second. We take this into account in Eq. (2). The rate of production of the GM-factor depends also on the concentration of nutrients, which can be phosphorus and nitrogen salts, supplied from the roots or sugars (mostly sucrose) produced in the process of photosynthesis in all green parts of the plant. We will not specify here the origin of nutrients coming to the apical meristem from lower parts of the plant and will denote their concentration by C. The concentration of the GM-factor determines the proliferation rate and, consequently, the rate of growth. We consider cell proliferation and subsequent cell elongation together, not separating them from each other. For the sake of simplicity, we assume that it is zero if its concentration is small, and it is a positive constant if it is large. These assumptions determine the form of the function f(r). We should consider that the process of plant growth strongly depends on the plant hormones, mainly on auxin, which is also involved in the regulation of plant organ formation, branching pattern, cell cycle progression, and many other processes. We will begin with a simpler case where the dependence of cell cycle on the plant hormones is not taken into account. It will be considered in the more complete 1D model with branching where hormone production and transport are taken into account. In the 1D model with branching, we introduce concentrations of plant hormones to describe more precisely the branching mechanism. We need to specify conditions of the appearance of new buds and of their growth into new branches. The appearance of a new bud can be related to the transition from the G 0 phase of cell cycle to the G 1 phase

6 Modeling of Branching Patterns in Plants 873 (initiation of cell proliferation) and next to cell differentiation resulting in the formation and growing of new branch. We do not take into account a limited cell proliferation during bud formation; it can be related to consumption of a small amount of the GM-factor which is not self-renewed and stops proliferation. This transition can be determined by auxin and cytokinin. We will denote their concentrations by A and K, respectively. We suppose that all cells along the stem, except cells belonging to the apical meristem, are originally in the G 0 phase. If some of them undergo the transition to the G 1 phase and a subsequent differentiation, then we interpret them as a new bud. We need to specify the conditions on A and K for this transition. According to the experimental observations mentioned above, we can assume that the ratio of hormones (auxin/cytokinin) could determine the appearance and growth of a new bud. However, mathematical analysis of this problem and numerical simulations show that in this case, new buds would be not isolated, but would cover some intervals of the stem in a continuous manner. We come to the condition A = A 0, K = K 0,whereA 0 and K 0 are some given concentrations of the hormones that can be specific for each plant species. Moreover, bud formation can be accompanied by an additional production of A and K because of a possible cell proliferation. When a new bud is formed, it can stay in a dormant state or break up and give a new branch. The growth of the branch, which is related to cell proliferation, is determined by the concentration of the GM-factor at its apical meristem. The production of the GMfactor can be also influenced by A and K. Thus, auxin and cytokinin interfere twice in the 1D model with branching: in the condition of appearance of new buds and in the production of the GM-factor. 3. 1D model without branching 3.1. Model We consider in this section the 1D model without branching given by problem (1) (6) with F(A,K) 1. The one-dimensional model is justified if the length (or height) L of the plant is essentially greater than the diameter of its trunk. Hence, we consider the interval 0 x L(t) with the length depending on time. The left endpoint x = 0 corresponds to the root. Its role is to provide the flux of nutrients taken into account through the boundary condition. We do not model the root growth here. Therefore, the left boundary is fixed. The right endpoint, x = L(t) corresponds to the apex. Its width is much less than that of the plant. We suppose in the model that it is a mathematical point. The value L(t) increases over time. According to the assumption above, the growth rate is determined by the concentration of metabolites at x = L(t), Eq.(4). The function f(r)will be specified below. We recall that the interval 0 <x<l(t)corresponds to differentiated cells that conduct nutrients from the root to the apex. We suppose that they are in a liquid solution. Denote by C their concentration, which is a function of x and t. Its evolution is described by the diffusion advection equation (1). System of Eqs. (1) (6) is a generic one-dimensional model of plant growth based on: (a) continuous medium assumptions of mass conservation (for C + R) and of the proportionality of the flux C/ x at the boundary to the value of C; and (b) a biological

7 874 Bessonov et al. Fig. 1 Functions f, g, andf 1 (Section 4). assumption that there is a chemical species R, the plant growth and mitosis factor, which is produced in the meristem and which determines the plant growth. We note that the conservation of mass in the case σ = 0 implies that the term g(r)c enters both the boundary condition and Eq. (2). Therefore, the assumption that the rate of the plant growth factor production depends on its concentration R makes the boundary condition depend on it also. We will see below that properties of the function g can be crucial for plant growth. In particular, if it is constant (the production rate is not autocatalytic), we will not be able to describe the essential difference in plant sizes. We now specify the form of the functions f and g. We will consider f as a piecewise constant function equal to 0 if R is less than a critical value R f and equal to some positive constant f 0 if R is greater than R f (Fig. 1a). This means that the growth begins if the concentration of the plant growth factor exceeds some critical value. The production of the growth factor R is assumed to be auto-catalytic. To simplify the modeling, we consider a piece-wise linear function g(r) (Fig. 1b). In some cases, we also consider smooth functions f and g. These assumptions are consistent with plant morphogenesis. It is well known, for example, that auxin, produced in the apex, stimulates mitosis and cell proliferation. Kinetin is also known to stimulate cell proliferation. Production of mitosis factors can be selfaccelerating through cyclinb-cdc2 complex and activation by cdc25 (Hoffmann et al., 1993). For large concentrations, cyclin can also inhibit its own production (Alberts et al., 1995, p. 876) Stationary solutions In this section, we study stationary solutions of the model described in the previous section. Since f(r)= 0 in this case, we obtain from (1) Then from (6) and(2) C(x) = C 0 C 0 C(L) L C(L) = C 0 σl d R. x.

8 Modeling of Branching Patterns in Plants 875 Finally from (2) σr C 0 σl d R = g(r). (7) This equation should be completed by the condition R<R f (8) such that L (t) = 0. We assume in what follows that σ<g (0). Then for all L sufficiently large, there exists a solution R of Eq. (7) with condition (8). One of the solutions is always R = 0. Depending on the function g(r), there can also exist, solutions with R>0. Denote by Φ(R) the left-hand side in (7). The standard linear stability analysis shows that the stationary solution is stable if Φ (R) > g (R) for a solution R Numerical simulations The functions f and g are characterized by two critical values: the length L(t) increases if R>R f, and the production of R is strongly accelerated if R>R g. The behavior of the system is different in two cases, R f >R g and R f <R g. All simulations are carried out for d = su 2 /tu, σ = su/tu.heresu is a space unit, and tu is a time unit. We will vary h and the initial length L 0. We briefly describe the numerical method for the 1D model without branching. The diffusion-advection equation is solved by an implicit finite difference method on a fixed grid except for the moving boundary which is considered as one of the grid points. The resolution of the equation for R is standard though it requires high accuracy because of the exponentially small values of R in the beginning of the rest periods. The model with branching demands the development of the special numerical algorithm since the computation of the diffusion-advection equation is carried out on a number of interconnected intervals whose structure is not a priori given. The details of the method are presented in Bessonov and Volpert (2006) Linear growth If R f >R g, then the length increase is close to a linear function of time. It reaches its stationary value, and then does not change (Fig. 2). The final length L f weakly depends on h and L 0.ForL 0 = 0.1 andh from to 0.05, L f changes from 2.54 to If h = and L 0 increases from 0.05 to 0.5, the final length decreases from L f = 2.56 to L f = The value of R g assumed in the simulations is The concentration R is monotonically decreasing over time, approaching its final value Therefore, the results of the simulation remain the same if the function g is identically constant (g 0.01) Periodic growth The behavior of the solution to problem (1) (6) is different if R f <R g. In this case, the growth is periodic in time (Fig. 3a). Short periods of growth are separated by long time intervals where the length does not change. The length change is approximately the same

9 876 Bessonov et al. Fig. 2 Linear growth. Fig. 3 Periodic growth. during all periods of growth except for the first one, where it is about two times greater. The periods without growth become longer over time. This is related to the growing length of the interval. For larger L, it takes more time for the concentration C(L,t) to become large enough for R(t) to increase. Fig. 3b shows the function R(t). Contrary to the previous case, the final length L f is very sensitive to the value of h (Fig. 4). For h = L f = 14.56, for h = 0.003, L f = The number of periods of growth also varies with h. If for two different values of h, the number of periods of growth is the same, then the final length depends on h weakly. The dependence of the final length on the initial length remains weak. For h = 0.003, as L 0 changes from 0.1 to1.0, L f changes from 4.19 to We recall that the first boundary condition in (6) determines the amount of nutrient available for the plant. The value of the concentration at the left endpoint influences the

10 Modeling of Branching Patterns in Plants 877 Fig. 4 Dependence of the final length on h for different C 0. number of growth intervals and the final plant length. If we decrease the boundary condition, the length also decreases (Fig. 4). 4. 1D model with branching 4.1. Formulation of the model In this section, we discuss conditions of appearance of branches in the framework of the one-dimensional model of plant growth. We use the experimental observation on shoot and root growth from callus: if the concentrations of two hormones, auxin and cytokinin (which we denote by A and K, respectively) are in a certain proportion, then shoots will appear. For a different proportion, not shoots but roots will grow (Heisler et al., 2005; Tremletal.,2005). Hormone A is produced in growing parts of the plant (leaves, shoots); hormone K in either roots or in growing parts. In our model, A will be produced at the moving boundary x = L that corresponds to the apical meristem. The rate of production is proportional to the growth rate. Hormone K will either be supplied solely through the stationary end of the interval x = 0 (the root) or will also be produced at the moving boundary. The concentrations of nutrients C, and of hormones A and K are described by the diffusion equations with convective terms: C t K t + V C x = d C V K K x = d K 2 C γc, (9) x2 2 K x2 μk, (10)

11 878 Bessonov et al. A t A V A x = d 2 A A νa. (11) x2 The convective speed V in the first equation is determined as the speed of growth: dl = V, V = f(r). (12) dt Here d C, d K, d A and γ, μ, ν are parameters; the space variable x is defined independently for each branch. The convective speeds V A and V K in Eqs. (10) and(11) can be different in comparison with Eq. (9). It corresponds to transport in the phloem in the direction from top (meristem) to bottom (root). The last term in the right-hand side of Eq. (9) describes a possible consumption of nutrients along the stem. This assumption is biologically relevant. We did not consider it in previous section in order to simplify the basic model. The rate of production of the GM-factor at x = L(t) is given by the equation h dr = F(A,K)g(R)C σr. (13) dt Here F(A,K)= F 1 (A)F 2 (K), ka, 0 A 1/k, 1, 1/k A A F 1 (A) = 0 1/k, k(a 0 A), A 0 1/k A A 0, 0, A 0 A (see Fig. 1c), and the function F 2 (K) is defined similarly. The form of the functions F 1 (A) and F 2 (K) is chosen in accordance with the biological observations that there are optimal concentrations of plant hormones. Proliferation of plant cells can decelerate if the concentrations are too low or too high. In particular, it is the case of auxin playing an important role in apical domination. The initial and boundary conditions for L, R and C are the same as for the 1D model without branching (Section 2), the concentrations A and K are initially zero. The boundary conditions for K describe its possible production in the root, and its production in the meristem with rate proportional to the rate of growth: K x=0 = K 0, d K K x = ε K V. (14) x=l Finally, the boundary conditions for A are similar, except that the boundary condition at x = 0 takes into account that this hormone can be transported from the stem to the root: A x βa A = 0, d A x=0 x = ε A V. (15) x=l We define next the branching conditions. A new branch appears at x = x 0 and t = t 0 if A(x 0,t 0 ) = A b, K(x 0,t 0 ) = K b, (16) where A b and K b are some given values. Appearance of a new branch means that there is an additional interval connected to the previous one at its point x 0.Thevariables

12 Modeling of Branching Patterns in Plants 879 C n,a n,k n and R n are described at the new interval by the same equations as above. Here the subscript n determines the number of the branch. We should complete the formulation by the initial value of the concentration R n = R n (t 0 ). It cannot be found as a solution of the problem but should be considered as a parameter. Formation of a new branch is accompanied by production of A and K. It is not describedbyeqs.(10) and(11). It is an additional source term localized in the vicinity of each point x 0 where condition (16) is satisfied during a short time after t = t 0. Therefore, condition (16) will not be satisfied in the vicinity of the new branch any longer, and the set of branch will be discrete. To complete the model, we impose additional conditions at the branching points: continuity of concentrations and conservation of fluxes. The branching conditions are formulated above in terms of plant hormones, auxin and cytokinin. In reality, the role of these hormones is essentially more complicated and not completely understood. In particular, it is known that auxin plays an important role in formation of plant organs and in plant diversity. We recall that the local growth rate of the apical meristem is determined by the function f(r). It is characterized by two parameters R f and f 0. The first one determines the critical value of the GM-factor for which the plant growth begins. The second one determines the rate of growth. Both of them can influence the growth pattern. In the 1D model without branching, the value of R f determines the mode of growth, oscillating or linear. In the 2D model, the value of f 0 determines the plant form Bessonov and Volpert (2006). Based on these considerations, we can suppose that R f and f 0 maydependontheauxin concentration. Growth conditions and growth rate may also depend on some other metabolites, in particular, on sugars produced by plant leaves and transported to the apical meristem. Since we do not consider leaves in the model, we can suppose that they are produced inside all existing branches with a constant rate at each unit length. The equation for the sugar concentration C s becomes C s t C s + V s x = d 2 C s s x + a s, (17) 2 where V s is the speed of the convective transport of sugars, d s is the diffusion coefficient, a s is the rate of production per unit length. The boundary conditions should take into account that sugars are consumed in growing plant organs with the rate proportional to the growth rate. In our case, sugars are consumed in the apical meristem, C s x = ε s V. (18) x=l We should specify how the concentration of sugars can influence the growth rate. This influence can be realized directly through the function f and indirectly through the production of the GM-factor, that is through the function g. Thus, the function f can be considered in the form f(r,a,c s ) = { 0, 0 R<Rf (A), f 0 (C s ), R f (A) R, where R f (A) is a decreasing function of A (auxin acts to stimulate growth), f 0 (C s ) is increasing as a function of C s being equal 0 for small values of C s and some constant

13 880 Bessonov et al. for large values (no growth for small concentrations of sugars). In this work, we restrict ourselves to the case where R f (A) and f 0 (C s ) are constants Results A typical example of plant growth in the 1D model with branching is shown in Fig. 5. In the beginning of the evolution, there is a single branch which grows with an approximately constant speed until the growth period is finished. The apical meristem is located at the upper end of the interval. The plant hormones auxin and cytokinin are produced there and are transported along the whole branch. It can be diffusive or convective transport. Cytokinin can also be produced in the roots. This is taken into account through the boundary condition at x = 0. If at some point of the branch the concentrations of auxin and of cytokinin take on some prescribed values, then a new bud appears. In the simulation shown in Fig. 5, there are five buds that appear one after another at an approximately equal distance. Each of the buds contains its own apical meristem with some value of the GM-factor R. Whena new bud appears, the initial value R 0 is prescribed. It can be some given constant or it can depend on some factors (on plant hormones or on the value of R in the apical meristem of the main growing branch). After that, the value of R i in the bud evolves according to thesameequation(see(13)). When R i becomes larger than R f (see (12) and Fig. 1), proliferation begins and a new branch starts growing from the bud. As it is shown in the Section (Fig. 3), growth of branches can be stepwise, that is, a branch has periods of growth and of rest during which the speed of its growth equals zero. When the main branch stops growing, under appropriate conditions, a new branch can appear from another bud. As it is discussed above, we consider diffusive and convective fluxes of nutrients. Diffusive flux acts throughout all branches, whereas convective flux is directed to growing branches. This is related to the continuity equation for the incompressible fluid. The routes of convective flux of nutrients is shown with a dashed line in Fig. 5. Thus, initiation of growth of new branches is determined by the interplay between the concentrations of nutrients and of the growth factor. In the example, presented in Fig. 5, there are five buds formed on the main branch. Three of them give branches of the second generation with three new buds on each of them. Branches of the second generation appear one after another when the main branch stops growing. It is interesting to note that one of the buds on the main branch gives a rudimentary branch which stops growing right after it appears. There are also some branches of the third generation. The total plant length as a function of time, that is, the sum of the lengths of its branches at each moment of time, is shown in Fig. 6. It has specific stepwise form similar to that in the 1D case without branching. In the case with branching, each step corresponds to a new branch. This function is different in the case h = shown in Fig. 7. The branches appear pairwise resulting in the alternation of short and long steps. Similar to the 1D case without branching, the final plant size decreases for larger h.in the case with branching, the final length is determined basically by the number of branches and not by their length (Fig. 8). There are four generations of branches for h = , threeofthemforh = and h = 0.001, and only two generations for greater values of h. A possible explanation of the influence of h on the final length from the point of view of nonlinear dynamics is given in the Discussion.

14 Modeling of Branching Patterns in Plants 881 Fig. 5 Time evolution of the plant. The figure shows the moments in time when new buds or branches appear. Convective flux of nutrients is shown with a dashed line.

15 882 Bessonov et al. Fig. 6 Total length as a function of time, h = Fig. 7 Total length as a function of time, h = Fig. 8 Final plant forms for different values of h. Plant evolution in time can be influenced by the initial value of the GM-factor concentration in a new bud. We have described it in the case where R 0 = 0.12 (Fig. 5). In fact, it is the same for all values of R 0 between 0.01 and Further increase of this parameter changes the plant evolution (Fig. 9). When the first bud appears, it does not stay dormant but gives a new branch right away. It grows at the same time with the main branch. The difference between the two cases is determined by the behavior of solutions of Eq. (13). As we have already discussed, when a new bud appears, we prescribe it an initial value R 0 of the GM-factor concentration. We recall that the concentration of nutrients C is a func-

16 Modeling of Branching Patterns in Plants 883 Fig. 9 Evolution of the plant structure in time, h = , R 0 = Increase of R 0 makes the first bud to break up and changes completely the plant structure. tion of space and time. Its value C(x 0,t 0 ) at the new bud determines the right-hand side of this equation: it equals zero at R = 0 and at two positive values of R, it is negative between first two zeros and positive between the second two. If the initial value R 0 is at the interval of negativity, then the solution rapidly decreases, and the bud remains dormant. When the main branch stops growing, the concentration of nutrients increases. When its value C(x 0,t) at the bud approaches 1, the right-hand side of Eq. (13) becomes positive for small positive values of R. The concentration of the GM-factor starts growing. After some time, it can reach the critical value which is necessary for the new branch to grow. However, it may happen that another branch will start growing before this one. Then the concentration of nutrient can drop down again, and the concentration of the GM-factor may also decrease. The situation shown in Fig. 9 is different. The initial concentration R 0 of the GMfactor appears to be at the interval of positivity of the right-hand side of Eq. (13). Its solution grows, and there is a sufficient amount of the GM-factor to initiate and to support growth of the new branch. Thus, formation of new buds and growth of new branches is determined by a complex interaction of plant hormones, nutrients and mitotic factors. We will return to this question in the Discussion below. Figures 10 and 11 show final plant forms for different values of parameters. We recall that V A is the velocity of convective motion of auxin from above to below. It can be related to the polar auxin transport. In all previous simulations this value was zero. A small increase of this parameter leaves the plant structure approximately the same though two branches are not developed in the case V A = 10 4 in comparison with V A = 0. For larger values of this parameter, the plant structure becomes very different. There are less buds on the main branch because the auxin flux prevents the realization of the conditions for new buds to appear.

17 884 Bessonov et al. Fig. 10 Final plant forms for different values of V A. Fig. 11 Final plant forms for different values of Ad. Appearance of new buds is accompanied by production of some additional quantity of auxin and of cytokinin characterized by the parameters A d and K d. Figure 12 shows the final plant forms for different values of A d. Even small changes of this parameter can essentially influence the plant evolution and final forms. Finally, we show some examples of the simulations where R 0 in new buds is not prescribed (Fig. 12). It takes the same values as the value of R at this moment of time in the apical meristem of the branch where the bud appears.

18 Modeling of Branching Patterns in Plants 885 Fig. 12 1D model with branching. Branching patterns for different values of parameters in the branching conditions. 5. Discussion 5.1. Nonlinear dynamics The typical behavior of the evolution problem in the 1D case consists in the alternation of growth periods during which the interface moves (i.e., the length of the interval grows), and of rest periods where it is unmovable. After several such cycles, the solution converges to a stationary solution. This stationary solution is stable and the oscillations occur far from it. So the appearance of the oscillations is not related to a Hopf bifurcation, in which a stationary solution loses its stability, resulting in the bifurcation of a limit cycle near it. To explain this dynamic, we should remark that the problem possesses two continuous families of stationary solutions. We will denote them by v s (L) and v u (L). Here the subscript s corresponds to stable solutions, u to unstable solutions and L to the length of the interval; in both cases solutions exist for all L (L should be sufficiently large in the case of stable solutions). Stationary solutions are in fact couples (C(x), R),whereC(x) is the concentration distribution in the interval [0,L], and R is the value of the concentration of the growth and mitosis factor at x = L. For stable stationary solutions,c(x) is a linear function and R is some positive constant for unstable solutions; C(x) is constant, R = 0. Stability of stationary solutions is determined by the stability of R considered as a stationary solution of the equation h dr dt = Cg(R) σr. (19) If C = C(L) in this equation is fixed, then it can have from one to three stationary solutions (for the functions g(r) considered in this work). For small C, the right-hand side of this equation is negative, and R = 0 is its stable stationary solution, for C close to 1, it is unstable. These properties of the problem explain why solutions v u (L) are unstable or, more precisely, have stable and unstable manifolds. In the evolution problem, there is a coupling between the concentration distribution inside the interval, the value R, and the growth of the interval length. The value of C at x = L is a function of time. During growth periods,

19 886 Bessonov et al. Fig. 13 Schematic representation of solutions of the 1D evolution problem. it becomes small because nutrients are consumed. Therefore, R = 0 becomes stable with respect to Eq. (19). The solution of the evolution problem approaches one of the stationary solutions from v u (L). During rest periods, the concentration C(x,t) gradually increases, approaching 1 everywhere inside the interval. When C(L,t) becomes sufficiently large, R = 0 becomes unstable with respect to Eq. (19). The value of R grows, and when it is sufficiently large, a new growth period begins. This is the mechanism of oscillations in the 1D problem without branching. It is shown schematically in Fig. 13 with two families of stationary solutions and a trajectory corresponding to the solution of the evolution problem. In the 1D case with branching, the qualitative behavior is the same. It becomes more complex because of the interaction of several growing branches Transport of nutrients and growth modes In spite of the complexity of the growth mechanisms, it becomes clear that growth pattern is basically determined by the interaction of nutrients with plant hormones and growth factors. Already in the 1D model without branching, we observe time oscillations resulting from the interaction of nutrients with the GM-factor. Nutrients are consumed during the period of growth, their concentration in the apical meristem drops down, production of the GM-factor stops, and after some time growth also stops. During the period of rest, nutrients are accumulated, their concentration in the apical meristem increases, production of the GM-factor starts again. After some time, it results in a new growth period. This mechanism of oscillations can be related to endogenous rhythms in plants. The number of growth periods determines the final plant size. It depends on the values of parameters that can be related to the plant genotype. Numerical simulations show that the final plant length in the 1D model without branching increases when the width of the apical meristem, considered as a parameter decreases.

20 Modeling of Branching Patterns in Plants 887 If both, the width of the apical meristem and the final length are determined by plant genome, that is, they are specific for each plant species, we can put the question about correlation of these two quantities in real plants. We are not aware about detailed biological investigation of this question. Moreover, we do not know at the moment whether we should compare the entire plants or single branches, or within plant genders. Here are some examples available in the biological literature (the characteristic size of the meristem is indicated) (see, e.g., Clowes, 1961): Cycas revoluta (sago-palm tree) mkm, Picea (fur tree) and Pinus (pine) mkm, Lupinus albus (lupine) mkm, Zea mays (maize) 130 mkm. The comparison between the first two lines (trees) and between the second two lines (herbs) shows that the plant size is bigger if the meristem is smaller. 1D model with branching shows time oscillations and spatial branching pattern. New branches appear at two stages. At the first stage, there are new buds and then new branches growing from them. Conditions of appearance of new buds are formulated in terms of concentration of auxin and cytokinin. It should be emphasized that it is not their ratio, as it is sometimes suggested in the biological literature, but the concentration of each hormone that should take some given value. A new bud gives a new branch if the concentration of the GM-factor at the bud is sufficiently high to start cell proliferation. The GM-factor is related to proteins responsible for cell cycle. Its production depends on the concentration of nutrients and can be influenced by plant hormones. Transport of nutrient considered in the model can be diffusive and convective. A flux of nutrients comes to the growing branch because of the continuity of the fluid (1D model) and pressure difference (2D model). When growth stops, convective flux disappears, accumulation and redistribution of nutrients occur due to diffusion. It may be not exactly the same in real plants: Water evaporation through plant leaves can provide convective flux independent of plant growth Auxin transport and apical domination The role of auxin in formation of plant organs is largely discussed in the recent biological literature (see the Introduction). It is now established that auxin is produced in the growing (proliferating) parts of the plant and transported from them to other plant tissues; auxin concentration for bud formation is less than for proliferation; and, finally, an excess of auxin can stop proliferation (Clowes, 1961; Lyndon, 1998). These experimental observations are important to understand and to model plant development. However, there is still some uncertainty about the role and the mechanisms of auxin signaling. For example, some biological observations show that auxin concentration is greater in the proliferating parts of the plant tissue but there are also some others (Skoog, 1954) showing that auxin, except for very low concentrations, inhibits bud formation and bud development. The idea that the places of new bud formation correlate with higher auxin level (Zhu and Davies, 1997;Went,1944;Reinhardtetal.,2000, 2003; Blilou et al., 2005; Vernoux et al., 2000; Galweiler et al., 1998; Aidaetal.,2002; Stieger et al., 2002; Himanen et al., 2002) exists together with the observation that auxin concentration in these places is lower (Clowes, 1961; Lyndon, 1998; Skoog, 1954; Heisler et al., 2005).

21 888 Bessonov et al. The discovery of the families of PIN and of AUX proteins responsible for active auxin transport allowed a possible explanation of the mechanism of plant organ formation based on auxin transport and on its nonuniform distribution (Treml et al., 2005; Reinhardt, 2005; Smith et al., 2006; Jonsson et al., 2006; de Reuille et al., 2006; Jürgens and Geldner, 2002). This approach is mostly applied for the explanation of the formation of primordia within the apical meristem but there is a number of other questions about auxin involvement in the processes of plant differentiation and growth. Nonuniform auxin distribution inside plant organs can be at the origin of some contradictions in the experimental observations mentioned above. One of the important processes in which auxin is involved is apical domination suppression of growth of lateral branches by the apical meristem of the main branch. As it is shown in many investigations, auxin plays a major role in this process (see Lyndon, 1998 and the references therein). In particular, it is shown that its concentration is greater in the proliferating parts of the plant. The main place where auxin is synthesized is the apical meristem from which it is transported to other plant tissues (Clowes, 1961; Lyndon, 1998). The release from apical domination (e.g., after cutting off the apex of the main branch) is correlated with a decrease of the auxin level in lower parts of the plant. This can initiate growth of lateral branches. Application of the external auxin to the place of cut apex can stop again growth of the lateral branches. Obviously, this mechanism should be interconnected with the role of auxin in the process of bud formation. The main current paradigm is that auxin produced in the apical meristem of the main branch prevents growth of new branches from lateral buds. But is it really possible? Suppose that this is the case. Therefore, if its concentration in the lateral bud is too high, such that it prevents growth of the new branch, then its concentration in the apical meristem is even higher and should prevent growth of the main branch. This contradicts the assumption about growth of the main branch. Thus, we come to the conclusion that auxin produced in the apical meristem cannot prevent growth of new branches unless there are some other factors that should be taken into account. We will discuss them below. In the framework of the model developed in this work, apical domination could be explained on the basis of cell cycle regulation connected to nutrient supply. There are some biological observations confirming that the mechanism of apical domination is rather related to the interaction of the GM-factor with nutrients than to auxin concentration. For example, transition to flowering (evocation) can result in the development of axillary buds indicating a release from apical domination. But auxin production or its level does not seem to be essentially changed during evocation. The most critically important characteristics of the apex during evocation is an increased growth rate and a shortened cell cycle (Clowes, 1961). An important role of the flux of nutrients on the interconnection of plant organs is confirmed by the fact that removal of lateral branches stimulates growth of fruits. Another observation is that while the application of the external auxin to the apical meristem can stop growth of lateral branches, the application of external cytokinin to the dormant buds can initiate growth of new branches though the auxin level in the buds does not change. Both effects related to application of the plant hormones can be simulated in the framework of the modeling presented above. We will now discuss in more detail possible mechanisms underlying apical domination from the point of view of the model suggested in this work. The model is based on