TREEPH_030_10toc.qxd:Layout 1 - Oxford Academic

Tree Physiology 30, 1235–1252 doi:10.1093/treephys/tpq068

A physiological model of softwood cambial growth TEEMU HÖLTTÄ,1,2 HARRI MÄKINEN,3 PEKKA NÖJD,3 ANNIKKI MÄKELÄ1 and EERO NIKINMAA1 1

Department of Forest Sciences, University of Helsinki, PO Box 27, 00014 Helsinki, Finland

2

Corresponding author ([email protected])

3

Vantaa Research Unit, Finnish Forest Research Institute, PO Box 18, FI-01301 Vantaa, Finland

Received January 8, 2010; accepted June 27, 2010; published online July 26, 2010; handling Editor David Whitehead

Summary Cambial growth was modelled as a function of detailed levelled physiological processes for cell enlargement and water and sugar transport to the cambium. Cambial growth was described at the cell level where local sugar concentration and turgor pressure induce irreversible cell expansion and cell wall synthesis. It was demonstrated how transpiration and photosynthesis rates, metabolic and physiological processes and structural features of a tree mediate their effects directly on the local water and sugar status and influence cambial growth. Large trees were predicted to be less sensitive to changes in the transient water and sugar status, compared with smaller ones, as they have more water and sugar storage and were, therefore, less coupled to short-term changes in the environment. Modelling the cambial dynamics at the individual cell level turned out to be a complex task as the radial shortdistance transport of water and sugars and control signals determining cell division and cessation of cell enlargement and cell wall synthesis had to be described simultaneously. Keywords: cell enlargement, phloem transport, wood formation, xylem transport.

Introduction The timing and rate of wood formation during the growing season are key processes in determining the amount and properties of wood produced. Wood formation depends on genetic signalling, availability of resources, temperature, tree water and nutrient status and stage of ontogenic development. Xylem formation has been a subject of modelling over decades (Wilson and Howard 1968, Howard and Wilson 1972, Wilson 1973, Stevens 1975). However, the early models were mostly conceptual in nature, which limits their applicability to a particular range of environmental conditions (Ford and Linder 1981). Tree level and local processes in the cambium can be described as delayed and instantaneous functions of the envir-

onment. Process-based models of tree growth derive cambial growth from water relations (Steppe et al. 2006) or photosynthetic production and the transport of substrate to the growth sites (Daudet et al. 2002, De Schepper and Steppe 2010), thus linking growth with variations in weather (e.g., Thornley 1999) and also in local hormonal concentrations (Drew et al. 2010). In some process-based models, cambial growth has been explicitly divided into the processes of cell division, cell enlargement and cell wall synthesis, and the rates of these phases have been described as a function of environmental conditions (e.g., Deleuze and Houllier 1995, 1998, Fritts et al. 1999, Vaganov et al. 2006). However, it has been difficult to link cambial growth with short-term carbon dynamics, partly due to the scarcity of direct observations of cambial dynamics and partly due to the large number of different processes influencing the growth rate. Furthermore, many of the model parameters do not represent actual measurable quantities. Cause–effect relationships are being increasingly explored and recent studies provide new knowledge on many of the physiological processes underlying wood formation at the tissue level. The rate of cell enlargement and substrate supply, i.e., long- and short-distance sugar and water transport (Thompson and Holbrook 2003, Hölttä et al. 2006, Lacointe and Minchin 2008) to the cambium can be readily quantified by measurable physical quantities such as cell elasticity, cell plasticity, hydraulic conductance, water potential, turgor pressure and sugar concentration. These factors directly influence or indirectly contribute to cell production and enlargement (Cosgrove 1993, Taiz and Zeiger 2002). These processes are directly linked to each other through water potential gradients and have clear physical constraints influenced by the tree size and structural properties of stems. Within these physical constraints, there are only a limited number of possibilities of how osmoregulation can actually move assimilated sugars from the source leaves and drive the growth of new cells downstream in the stem. The purpose of this study was to model softwood formation, building upon a detailed description of the physiological processes related to cambial growth from earlier works by Génard et al. (2001), Steppe et al. (2006) and De Schepper

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and Steppe (2010). The goal was to broaden the understanding about cambial activity at the whole-tree level in connection with xylem and phloem transport and leaf gas exchange and also to add a description of cambial dynamics at a cell level. However, environmental conditions, e.g., light, temperature and relative humidity, were not used directly as input to the model. Instead, the whole-tree photosynthesis and transpiration rates were used as model input, as this decreases the number of parameters in the model and also allows us to avoid using a stomatal conductance model. Under most circumstances, the photosynthesis rate is a saturating function of light and transpiration rate is a saturating function of water vapour pressure deficit in the ambient atmosphere (Mäkelä et al. 1996). The other model inputs influencing cambial growth were xylem, cambium and phloem structural parameters which, together with the photosynthesis and transpiration rates, translate into physical/physiological quantities in the tree, such as turgor pressure and sugar concentration inducing cell enlargement and cell wall synthesis. The model predictions of stem diameter change were also tested against measured field data from Scots pine trees in central Finland. Materials and methods We used detailed field measurements from SMEAR II station (Vesala et al. 1998) for model parameterization and testing. A 12-m, 40-year-old Scots pine (Pinus sylvestris L.) with a breast height diameter of 16 cm was used. Stem diameter variation was measured at a 5-min frequency with a linear displacement transducer (Solartron Inc., Model AX/5-0/5, Bognor Regis, West Sussex, UK; accuracy of 1 μm) used as a point dendrometer at breast height. Stem temperature (1-cm depth) and the metal frame temperature were measured using copper– constantan thermocouples and the data were corrected for the effects of thermal expansion of wood and frame (Sevanto et al. 2005). Measured photosynthesis and transpiration rates were fed as input to the model when model comparison was done against measured stem diameter change data, i.e., in simulations as shown in Figure 5 (in other simulations, photosynthesis and transpiration inputs were hypothetical). Transpiration rate was taken to be the sap flow rate, which was measured with the Granier-type heat dissipation method (Granier 1987) at a 5-min frequency. Two probes were inserted into the sapwood ~ 10 cm apart. The upper probe was heated with constant power and the sap flux density was calculated from the temperature difference between the two probes using the standard protocol (see, e.g., Granier 1987). It should be noted that the Granier-type method can sometimes underestimate sap flux (e.g., Clearwater et al. 1999). Measured photosynthesis rates (half-hour averages) from Eddy covariance (e.g., Markkanen et al. 2001) of SMEAR II were downscaled to the tree level and fed as input to the model. The absolute values for the whole-tree photosynthesis rates were estimated from Sevanto et al. (2003) for the same site and same trees.

Model description In the model, cambial activity includes the production of new tracheids by the vascular cambium and the subsequent enlargement, wall formation and maturation of tracheids into the secondary xylem. Cambium also produces the phloem, but phloem production was not included in the model. The transport of water and sugars to the cambium along the stem is described in terms of pressure gradient-driven mass flow in the xylem and phloem and the radial diffusion-like process of sugars in the cambium. Sugars are unloaded to the cambium from the phloem. Water and sugars then induce cell enlargement and cell wall synthesis. The model is depicted schematically in Figure 1. Photosynthesis and transpiration Photosynthesis and transpiration rates are two important inputs influencing the water and sugar status of the tree. When studying the overall model behaviour, they were both given diurnally varying values that followed the positive part of a sine wave. Their maximum values were determined from measurements, except in the simulations shown in Figure 5 where measured photosynthesis and transpiration data were used. The daily maximum values of photosynthesis and transpiration rates were given constant values during the whole simulation period, i.e., the period when cambial growth was considered to occur. This approximation is close to reality for the net CO2 assimilation (photosynthesis minus foliage respiration) from mid-May to mid-August in Southern Finland (soil water is rarely limiting to gas exchange; Kolari et al. 2009). Long-distance xylem and phloem transport The xylem water transport and phloem sugar transport models described in Hölttä et al. (2006) were used to model the long-distance transport of water and sugars to the cambium. The transport model uses soil water potential, transpiration rate, photosynthesis rates and tree structural parameters such as the cross-sectional areas of the xylem sapwood and phloem and their hydraulic conductivities as input. Photosynthates are instantaneously converted into sucrose and loaded to the phloem following photosynthesis. The model calculates the transient axial profile of xylem water potential and phloem sugar concentration and turgor pressure. See the Supplementary data for the model description of the axial water and sugar fluxes. Radial (short-distance) transport of water and sugars The unloading rate of sugars per unit length of axial distance (U, in moles per metre per second) from the phloem to the cambium is concentration dependent (Goeschl et al. 1976, Thompson and Holbrook 2003): U = As C p

ð1Þ

where Cp is the phloem sugar concentration and As is a sugar ‘unloading constant’ (in square metres per second). Sugars

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A PHYSIOLOGICAL MODEL OF SOFTWOOD CAMBIAL GROWTH

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Figure 1. Schematic presentation of the model. (A) Xylem water flux (black arrows) is induced by the axial water pressure gradient in xylem, which is caused by the pressure decrease at the top of the tree due to transpiration. Phloem sugar flux (grey arrows) is induced by the axial water pressure gradient in the phloem, which is caused by sugar loading and unloading to and from the phloem. Sugars are unloaded at the growing meristems from the phloem to the cambium and transported radially (Eqs. 1 and 2). Sugars in the cambium draw water osmotically mainly from the xylem according to the local radial water potential gradient (Eq. 3). The equations governing the axial and radial sugar and water fluxes and the resulting effects on pressure and concentration are from Hölttä et al. (2006) and are also shown in the Supplementary data. (B) Sugar and subsequent water flux to a cambial cell increase the local turgor pressure, which causes cell expansion (Eqs. 4–8). If the cell expansion rate is large when the threshold size for cell division is reached, the cells divides into two (Eq. 11). If the expansion rate is low, the cell enters the cell enlargement phase. Cell enlargement ceases and the cell enters the cell wall synthesis phase when the enlargement rate decreases below a threshold value or when the maximum cell size is reached (Eq. 9). Cell wall is deposited until a threshold value in cell wall thickness is reached (Eqs. 12–15).

are unloaded from the phloem to the outermost cambial cells and from thereon transported radially inwards. To simulate the demand of sugars for local sugar usage in cambial respiration, only a fraction (Fs, which equals 0.5 in the base case) of the sugars unloaded from the phloem are loaded to the cambium. The radial transport rate of sugars (Sr) is: Sr = dðCi −Ci + 1 Þ

ð2Þ

where Ci is the sugar concentration in cambial cell i (from the cambial initial) and d is a radial sugar transport coefficient describing the activity of the transport process. The approach corresponds to a ‘diffusion model’ (Wilson 1973) where sugar concentration is always highest near the phloem and decreases inwards. Also, De Schepper and Steppe (2010) used this equation in their approach. The actual transport mode of sugars could be diffusion (Patrick 1997), but we do not necessarily have to assume that here.

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Radial water flux (Jr) is towards the water potential gradient: Jr = LAc ðψi −ψi + 1 Þ

ð3Þ

where ψi is the water potential, L is the (area-specific) hydraulic conductance and Ac is the radial cross-sectional area of the cambium, i.e., the area through which water exchange takes place. Cell enlargement Tracheids are assumed to enlarge as a result of tensile forces generated by cell turgor. The irreversible volume increase of the enlarging cell is (Lockhart 1965, Génard et al. 2001, Steppe et al. 2006): dV0 = ϕðP−P0 Þ if P > P0 dt

ð4Þ

ð5Þ

where V is the cell volume, ψ0 is the water potential in the adjacent tissue (xylem and phloem) and ψ is the water potential in the cambium. Note that V0 increases due to plastic wall yielding and cannot decrease, while V increases due to the exchange of water and can also decrease when water potential in the surrounding tissue decreases (Cosgrove 1981). In steady-state conditions when water potential of the surrounding tissue is constant, changes in V and V0 are equal. The water potential of the xylem is dependent on the axial position along the stem, whereas the water potential in the growing cells varies according to both axial and radial positions. Water potential of a cell is calculated in terms of turgor and osmotic (Pπ) pressures: ψ = P−Pπ = P−CRT

ð6Þ

where C is the sugar concentration, R is the universal gas constant and T is the temperature. The turgor pressure in the enlarging cells is (Philip 1958, Gardner and Ehlig 1965, Cosgrove 1981): V −1 P=E V0

dP dP dV dP dV0 E dV EV dV0 = + − = : dt dV dt dV0 dt V0 dt V02 dt

ð7Þ

ð8Þ

Cell enlargement ceases and the cell enters the phase of cell wall synthesis when the cell enlargement rate decreases below a threshold value Te or when the cell reaches the maximum diameter (Dfinal) (Fritts et al. 1999): Dfinal ðrÞ = Dmax −ðDmax −Dmin Þexpð−b44 ðr−b45 ÞÞ

where V0 is a ‘reference volume’, i.e., the volume of the cell at zero turgor pressure (Cosgrove 1981), ϕ is the extensibility of the cell wall, P is the turgor pressure and P0 is a threshold turgor pressure above which cell enlargement occurs. This threshold turgor pressure has been found to vary between 0.1 and 0.9 MPa in a variety of plant tissues (Génard et al. 2001). Génard et al. (2001) used a value of 0.9 MPa in their study. The actual volume change V of the enlarging cells is caused by exchange of water with the adjacent tissue according to the water potential difference: dV = LAc ðψ0 −ψÞ dt

where E is the elastic modulus of the cell. We assumed a linear relationship between turgor pressure and cell volume which is a good approximation when changes in turgor pressure are small (Nobel 1991). Differentiating the above equation with respect to change in the elastic and plastic volume changes leads to the change in turgor pressure in the enlarging cells as follows:

ð9Þ

where r is the distance from the phloem and Dmax, Dmin and bij are parameters. The parameter bij has the same numbering as in the user manual (Fritts et al. 2000) for the model of Fritts et al. (1999). Cambial cell radial diameter D was calculated from cambium volume using the geometry: D=

V πdsapwood l

ð10Þ

where dsapwood is the sapwood diameter and l (= model tree height/N) is the height of the numerical element. Equation (10) assumes that cell diameter is small in comparison with sapwood diameter. Cell division To describe cell division, we used an approach described in previous papers (Fritts et al. 1999, Vaganov et al. 2006). The size of cambial cells increases from their initial size described by the cell enlargement Eqs. (41–10) until they reach a threshold diameter, Dt. At this point, they either divide into two daughter cells of one-half the diameter of the dividing cell or enter the enlargement phase where cells do not divide. The latter occurs if the growth rate of the cell is lower than a threshold growth rate, Td(r); otherwise, the cell continues to divide. The threshold growth rate Td(r) increases linearly with cell distance from the cambial initial as follows: Td ðrÞ = b22 expðb26 ðre −b27 ÞÞ

ð11Þ

where re is the distance from the phloem when the cell started enlarging (Fritts et al. 1999). Cell wall synthesis After cell enlargement has ceased, sugars are used for cell wall synthesis. The cell wall synthesis rate (W) [mol (sucrose) s −1] is calculated by: W = Aw C

if P > 0:5 MPa


ð12Þ


where Aw is a constant of proportionality between sugar concentration and the rate of cell wall synthesis. Cell wall synthesis was assumed to occur only above a threshold turgor pressure (P > 0.5 MPa) which was made high enough to ensure that the cell does not lose turgor when xylem water potential decreases. Cell wall volume (Vw) was made to increase with the cell wall synthesis rate times a conversion factor (Csw) which describes the relationship between cell wall volume and the amount of sucrose used (Fritts et al. 1999): dVw = Csw W : dt

ð13Þ

The cell wall was deposited until a threshold value (Tw,final) was reached (Fritts et al. 1999): Tw;final = MIN fwmin −ðwmin −wmax Þ expð−b63 ðd−dmin ÞÞ; d ð1−lmin Þ=2Þg

ð14Þ

where wmin and wmax are the minimum and maximum cell wall thicknesses, respectively, d is the lumen diameter, dmin is the minimum lumen diameter and lmin is the minimum fraction of lumen diameter of the total cell diameter. The relationship between cell wall thickness (T) and cell wall volume (Vw) was (Fritts et al. 1999): Vw = 2Tw ðD + DT −2Tw Þl

ð15Þ

where DT is the tangential cell diameter which is constant. Stem diameter change Stem diameter varies due to irreversible wood formation and reversible changes caused by xylem water potential variation as shown in earlier works by Génard et al. (2001) and Steppe et al. (2006). We adapt a similar formulation. The rate of stem diameter change (dd/dt) was calculated from the change in water mass in the xylem (mx), cambium (mc) and phloem (mp) using the geometry: dd 2 dmx dmc dmp = + + dt πhDstem ρ dt dt dt

ð16Þ

where h is the height over which the change in mass is calculated, Dstem is the stem diameter (which varies in time) and ρ is water density (1000 kg m −3). Scaling from small to large trees Sensitivity of the model to changes in tree size was also explored. Leaf area (photosynthesis rate and transpiration are assumed to be proportional to leaf area), xylem and phloem cross-sectional area and the permeability of both xylem and phloem were all related to tree height. Leaf area and xylem and phloem cross-sectional area were assumed proportional to crown length raised to the power of 2.5 (Mäkelä and Sievänen 1992) and crown length was assumed to equal 100, 50 and 33% of the tree height in the 1.2-, 12- and 120-m

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trees, respectively (Valentine and Mäkelä 2005). The wholetree xylem and phloem conductances were assumed to be invariant of tree height (West et al. 1999). The sugar unloading coefficient (As in Eq. 1) was also assumed proportional to the cross-sectional area between the phloem and cambium. Model realization and model runs For the numerical solution, the model tree was radially divided into the functional components of xylem, cambium and phloem (Figure 1). The cambium was further radially divided into M elements (M is the transient number of cambium cells). Axially, the model tree was divided into N elements (N = 20). Explicit Euler finite difference scheme (e.g., Kreyzig 1993) was used for the numerical solution with a time step of 0.1 s. Larger time steps would result in numerical instability. Model parameters were chosen to represent a typical 12-m Scots pine tree in Southern Finland. When possible, the model parameters were obtained from the measurements on the 12-m Scots pine tree at the SMEAR II station. When this was not possible, the values were estimated from the literature. The parameter values describing cell division, cessation of cell enlargement and cessation of cell wall synthesis were taken from Fritts et al. (2000). All parameter values are shown in Table 1. The model was first run as a simplified version without a radial dimension in the cambium, i.e., there was no distinction amongst individual cells (the ‘One-dimensional case’ subsection in the ‘Results’ section; Figures 2–5). This was done in order to simulate the general trends of growth without going into the complex details of growth at the individual cell level. Cambial growth was the only sugar sink in these simulations, except for the fraction of sugars (Fs) that was used locally in the cambium for respiration. In later simulations, a complete version of the model was run where the cambium was divided into individual cells (the ‘Two-dimensional case’ subsection in the ‘Results’ section; Figures 6–9). In this case, the seasonal dynamics of allocation of photosynthates to cambial growth was also considered in detail as this was found to distinctly affect the results. An alternative sink for sugars, simulating sugar consumption by shoot and root growth and their respiration, was included in the model. The cambial sugar sink was assumed to be small at the beginning of the growing season, then increasing towards mid-summer and again the late summer and autumn. In early and late summer, shoot and root growth, respectively, were assumed to consume a large proportion of the available sugars (e.g., Kaufman 1977). See the inset of Figure 6A for the dynamics of sugar partitioning between cambial growth and other processes over the growing season. The two-dimensional model simulations were run with only one axial numerical element as the simulation time increased very much with discretization into 10 axial elements as in the one-dimensional case. However, it was verified that the results presented for the two-dimensional case represent the cambial growth dynamics at the middle of the tree stem (6 m).


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Table 1. Model parameters. ‘Iterated’ means that the parameter value has been found by trial to get reasonable and stable results. ‘Estimated’ means that the magnitude of the parameter is obtained from the literature. ‘Measured’ means it has been measured in the Helsinki University research station in Hyytiälä, Southern Finland. Parameter

Value

Reference

One-dimensional case Sugar unloading coefficient from phloem (As in Eq. 1) Fraction of sugars unloaded from the phloem which are loaded to the cambium (Fs) Cell wall synthesis parameter (Aw in Eq. 12) Maximum photosynthesis rate Maximum transpiration rate Tree height Phloem axial permeability Xylem axial permeability Radial water permeability (L in Eq. 5)

8.3 × 10 −10 m2 s −1 0.5

Iterated Estimated

1 × 10 −15 s −1 15 × 10 −6 mol (of sucrose) s −1 5.5 g s −1 12 m 0.5 × 10 −12 m2 1 × 10 −12 m2 2 × 10 −12 m3 m −2 Pa −1 s −1

Cell wall extensibility (ϕ in Eq. 4) Turgor threshold value for cell expansion (P0 in Eq. 4) Xylem sapwood diameter Xylem heartwood diameter Phloem thickness Xylem sapwood diameter (1.2-m tree)

3 × 10 −16 m3 Pa −1 0.5 MPa 16 cm 8 cm 4 mm 1.85 cm

Xylem heartwood diameter (1.2-m tree)

0 mm

Phloem thickness (1.2-m tree)

1.24 mm

Xylem sapwood diameter (120-m tree)

6m

Xylem heartwood diameter (120-m tree)

5.815 m

Phloem thickness (120-m tree)

25 mm

Initial phloem and cambium osmotic sugar concentration (π in Eq. 4) Initial cambial cell size

2 MPa

Iterated Measured Measured Measured Estimated Perämäki et al. (2001) Génard et al. (2001), Sevanto, personal communication Estimated Taiz and Zeiger (2002) Measured Measured Measured Calculated by scaling assuming no heartwood Calculated by scaling assuming no heartwood Calculated by scaling assuming no heartwood Estimated, but sapwood cross-sectional area calculated by scaling Estimated, but sapwood cross-sectional area calculated by scaling Estimated, but phloem cross-sectional area calculated by scaling Estimated

1 × 10 −6 m

Fritts et al. (2000)

Additional parameters for the two-dimensional case Sugar transport coefficient (d in Eq. 2) Threshold diameter for division (Dt) Parameter b22 in cell division (Eq. 11) Parameter b26 in cell division (Eq. 11) Parameter b27 in cell division (Eq. 11) Threshold growth value to stop cell enlargement (Te) Maximum cell size (Dmax in Eq. 9) Minimum cell size (Dmin in Eq. 9) Parameter b44 in cell size (Eq. 9) Parameter b45 in cell size (Eq. 9) Maximum cell wall thickness (wmax in Eq. 14) Minimum cell wall thickness (wmin in Eq. 14) Parameter b63 in cell wall thickness (Eq. 14) Parameter lmin in cell wall thickness (Eq. 14) Conversion factor between sugar amount and cell wall volume (Csw in Eq. 13) Tangential cell diameter (DT)

5 × 10 −11 mol s −1 10 × 10 −6 m 32 μm day −1 0.06 μm −1 32.55 μm 0.1 μm day −1 49.0 μm 16.3 μm 0.0169 μm −1 9.5 μm 15.0 μm 2.0 μm 0.2 day −1 0.5 0.00026306 m3 (cell wall) mol −1 (sucrose) 30 μm

Iterated Fritts et Fritts et Fritts et Fritts et Fritts et Fritts et Fritts et Fritts et Fritts et Fritts et Fritts et Fritts et Fritts et Fritts et


al. al. al. al. al. al. al. al. al. al. al. al. al. al.

(2000) (2000) (2000) (2000) (2000) (2000) (2000) (2000) (2000) (2000) (2000) (2000) (2000) (2000)

Fritts et al. (2000)


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Figure 2. The transpiration and photosynthesis rates (sugar loading) fed as input to the model (A) and the resulting diurnal dynamics in xylem water potential (B), phloem osmotic pressure (C), phloem turgor pressure (D), irreversible cambium volume (V0) (E), irreversible cambial growth rate (F), cambium turgor pressure (G) and change in cambium volume (V) (in cubic millimetres per second per 1 m tree height) (H) for a 2-day period 30 days after the initiation of cambial growth. Values are shown at 1-, 6- and 11-m height of the 12-m model tree. TREE PHYSIOLOGY ONLINE at http://www.treephys.oxfordjournals.org

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Figure 3. Predicted irreversible growth rate of cambium volume in response to changes in environmental conditions and structural parameters (A). Parameter values and growth rates are expressed in relation to their base case values (Table 1). In the case of the ‘drought’ soil, water potential was reduced to −2 MPa with a simultaneous decrease in the hydraulic conductance between the soil and the bottom-most xylem element to one-tenth of its base case value. Diurnal pattern of irreversible growth rate of cambium volume (in cubic millimetres per second per 1 m height) (at 11 m) in response to changes in cell wall extensibility (ϕ) (B).

Results One-dimensional case The daily patterns in transpiration and photosynthesis rate (Figure 2A) were the main drivers for xylem water potential (Figure 2B), phloem osmotic pressure (Figure 2C) and phloem turgor pressure (Figure 2D), the key variables influencing sugar and water uptake to the cambium. Phloem turgor pressure mainly followed xylem water potential but there were also small diurnal changes in phloem osmotic pressure. Irreversible cambium volume (V0 in Eq. 4) increased (Figure 2E) mainly

during the night (Figure 2F) when cambium turgor pressure (Figure 2G) was above the threshold for cell enlargement. The actual or ‘reversible’ cambium volume (V in Eq. 5) decreased during the morning when xylem water potential decreased and increased during the afternoon when xylem water potential increased (Figure 2H). The changes were more pronounced at the tree top due to larger variation in xylem water potential. Cambium sugar concentration remained diurnally very constant (not shown). Diurnal changes in the cambium turgor pressure and growth rate were larger near the top of the tree due to larger xylem water potential



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Figure 4. Predicted relative cambial volume after 30 days of growth for trees with heights of 120 (A), 12 (B) and 1.2 m (C) subjected to varying photosynthetic rates, initial phloem sugar concentrations and ‘drought’. In the ‘drought’, soil water potential was reduced to −1 MPa with a simultaneous decrease in the hydraulic conductance between the soil and the bottom-most xylem element to one-tenth of its base case value (Table 1). Values for each height are expressed in relation to the base case photosynthesis or transpiration rate or initial sugar concentration.


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variation closer to the transpiring needles. All the results are shown 30 days after the initiation of cambial growth, i.e., 30 days after sugar unloading to the single cambial initial began. The model parameters and inputs were varied one at a time to examine their influence on cambial volume growth (Figure 3A). Higher transpiration rate or lower xylem hydraulic conductance decreased xylem water potential, which reduced water availability for growth. Reduction in xylem hydraulic conductivity by the same proportion as raising the transpiration rate yielded the same results, as both resulted in the same values for xylem water potential (the same is true for all the results presented later). Higher transpiration rate and low xylem hydraulic conductivity also led to a larger water potential gradient along the stem (not shown), which resulted in a larger gradient in phloem sugar concentration and, therefore, larger sugar unloading from the phloem at the top of the tree. Cambial growth was also decreased by a ‘drought’, i.e., a decrease in soil water potential with a simultaneous decrease in the conductance between the soil and the bottom-most numerical xylem element. Cambial growth rate responded to a change in the photosynthesis rate, but also the initial osmotic concentration in the phloem had a big influence on cambial growth still after 30 days. If the cambium sink strength (As in Eq. 1) decreased or the photosynthesis rate increased much, the phloem was not able to transport all sugars, and sugars accumulated at the tree top (not shown). In contrast, higher cambium sink strength (As in Eq. 1) led to faster cambial growth as more sugars were taken from the phloem to the cambium. As the parameter values were not varied axially, the cambial growth rate was higher at the top of the tree (Figure 2G). As loading of sugars to the cambium was dependent on phloem concentration, sugar unloading to cambium was higher at the tree top as phloem transport necessarily demands a decreasing sugar concentration from the leaves downwards. The difference in sugar concentration between the tree top and bottom was larger when the cambium sink strength was higher, as then a larger proportion of the sugars was unloaded at each height and less sugars were left for the tree bottom. The cambial growth rate at the tree bottom could be made larger compared with the apex when cambium sink strength increased sufficiently from top to bottom (not shown). In contrast, water potential decreased upwards, i.e., larger amounts of sugars are needed to ‘pull’ water to the cambium closer to the apex. Thus, the cambial growth rate results from the combination of sugar and water availability with more sugars at the tree top and more water at the bottom. Higher cell wall extensibility (ϕ in Eq. 4) led to faster cambial growth as less turgor pressure (and less sugars to osmotically draw in the water) was needed to induce a certain growth rate. Larger cell wall extensibility also resulted in larger diurnal variation in the cambial growth rate (Figure 3B). On the other hand, low cell wall extensibility produced a steadier cambium growth rate, which was less dependent on diurnal variations in xylem water potential and turgor pressure.

Scaling from small to large trees Scaling of the structural properties with tree size led to very similar values of xylem water potential, phloem turgor pressure and osmotic concentration at the tree top and bottom regardless of tree height (as in Figure 2), the only main difference resulting from gravity for the very tall trees. Our modelling results revealed that the largest trees were less dependent on current year photosynthetic production for cambial growth (Figure 4A). Instead, they were more dependent on the initial amount of sugars in the phloem (Figure 4B). The opposite was true for the small trees (Figure 4C). Their growth was mainly determined by current photosynthetic production. The dependency of growth on current photosynthetic production also increased in all cases when the length of the period in question was increased (not shown). The cambial growth of smaller trees was more affected by ‘drought’ than that of the larger ones. In the taller trees, relative change in xylem water potential with declining soil water potential was lower, as gravity already accounts for much of the water potential loss, and also due to the larger internal water stores of the tree. Cambial growth relative to initial stem volume was larger for the smaller than larger trees. The smallest tree (1.2 m) grew ~30 times faster than the 12-m tree and ~4000 times faster than the 120-m tree in relation to their initial stem volume (not shown). Stem diameter change The model was able to predict stem diameter changes quite well (Figure 5A). However, the fraction of sugars loaded to the cambium for cambial growth (Fs) had to be decreased from 0.5 to 0.25 to get the model to fit the measurements. The diurnal reversible variation in stem diameter was ~0.1 mm day −1, and the irreversible cambial growth rate was approximately a quarter of this. Sensitivity of the model was tested by varying photosynthesis and transpiration rates, as well as cell wall extensibility and soil water potential. Changing the value of the photosynthesis rate resulted in an increase in diameter growth (Figure 5B). The effect of photosynthesis rate became more pronounced as time passed, as the effect of initial phloem sugar concentration declined. Increasing the transpiration rate increased the diurnal variation in stem diameter and also slightly decreased irreversible growth (Figure 5C). Increased cell wall extensibility (ϕ in Eq. 4) accelerated diameter growth as cambial growth increased. Two-dimensional case Figure 6 illustrates the predicted dynamics of cell division and enlargement (Figure 6A) and cell wall formation (Figure 6B) throughout the growing season for the base case at the middle of the model tree. All new conduits had undergone all the stages of cell formation and were mature at Day 120 after the initiation of cambial growth. Cell division, cell enlargement and cell wall formation occurred simultaneously throughout the growing season (Figure 6C) except at the very



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Figure 5. Measured and modelled stem diameter change from 29 June to 10 July 2006. The inset shows the measured transpiration (grey line) and photosynthesis (black line) rates fed as input to the model (A). Sensitivity of the modelled stem diameter to changes in photosynthesis rate and initial sugar concentration (B) and in transpiration rate and cell wall extensibility (C).

beginning and end of the growing season. The relationship between the cell wall extensibility (ϕ in Eq. 4) and the constant of proportionality between sugar concentration and the rate of

cell wall synthesis (Aw in Eq. 12) determined the partitioning of sugar between cell wall enlargement and cell wall synthesis when there were cells present in both phases simultaneously (a


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Figure 6. Predicted dynamics of cambial growth (inset shows the ratio of the cambial sink strength to all sinks) (A), cell wall thickness (B), number of cells in different phases of development (C) and final cell lumen diameter to cell wall thickness ratio for the base case (D) at the middle of the tree. ‘Cell number’ on the x-axis refers to the succession order of the cambial cell from phloem; t refers to the time after initiation of growth.

high ϕ/Aw ratio led to increased cell enlargement in relation to cell wall synthesis; not shown). The conduits formed in the beginning of the growing season grew larger than those created later in the growing season. Cell division and enlargement rates decreased with increasing number of cells as less sugars and water were available for each dividing cell. Also, the total resistance of water flow from the xylem to the cell division zone increased then as there were more conduits and the radial transport distance was longer. The early wood conduits also had larger cell lumen to cell wall thickness ratios compared with the latewood conduits (Figure 6D) resulting from Eq. (14). The turgor pressure (determining the cell enlargement rate; Figure 7A), sugar concentration (determining sugar movement and cell wall synthesis rate; Figure 7B) and water potential gradients (determining water movement; Figure 7C) in the cambium were influenced by the radial resistances to water and sugar flow. Low radial water permeability induced a steep gradient in the xylem water potential (Figure 7C), reducing the turgor pressure and thus enlargement rate for conduits near the phloem which are furthest from the primary source of water

in the xylem. This led to a low number of cells (Figure 7D), as cell division is determined by the local cell enlargement rate adjacent to the phloem, and a large average conduit size (Figure 7D), as sugar and water were distributed to fewer conduits. Low radial water permeability also resulted in relatively larger early wood conduits, as the cambium was relatively closer to the xylem at the beginning of the growing season. In contrast, high radial water permeability led to more, but on average smaller, conduits. Low radial sugar transport coefficient accumulated sugars near the phloem where high turgor pressure and enlargement rates were induced. Similarly to the case of high permeability to water, a low radial sugar transport coefficient led to a large number of small cells because both sugar concentration and turgor pressure decreased further away from the phloem. A high radial sugar transport coefficient had the opposite effects. Changes in water and sugar permeabilities also affected the time needed to complete all xylem conduits. If the sugar transport coefficient was low or, similarly, the water permeability was high, all cell walls could not be completed during the growing season, as carbon availability was insufficient for the large number of conduits (not shown).



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Figure 7. Predicted radial profile of turgor pressure (A), osmotic concentration (B), water potential (C) at 30 days after the beginning of cambial growth and final cell size distribution (D) for different radial water permeabilities and radial sugar transport coefficients at the middle of the tree. ‘Cell number’ on the x-axis refers to the succession order of the cell from phloem.

Accordingly, if the threshold values for cell division and cessation of cell enlargement were set too low or the maximum size for conduits too high (Eq. 9) in relation to the sugar and water availability and transport properties, many

and large cells were produced but the cell walls could not reach their required thickness. An increase in the photosynthesis rate led to increased availability of sugars and hence to more cells (Figure 8); a reduction in photosynthesis did the

Figure 8. The sensitivity of the final mature cell diameter to changes in the photosynthesis and transpiration rates at the middle of the tree. ‘Cell number’ on the x-axis refers to the succession order of the cell from phloem. TREE PHYSIOLOGY ONLINE at http://www.treephys.oxfordjournals.org

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Figure 9. Final mature cell diameter distribution at the middle of the tree with different parameterization. Here, the radial sugar transport coefficient was increased and radial water permeability decreased by a factor of 5 with a simultaneous increase in the threshold rate for cell division (Eq. 1) and cessation of cell enlargement by factors of 10 and 3, respectively, in relation to the base case (Table 1). Also, parameter b44 in Eq. 9 that determined the maximum cell diameter was doubled in relation to the base case. ‘Cell number’ on the x-axis refers to the succession order of the cell from phloem.

opposite. An increase in the transpiration rate led to fewer cells for the same reason as in the one-dimensional case: water potential and turgor pressure were lowered everywhere in the cambium. A reduction in the transpiration rate had the opposite effect (Figure 8). Increasing the threshold growth rate for cell division (b22 in Eq. 11) naturally led to fewer conduits (not shown) and faster cell enlargement, as sugars and water were distributed to fewer cells. This also led to the early cessation of cambial growth at 80 days, even though the sugar sink for cambial growth was still present (see inset of Figure 6A). The cell size distribution predicted by the model appeared to be sensitive to changes both in radial sugar and water transport capacity. However, the model was also sensitive to many other parameters which could compensate the effects of the short-distance radial sugar and water transport properties. For example, low sugar transport coefficient and high water permeability producing a large number of small conduits (Figure 7D) could be compensated by increasing the threshold enlargement rate for cell division (Eq. 11). This would have an effect in the opposite direction by reducing the number of conduits and increasing their size. In general, a similar final cell size distribution as in the base case could be achieved with very different parameterization (Figure 9; see the figure caption for the parameterization). Discussion A model was presented that describes cambial growth based on the physiological processes of a tree. The model links softwood formation explicitly to transpiration and photosynthesis rates and the within-tree transport of water and sugars. Transpiration and photosynthesis rates can, in turn, be linked to environmental conditions. The effect of water availability, a concept used in some earlier models, was translated into

xylem water potential using a whole-tree transport model (e.g., Daudet et al. 2002, Hölttä et al. 2006), which calculates xylem water potential from soil water potential, transpiration rate and xylem and soil hydraulic conductance. Sugar availability was determined by the photosynthesis rate and the transport and release from the sugar pool in the phloem (e.g., Thompson and Holbrook 2003). Temperature implicitly affects all metabolic processes such as photosynthesis, transpiration, sugar unloading from the phloem and radial sugar transport rate, thus affecting cell division, enlargement and cell wall synthesis. Accurate modelling of secondary growth in trees requires considering the carbon and water balances (Génard et al. 2001, Steppe et al. 2006, De Schepper and Steppe 2010), as climate variations may have different influences on secondary growth and GPP. As an extension to the previous work, we provide a detailed growth description at the cell level using existing phenomenological growth models. The results demonstrate that it is possible to derive realistic patterns of cell formation with this combined approach. This gives possibilities to link wood quality with tree level processes, which could potentially lead to valuable silvicultural applications. In the model, the transport capacity of the xylem and phloem, as well as the elastic and plastic properties of the cells, had a large effect on cambial growth. Turgor pressure in the cambial cells is the exclusive driving force for cell enlargement (Lockhart 1965, De Schepper and Steppe 2010). Maintenance of turgor pressure above the threshold for cell enlargement was dependent not only mainly on photosynthetic production and water potential in the tree, but also on the long- and short-distance transport capacity of water. Growth was found to occur mainly during the night, and larger cell wall extensibility resulted in larger diurnal variability in growth, as Steppe et al. (2006, 2008) also concluded. To draw water to the growing cells, the water potential in the cells needs to be lower than that of the surrounding tissue. This



is achieved by simultaneously transporting sugars to the cambium. In addition to the direct effects described by the model, indirect effects of the environmental conditions, not explicitly taken into account in the model, mediate their effects on cambial growth by influencing stomatal control of transpiration and photosynthesis. For example, reduced soil water potential and soil hydraulic conductance (e.g., Duursma et al. 2008), decreased xylem conductance due to cavitation of xylem conduits (e.g., Hubbard et al. 2001) and reduced sink strength due to sugar accumulation in the leaves and phloem (Myers et al. 1999) would initiate stomatal closure and decrease both the photosynthesis and transpiration rates. The model became increasingly complicated when the radial dimension was included, i.e., when cambial growth at the cell level was considered. Cell division and the transition between the phases of enlargement to cell wall synthesis were not modelled truly mechanistically as the physiology of these processes is not fully understood (Fritts et al. 1999). Instead, cell division was made to be dependent on the enlargement rate and radial position of the cell (cf. Fritts et al. 1999, Vaganov et al. 2006). Cell wall synthesis was made to commence when the cells reached a threshold size determined by Eq. (9) or when the cell enlargement rate decreased below a threshold value. The approach produced qualitatively consistent results with the measured number of cells and cell size distribution (Mäkinen et al. 2003, 2008), but it requires many parameters that do not represent physically measurable quantities. In reality, cell division and final cell size are under hormonal control (e.g., Taiz and Zeiger 2002) and are influenced, for example, by local auxin concentrations. A radial auxin gradient has been found both in gymnosperm and angiosperm trees where auxin concentration peaked in the cambium and then decreased sharply to the xylem and phloem (Uggla et al. 1996, 2001, Tuominen et al. 1997). Also, an axial auxin gradient is typically found resulting from basipetal auxin transport, i.e., auxin concentration decreases from stem apex to base (faster cell division at the top) (Kramer 2002). High auxin concentration increases the cell wall extensibility and also stimulates cell division (e.g., Taiz and Zeiger 2002). A recent model development has incorporated an explicit formulation on auxin transport and its influence on growth (Drew et al. 2010). Furthermore, the radial permeabilities of water and sugar influence cell division and cessation of cell enlargement. So, the dynamics of growth allocation between conduits is a very complex process but can be understood from the interplay between the short-distance transport processes and hormonal control. The transport coefficient of sugar (d in Eq. 2) can be converted into the same units as a diffusion coefficient (although we are not necessarily assuming that the transport is purely a passive diffusive process here) and leads to a value of ~2 × 10 −12 m2 s −1, which is ~0.004 times the diffusion coefficient of sugar in bulk water (CRC Handbook of Chemistry and Physics 2001). Even such a low value was able to maintain the sugar concentration radially fairly constant (see Figure 7B), as the radial transport distances are short.

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We can also implicitly assume that the radial sugar gradient in the model of Fritts et al. (1999) is fairly small, as in their model cell growth rate increases monotonously with increasing distance from the phloem (as is also true for our base case; Figure 7A), and this would not be possible if sugar concentration decreased very rapidly with distance from the phloem. The radial water conductance from one cell to another was 2 × 10 − 12 m3 m − 2 Pa − 1 s − 1. For 30 cells, this equals a radial water conductance of 7 × 10 −14 m3 m −2 Pa −1 s −1, which corresponds to the magnitude estimated for the radial conductance between the xylem and phloem (Génard et al. 2001, Steppe et al. 2006). The results demonstrated that cambial growth is dependent not only on present-year conditions, but also on the initial state of the system, especially on the amount of sugars in the tree when growth initiates. The model tree was slow in reacting to changes in sugar production as the turnover time of one sugar molecule, i.e., the amount of sugar in the phloem divided by the photosynthesis rate, was quite long, 24 days in the base case simulation. Also, other studies have reported that the stored carbohydrates (e.g., Skomarkova et al. 2006) or previous year's photosynthetic production (e.g., Berninger et al. 2004) had a large role in the measured cambial growth rates. Rocha et al. (2006) did not find any correlation between the tree ring width and gross ecosystem productivity of the same year close to timberline in old black spruce forest. The turnover time of one sugar molecule increased with tree size, being 2.4 days for the 1.2-m tree and 240 days for the 120-m tree. Thus, a small tree was found to be much more dependent on current photosynthesis production compared with a large one. This was also found to be a general trend as scaling, for example, to the WBE theory (West et al. 1997, 1999) produced very similar results (not shown) as the scaling used here. Also, De Luis et al. (2009) and Vieira et al. (2009) reported smaller sensitivity of larger or older pines to weather variation compared with smaller ones. However, Carrer and Urbinati (2004) and Esper et al. (2008) reported the opposite or found no relationship between tree age and its climatic sensitivity. The contradictory results could be explained by size-related changes being different from age-related ones (De Luis et al. 2009) and other factors. For example, Skomarkova et al. (2006) showed a considerably different behaviour in relation to concurrent assimilation between trees depending on their competitive position within the canopy. In addition, larger trees have also larger internal water stores compared with their water use (Phillips et al. 2003). Hölttä et al. (2009) also predicted that larger trees can thrive longer during a drought by using their internal xylem sapwood water stores released gradually by cavitation (not included in this model) for transpiration. The new xylem cells build up a structure to fulfil the water transport needs of a tree. The xylem water transport system cannot be efficient and safe at the same time (Tyree 2003). Low water availability demands safety at the expense of efficiency, while efficiency may be sought during abundant water availability. Higher photosynthetic rate leads to larger water usage and the need for a larger hydraulic conductance


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in the xylem. On the other hand, when xylem water potential decreased, the model tree responded by producing smaller cells. Reduction in cell size can be beneficial, even at the expense of hydraulic conductance, when low water potentials are experienced, as smaller cells are less vulnerable to cavitation (Hacke and Sperry 2001). In addition, small cell size protects the xylem from the mechanical strains caused by low water potential (Hacke et al. 2001). Our model does not describe phloem formation, but the same general trends should also apply to the phloem. Accumulation of sugar in the phloem, a sign of inadequate phloem transport capacity or small sink strength, should accelerate new phloem conduit production and thus enhance phloem conductance, whereas low local sugar concentration should result in a decrease in the production of new phloem cells. Conclusions The model presented was designed for understanding and predicting general trends of how photosynthesis and transpiration rates, the activity of metabolic processes and stem structure (tree size and axial and radial permeabilities to water and sugar) influence cambial growth. Being based on the metabolic and physiological relationships between sugar and water transport to the cambium, water and carbon exchange with the atmosphere and wood formation, the approach is flexible and general for softwoods. Despite the physiological basis included into the model, the predicted cell size distributions still stand on several assumptions derived from empirical data, as the knowledge on the processes associated with cell formation is fragmentary. Modelling the dynamics of cell size distribution in a mechanistic way is a complex task as both the local cambial sugar and water status and the hormonal signals promoting cell division, enlargement and cell wall deposition have to be described simultaneously. Furthermore, water and sugar status and the signal controlling cell division and beginning of cell wall synthesis also interact with each other. In future work, it would be useful to evaluate the model more thoroughly against measured data on cambial activity. Supplementary data Supplementary data mentioned in the text is available to subscribers in Tree Physiology online. Acknowledgments We thank Alexander V. Shashkin for the constructive discussions which greatly improved this work. This research was funded by the Academy of Finland (#115650, #124390 and #1132561).

Funding The research was funded by the Academy of Finland (#115650), (#124390), (#32561).

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