Assimilate transport in phloem sets conditions for leaf gas exchange


E. Nikinmaa. E-mail:


Carbon uptake and transpiration in plant leaves occurs through stomata that open and close. Stomatal action is usually considered a response to environmental driving factors. Here we show that leaf gas exchange is more strongly related to whole tree level transport of assimilates than previously thought, and that transport of assimilates is a restriction of stomatal opening comparable with hydraulic limitation. Assimilate transport in the phloem requires that osmotic pressure at phloem loading sites in leaves exceeds the drop in hydrostatic pressure that is due to transpiration. Assimilate transport thus competes with transpiration for water. Excess sugar loading, however, may block the assimilate transport because of viscosity build-up in phloem sap. Therefore, for given conditions, there is a stomatal opening that maximizes phloem transport if we assume that sugar loading is proportional to photosynthetic rate. Here we show that such opening produces the observed behaviour of leaf gas exchange. Our approach connects stomatal regulation directly with sink activity, plant structure and soil water availability as they all influence assimilate transport. It produces similar behaviour as the optimal stomatal control approach, but does not require determination of marginal cost of water parameter.


The response of plant gas exchange to the changing climate will determine how vegetation contributes to the global carbon sink in the future (Canadell & Raupach 2008). Recent findings suggest a decline in global land evaporation trend due to soil moisture limitation in Southern hemisphere (Jung et al. 2010) with potentially important consequences to the entire earth system. Climate and atmospheric composition influence the driving gradients of the linked carbon dioxide and water vapour fluxes at the vegetation-atmosphere interface (Nemani et al. 2003). Plants also regulate this gas exchange and depletion of soil moisture levels through adjusting the size of stomatal opening according to their physiological and structural properties (Nobel 2005).

The closing and opening of the guard cells of stomata involve changing osmotic pressure and ion fluxes between epidermal cells and guard cells (Buckley 2005). This requires production of oxidizing species in the guard cells (Wang & Song 2008) and activation and deactivation of ion channels and transporters in the guard cell plasma membrane (Pandey, Zhang & Assmann 2007). The opening of the stomata has been shown to depend on the atmospheric CO2 concentration, water vapour pressure deficit (VPD), irradiation and soil water availability (Landsberg 1986). Most plant physiological and ecosystem models indeed describe stomatal action to be dependent on assimilation rate and environmental factors influencing CO2 and water vapour fluxes (Ball, Woodrow & Berry 1987; Leuning 1995). At plant level and over the developmental cycle, however, the regulation of the stomatal opening is only partially understood. There are clear indications of hormonal signalling (Christmann et al. 2007) and tree level hydraulic control to avoid excessive water loss (Sperry et al. 2002; Brodribb 2009). On the other hand, stomatal control that optimizes carbon uptake relative to water loss in transpiration (e.g. Mäkelä, Berninger & Hari 1996) has been shown to describe stomatal behaviour adequately (Hari & Mäkelä 2003).

Transpiration has a direct influence on the carbon transport from source leaves to sinks, but that is seldom considered in studies of stomatal action. Current understanding is that phloem transport is driven by osmotically regulated turgor pressure gradient that sugar loading at sources and unloading at sinks maintain (Van Bel 2003; Thompson & Holbrook 2003). However, there are a number of details that still need to be clarified before complete understanding is reached (Thorpe & Minchin 1996; Van Bel 2003; Mullendore et al. 2010; Turgeon 2010). According to the above Münch hypothesis, the phloem draws water osmotically from the transpiration stream at the source as assimilated sugars are loaded into sieve cells. This means that phloem transport competes with transpiration for the water in leaves. At high transpiration rates, high xylem water tension effectively hinders the water flux to the phloem (Hölttäet al. 2006). This lowers the turgor pressure gradient in the phloem and slows down phloem transport. Increased sugar loading at the source could compensate for the higher leaf water tension. However, excessive loading rate of the most commonly transported sugar, sucrose, causes rapid viscosity build-up that eventually blocks phloem transport (Hölttä, Mencuccini & Nikinmaa 2009).

Assimilate transport from leaves is a necessary requirement for continuous photosynthetic production, particularly in trees where loading is often passive (Turgeon 2010). Accumulation of excess sugars in leaves leads to sugar turnover to starch and down-regulation of photosynthesis. An active regulatory linkage between carbohydrate source and sink has been suggested (Ainsworth & Bush 2011). Such regulation would be connected to the long-distance transport capacity as well, because of the viscosity linkage that excess loading of the most common transport carbohydrate, sucrose, could cause (Hölttäet al. 2006, 2009; Thompson 2006).

Model of whole tree carbon dynamics has been previously used to predict behaviour of photosynthetic light use efficiency and optimal leaf protein content (Dewar, Medlyn & McMurtrie 1998). Here we use a detailed formulation of water and assimilate transport in trees to hypothesize that the stomata operate to maintain maximal assimilate flux from the source to the transporting tissue, the phloem, under given conditions. When phloem transport capacity is high enough to avoid viscosity limitation, this leads to opening of stomata, such that the plant still maintains phloem turgor and its axial gradient to transport the photosynthesized sugars away from the leaves. To test this hypothesis, we (1) modelled the impact of stomatal conductance on water and assimilate transport; (2) determined the stomatal opening that gave the highest assimilate transport rate under different environmental conditions; and (3) compared the modelled response against observed behaviour of leaf stomata.


Outline of the study

We combined a simple model of leaf photosynthetic rate (Mäkeläet al. 1996), a tree level coupled xylem-phloem transport model (Hölttäet al. 2006) and a model of leaf carbon storage (Berninger et al. 2000) to investigate our hypothesis. The main features of the model are: (1) leaf transpiration is linearly proportional to stomatal conductance and VPD of surrounding air; (2) the photosynthetic rate is a saturating function of irradiance and a linear function of leaf internal CO2 concentration under low leaf sugar concentrations; (3) down-regulation of photosynthesis is a piecewise function of sugar concentration in leaves; (4) sugar turnover to starch follows Michaelis–Menten kinetics relative to sugar concentration and starch turnover to sugars is a linear function of starch concentration (Escobar-Gutiérrez et al. 1998); (5) the sugar loading in the leaf is proportional to the photosynthetic rate minus the sugar turnover rate to starch mimicking the passive loading mechanism often observed in trees (Turgeon 2010; Liesche, Martens & Schulz 2011); (6) phloem sap viscosity is exponentially proportional to phloem sap sucrose concentration (Morrison 2002); (7) the sugar unloading rate at the sink is proportional to phloem sugar concentration exceeding turgor maintenance requirement (Patrick 1997); (8) water movement and uptake from soil to root to leaf is directly proportional to pathway conductivity and the pressure difference between leaf and soil; (9) water movement in radial direction is directly proportional to the radial conductivity and the radial water potential difference that results from hydrostatic and osmotic pressure differences between the xylem and the phloem; (10) phloem transport of sap is directly proportional to the turgor pressure differences between the source and sink parts of the tree, the phloem conductivity and the sap viscocity; and (11) phloem transport of sugars is determined by multiplying water flow rate with the sugar concentration.

We ran the model with variable VPD, solar radiation and soil water potential, and we calculated stomatal opening, transpiration and photosynthetic rate and the resulting flows of water and carbon. We then transiently varied stomatal conductance to give maximum phloem sugar flow from the leaves at all times and studied how this stomatal conductance level varied relative to environmental driving variables [VPD, photosynthetically active radiation (PAR), soil water potential] and xylem and phloem hydraulic conductivities.

The model structure

We used the model described in Hölttäet al. (2006) to calculate sap fluxes in the xylem and phloem. The model is a numeric formulation of the Münch pressure-flow hypothesis (e.g. Nobel 2005). In the model, sap flow both in the xylem and the phloem is due to axial pressure differences and radial water exchange arising from water potential differences between these tissues. A brief description of the model is in the Appendix 1. Here we describe the new model components we added to be able to address our research question. These included the formulations for leaf gas exchange, source-sink sugar dynamics and water uptake from soil. The equations are formulated explicitly in a dynamic form.

Gas exchange

Transpiration (J) [g (H2O) m−2 s−1] and photosynthesis rates (A0) [g (CO2) m−2 s−1] are calculated from stomatal conductance (g) (ms−1) and ambient VPD (dw) [g (CO2) m−3], PAR (I) (µmol m−2 s−1) and CO2 concentration (Ca) [g m−3] following a steady state-formulation by Mäkeläet al. (1996)




where 1.6 is the relation between stomatal conductance for water vapour and CO2 (as gs is expressed for CO2) and f (µmol m−2 s−1)is the following function of PAR


where β and γ are photosynthetic parameters (Mäkeläet al. 1996).

Down-regulation of photosynthesis at high leaf sugar concentrations

For the sake of simplicity, we modelled down-regulation of photosynthetic production with increasing leaf sugar accumulation (e.g. Ainsworth & Bush 2011) as a function of leaf sugar concentration (Cleaf) (mol m−3) in the following fashion:


where α and δ are constants (mol m−3). This made the photosynthesis rate decrease linearly as a function of sugar concentration when Cleaf increased from 1200 mol m−3 (which corresponds to 3 MPa osmotic potential) to 2800 mol m−3 (which corresponds to −7 MPa osmotic potential). The chosen threshold values were selected to avoid the viscosity build-up (Hölttäet al. 2009) but they had little influence on the simulation results as long as there was a feedback between the sugar accumulation and photosynthetic rate.

Respiration, needle sugar–starch dynamics, and phloem loading and unloading

Leaf respiration (Rleaf) that consumes photosynthates was modelled to be temperature (T) [°C] dependent according to field observations (unpublished data, see the description of measurements).


The rate of sugar to starch conversion (Sstarch) [mol s−1] in the leaves was made to follow Michaelis–Menten kinetics (e.g. Escobar-Gutiérrez et al. 1998) relative to leaf sugar concentration (Cleaf) (mol m−3)


while the rate of starch to sugar conversion (Ssugar) [mol s−1] was made to be linearly proportional to starch content (Nstarch, in units of moles equivalent to sucrose) (e.g. Escobar-Gutiérrez et al. 1998)


M1[s−1], M2[mol m−3] and M3[s−1] are constants (see Table 1).

Table 1. The parameter values used in the simulations
  • a

    In simplified cylindrical tree, trunk height was up to tree crown base.

  • b

    Assuming pipe model relationships between leaf area and sapwood area (Nikinmaa 1992) and phloem thickness of 2 mm.

  • c

    The model behaviour is not influenced by the choice for this parameter as long as it is large enough.

Tree height h12 mMeasureda
Xylem sapwood cross-sectional area, Ax4.7 × 10−3 m2Measuredb
Phloem cross-sectional area, Ap0.25 × 10−3 m2Measuredb
Cambium cross-sectional area0.05 × 10−3 m2Estimated to be one fifth of phloem
Xylem axial permeability, kx1 × 10−12 m2Estimated, based on transpiration rate and leaf water potential
Phloem axial permeability, kp1 × 10−12 m2Based on Thompson & Holbrook (2003)
Radial conductance, L1.25 × 10−13 m Pa−1 s−1Based on Thompson & Holbrook (2003)
The radial cross-sectional area between adjacent components (i.e. xylem to cambium to phloem), Arad0.5 m2 per meter length of treeGeometrical contact area
Leaf area, Aleaf40 m2Measured
Density of water, ρ1000 kg m3 
Molar mass of water, MH2O0.018 kg mol−1 
Gravitational constant, g9.81 m2 s−1 
Molar gas constant, R8.314 J mol−1 K−1 
Xylem elastic modulus, Ex1 Gpa Perämäki et al. (2001)
Phloem elastic modulus, Ep30 Mpa Hölttäet al. (2006)
Cambium elastic modulus, Ec10 MpaEstimated, Nobel (2005)
Ambient CO2 concentration, Ca350 ppmEstimated
Photosynthesis parameter, β0.00094 m s−1 Kolari et al. (2007)
Photosynthesis parameter, γ1100 × 10−6 µmol m−2 s−1 Kolari et al. (2007)
Photosynthesis down-regulation parameter, α1200 mol m−3Estimated
Photosynthesis down-regulation parameter, δ1600 mol m−3Estimated
Maximal rate of change of stomatal conductance5 × 10−6 ms−2 Buckley (2005)
Soil water potential, ψs−0.1 MPa (but variable in soil drying simulations)Measured to be typical at the study site
Saturated soil conductivity, Ksat5.7 × 10−6[mol m−1 s−1 Pa−1] Duursma et al. (2008)
Air entry water potential for soil, ψe−0.68 [kPa] Duursma et al. (2008)
Empirical coefficient related to the clay content of the soil, b4.14 Duursma et al. (2008)
Root length index, Rl5300 m−1 Duursma et al. (2008)
Root radius, rroot0.3 mm Duursma et al. (2008)
Radius of a cylinder of soil to which the root has access to, rcyl4 mm Duursma et al. (2008)
Constant in sugar to starch conversion, M10.001 s−1Based on Escobar-Gutiérrez et al. (1998)
Constant in sugar to starch conversion, M22400 mol m−3Based on Escobar-Gutiérrez et al. (1998)
Constant in starch to sugar conversion, M30.001 s−1Based on Escobar-Gutiérrez et al. (1998)
Sugar unloading coefficient, ω50 × 10−9 mol−1Chosen large enough not to affect resultsc
Period of iteration for stomatal conductance, titer0.45 sValue that maximized the daily cumulative sugar transport under the constraint that phloem turgor pressure was maintained positive
Numerical time step0.05 sThis was chosen small enough to keep the numerical solution stable.
Numerical axial element number, i40This was chosen large enough not to affect the results noticeably.

The sugar–starch dynamics was only included in simulations where their effect was specifically tested. It was not included in the other simulations for simplicity and because it has only minor effect on the results except at low soil water potentials (see Results for explanation).

Phloem loading rate (S) (mol s−1) was made equal to the CO2 assimilation rate. Phloem unloading rate (U) (mol s−1) was made proportional to sugar concentration (Thompson & Holbrook 2003)


where ω (s−1) is a coefficient of proportionality and Csink the sugar concentration at sink (mol m−3). The factor Psink,xyl/RT, xylem water potential at the sink divided by the gas constant and temperature, was included to ensure that the turgor pressure at the sink did not turn negative. This could happen if phloem osmotic concentration at the sink was so low that it resulted in a water potential below xylem water potential, which is unrealistic.

Water uptake from soil

Soil hydraulic conductance (Ks) was modelled based on an empirical relation (Campbell 1974)


where Ksat is saturated conductivity (mol m−1 s−1 Pa−1), ψe is the air entry water potential (Pa), ψs is soil water potential (Pa) and b (unitless) is an empirical coefficient related to the clay content of the soil. The values of the parameters were obtained from Duursma et al. (2008).

The effect of different soil layers and a varying water potential and soil conductivity as a function of distance away from the roots were not considered; instead, soil water potential and soil conductivity were given a spatially constant value for simplicity and also because soil water movement is not our main concern in this study. The water flux between the lowest compartment of tree and soil was modelled as


where ks[kg s−1 Pa−1] is ‘the effective hydraulic conductance for water uptake from roots’ (Duursma et al. 2008). ks is expressed in terms of Ks as


where MH2O is the molar mass of water (0.018 kg mol−1), Rl is root length index (m root m−2 soil surface area), rroot is root radius (m) and rcyl (m)is the radius of a cylinder of soil to which the root can access.

Iteration of the optimal stomatal conductance for maximizing assimilate transport from the leaves

Stomatal conductance was determined so that it would maximize the transport of sugars out of the leaves. The maximized parameter was the phloem sugar concentration times the radial inflow of water to the phloem at sugar source. This gave the same result as the actual sugar outflow but was numerically more stable than the latter. Because the model is dynamic with a time constant of several hours for reaching a steady state after a change in the driving variables (such as light and VPD), an instantaneous maximization of the transport rate of photosynthates will not necessarily correspond to a longer-term maximum. In principle, methods of dynamic control theory (e.g. Luenberger 1979) could be used for solving the dynamic optimum stomatal conductance. However, this was not considered a feasible option because of the complexity of the model. Instead, the problem was analysed in a hierarchical set-up, using the assumption that during a short time step, titer, an ‘instantaneous’ optimization in the leaf is carried out, keeping the boundary conditions constant. The time step length was further optimized to yield maximum daily cumulative transport of sugars out of the leaf. The latter optimization was conducted once for a given set of parameter values subject to the constraint that the turgor pressure in the leaf should remain positive.

For each time step, titer, stomatal conductance was constantly updated to a value which maximized the instantaneous sugar transport rate out of the leaves. The maximization was done iteratively so that different values for stomatal conductance were varied around its previous value for a period of titer seconds. The cumulative sugar flux out of the topmost phloem element was summed for each case of stomatal conductance, and the stomatal conductance which yielded the highest value was chosen for the next titer seconds when the optimization was carried out again to calculate the next stomatal conductance.

The daily cumulative transport of sugars out of the leaf was found to decrease monotonously with increasing time step, titer, because smaller titer always gave larger stomatal conductances (Fig. 1a). However, if titer was very small, the resulting stomatal behaviour led to non-physiological conditions of negative phloem turgor pressure at peak transpiration during the day and to xylem water potentials which were smaller than what can be sustained by the xylem without cavitation (Fig. 1b). In the second step of the optimization, the time step was therefore chosen as the value of titer that maximized the daily cumulative sugar transport under the constraint that phloem turgor pressure was maintained positive at all times.

Figure 1.

Influence of the time step length of local maximization (titer) on cumulative photosynthetic production and transpiration (a) and xylem and phloem water pressure components (b) with five different values for the time step length of local maximization (titer). titer was changed every day, and was given values (in order of progression of days) 0.1125, 0.225, 0.45, 0.9 and 1.8 s. The value used for titer was that which maximized production and maintained turgor pressure positive, that is, 0.45 s.

Modelled tree and boundary conditions

The model tree consisted of a stem that was connected to source leaves at the upper end and sink tissue at the lower end. The stem was discretized radially into xylem, cambium and phloem tissue. A semi-permeable membrane separated each of these tissues from the adjacent one. For the numerical solution, the model tree was divided vertically into 40 elements and xylem water pressure, phloem water pressure and sugar concentration were solved at each height. The phloem loading and unloading took place in the highest and lowest axial segments, respectively (Fig. 2). The model was parameterized according to literature values and measurement results from 50-year-old Scots pine trees (Pinus sylvestris L.) at SMEAR II station in University of Helsinki field station in Hyytiälä, Southern Finland (61°50′ 50.685″, 24°17′ 41.206″, 179 m a.s.l) (Table 1). The boundary conditions for simulations were irradiation, CO2 concentration and water VPD at the leaf surfaces, air temperature and soil water potential.

Figure 2.

Outline of the model tree. Qax and Qrad refer to axial and radial mass flow rate and S and U are assimilate loading and unloading rates in the highest and lowest axial segments, respectively. i index refers to the division of the stem in segments in axial direction and j index the division of stem radially into (1) xylem, (2) cambium and (3) phloem. A and J are rates of photosynthesis and transpirations. The above-ground boundary conditions for the model are water vapour pressure deficit (dw), solar radiation (I), ambient CO2 concentration (Ca) and air temperature (T), and the soil boundary condition is soil water potential (Ψs).

Measurements of leaf gas exchange and the model boundary conditions

The simulation results were compared against continuous measurements of photosynthesis from a Scots pine forest in 2007 at SMEAR II station (Hari & Kulmala 2005). The site was established in 1962 by sowing after prescribed burning and mechanical soil preparation. The soil is a Haplic podzol on glacial till. The stand is dominated by Scots pine (P. sylvestris L.) with sparse understorey of Norway spruce [Picea abies (L.) Karst.] and scattered deciduous trees. The dominant height of the stand in 2006 was 17 m and tree density (diameter at 1.3 m height > 5 cm) was 1400 per ha. The seasonal maximum of the all-sided leaf area index (LAI) was 7 in 2007. The SMEAR II stand is medium fertile Scots pine dominated boreal forests site. The forest floor vegetation consists mostly of blueberry (Vaccinium myrtillus L.) and cowberry (Vaccinium vitis-idaea L.) in the field layer. The ground vegetation consisted mainly of feather moss (Pleurozium schreberi) and other bryophytes.

The photosynthetic production and transpiration were measured continuously with automatic shoot chambers in the light crown that capture well the variation of whole tree sapflow (Duursma et al. 2008). The shoot chambers were acrylic plastic boxes with volume of 1 dm3. The chambers were open most of the time, exposing the chamber interior to the ambient conditions. For measuring fluxes, the chambers were closed intermittently for 1 min. Prior to chamber closure, the air flow was directed to infrared gas analysers (URAS 4, Hartmann & Braun, Frankfurt am Main, Germany) for CO2 and H2O concentration measurements. The gas measurements, air temperature inside the chambers and PAR outside of the chambers, were done 70–100 times a day. During the chamber closure, gas concentrations and environmental variables were recorded every 5 s. The flux calculation was based on the detection of the gas concentration change in the chambers during the closure (Hari et al. 1999). More detailed descriptions of the instrumentation were provided by Altimir et al. (2002).


The model captured the observed variation in photosynthetic production very well both over a day and over several weeks (Fig. 3). The stomatal conductance that gave the highest transport rate of sugars from the leaves followed well-known trends of stomatal conductance against light and vapour pressure (Fig. 4a,b). In low light, relatively closed stomata produced maximum transport rate. This is because low assimilation rate in dim light is not able to produce sufficient osmotic pressure to draw water from the transpiration stream if transpiration rate is high. At high light, assimilation capacity is often sufficient to compete with transpiration even with high VPD.

Figure 3.

Modelled CO2 assimilation rate versus measured CO2 assimilation rate (a), and modelled CO2 assimilation rate without stomatal control (stomata always fully open) versus measured CO2 assimilation rate (b). Data are shown a period of 3 d in (22–24) June 2007. (c) and (d) are same as (a) and (b), but for a period between 22 June and 22 July 2007.

Figure 4.

Stomatal conductance as a function of light (a) and vapour pressure deficit (VPD) (b), its daily pattern assuming sinusoidal light and temperature development with varying xylem conductance (kx) (m2) (c), (base case solid black, halved xylem conductance dark grey and doubled phloem conductance light grey), and with varying phloem conductance (kp) (d) (base case solid black, one-tenth phloem conductance light grey and doubled phloem conductance dark grey), as a function of soil water potential (e) and comparison of measured (solid black), modelled (grey) and modelled without stomatal control (dashed grey) net assimilation rate for 3 d (22–24) in June 2007 at Hyytiälä SMEAR II station (f).

When simulated against the diurnal variation in light, temperature and VPD, a typical daily pattern of stomatal conductance was observed (Fig. 4c). The model also predicted stomatal closure and decrease in transpiration rate in feed-forward fashion in high light conditions that has been difficult to explain with previous approaches (Buckley 2005).

With ample water supply, the phloem transport with used parameterization operated close to turgor maintenance limit. Lowering the soil water potential initially only caused a small change in stomatal conductance, but this was followed by a rapid drop as the soil water potential was reduced further (Fig. 4e), in line with observation (Duursma et al. 2008). Drying of soil increases the xylem water tension next to the transpiring surfaces if transpiration rate remains constant. Hydraulic coupling between xylem and phloem requires that more sugars are needed along the stem to maintain turgor pressure if xylem water potential drops. High phloem sugar concentration makes the system vulnerable to viscosity build-up and down-regulation of photosynthesis which both slow down phloem transport. In such conditions, higher sugar transport is obtained with smaller stomatal opening as the latter prevents xylem tension from rising too high. It was only in these simulations with drying soil that the down-regulation of photosynthetic rate with high sugar concentrations influenced the simulation results.

When phloem conductivity was low relative to assimilation capacity, simulations predicted hysteresis in the response of stomatal conductance against VPD (Fig. 4b). This has been observed in the field measurements of photosynthesis and attributed to diurnal changes in soil hydraulic conductivity (Schulze & Hall 1982). In our simulations, higher leaf sugar concentration in the afternoon caused the stomata to be more closed. Supporting our hypothesis, observations have shown that removal of hydraulic limitation by root pressurizing has smaller influence on stomata in the afternoon than in the morning (Mencuccini, Mambelli & Comstock 2000), suggesting that perhaps the limitation was in phloem transport and sugar accumulation in leaves rather than water transport from the roots. The stomatal opening that resulted in the highest assimilate transport rate in the phloem leads to realistic daily variation (minimum between −1.8 to 2.2 MPa Hölttäet al. 2006) in the shoot water potential (Fig. 5).

Figure 5.

Phloem osmotic pressure, phloem turgor pressure and xylem water pressure at the source (a), and sink (b). Phloem sugar flux rate at the source, middle of the tree and sink (c). Results are for the period 22 June–1 July 2007.

Decreasing the xylem and phloem conductivity decreased the stomatal conductivity (Fig. 4c,d). However, with our parameterization, for a young boreal Scots pine tree, a large decrease in phloem conductivity was required before any influence was seen. Changing xylem conductance affected stomatal conductivity more than phloem conductance, but changes in stomatal conductance were still smaller than changes in xylem conductance (Fig. 4c). This is because the decrease in leaf xylem water potential (Fig. 6a) was compensated by increasing sugar concentrations in the leaf phloem (Fig. 6b) as explained previously. If the sugar concentrations had been considerably higher than in Fig. 4c, as could be with lower soil water potential, the model would predict an approximately linear change in stomatal conductance as a function of xylem conductance. A similar logic follows for the changes in phloem conductance: Phloem conductance affects the stomatal conductance very little, as the ‘operating’ sugar concentration merely increases with decreasing phloem conductance (Fig. 6c) when there is sufficient phloem transport capacity. But, at a threshold when phloem sugar concentration is high and phloem conductance is low, a further decrease in phloem conductance will have a large effect on stomatal conductance as due to viscosity build-up it will lead to sugar accumulation in leaves and down-regulation of photosynthetic capacity.

Figure 6.

Xylem water potential (a) and phloem osmotic concentration (b) as xylem hydraulic conductance (kx) is varied, and modelled phloem osmotic concentration (c) as phloem hydraulic conductance (kp) is varied. (The same model run as in Fig. 4c,d).

The inclusion of the starch stores at the needles somewhat influences the diurnal dynamics of the stomatal behaviour. If sugar concentrations are low, as in the base case (Fig. 7a), then sugar turnover to starch reduces phloem transport and therefore also stomatal conductance because the sugar which is turned to starch during the day is not used to create the osmotic force to drive phloem transport. On the other hand, if sugar concentrations are high, as in Fig. 7b where soil water potential has decreased to −0.6 MPa, the starch conversion will allow for the stomata to remain more open during the day as this prevents the phloem sugar concentration from increasing to values where they down-regulate photosynthesis and/or increase phloem viscosity.

Figure 7.

Diurnal stomatal conductance with and without sugar stores (a), and diurnal sugar concentration with and without sugar stores (c) together with the diurnal pattern in the amount of starch in storarage (soil water potential at −0.1 MPa). (b) and (d) show same for a case where soil water potential is −0.6 MPa. Starch amounts are shown in molar equivalents of sugar.


Our study suggests that the control of gas exchange in leaves is very tightly connected to the transport rate of assimilates in the phloem. Because sugar storage in the leaves is limited, photosynthesis rate needs to be sensitive to increasing sugar concentrations and it seems that stomatal opening responds to that. Phloem transport rate is determined by the hydrostatic tension of water in leaf xylem and osmotic pressure in leaf phloem as these together set the limits for the turgor pressure driving phloem transport. Stomatal conductance influences both hydrostatic and osmotic pressures and thus influences the phloem transport rate. If phloem osmotic strength is low, as could be when assimilation rate is low, transpiration rate has to remain low as higher rates would result in too low xylem water potentials for maintenance of sufficient turgor pressure in phloem. When osmotic strength can be maintained high, as could be when assimilation rate is high, also transpiration rate can be high. On the other hand, very high sugar concentration in the phloem sap will lead to high sap viscosity and low transport and hence down-regulation of photosynthesis. In these conditions, again, smaller stomatal opening results in higher assimilation transport rate in the phloem. The overall approach does not depend on the actual loading mechanisms as long as some proportionality exists between the photosynthetic rate and loading rate as has been suggested (e.g. Ainsworth & Bush 2011). This is illustrated in Fig. 7 where the overall pattern of stomatal response remains the same with and without leaf sugar stores. In passive loaders, this connection is obvious, but also in apoplastic loaders negative hydrostatic water potential has been shown to increase phloem loading rates (Smith & Milburn 1980a). Also, down-regulation of sucrose carriers has been observed with sucrose accumulation as would follow with lowered hydrostatic pressure with consequent down-regulation of photosynthetic rate (Vaughn, Harrington & Bush 2002; Brodribb 2009) which both agree with our model assumptions.

Previously, stomatal closure has been attributed to hydraulic limitation of the xylem (Ryan & Yoder 1997; Buckley 2005). However, in girdling experiments that stop phloem transport, stomatal closure has been observed along with down-regulation of photosynthesis (Setter, Brun & Brenner 1980, Goldschmidt & Huber 1992, Nebauer et al. 2011). Also, carbon isotope discrimination studies have shown that leaf gas exchange and the isotope signal from the phloem transport are tightly linked (Keitel et al. 2003). Schulze (1993) suggested that maintenance of a sufficiently high turgor gradient at all times is an important mechanism in plant hydraulics. This can be achieved with stomatal regulation that controls both the tension gradient in xylem and the osmotic strength of the phloem sap. The obvious need for coordination between source and sink reactions and the hydraulic connection of the source reactions and long-distance transport have also lead other people to hypothesize about the connection of stomatal control and phloem transport (Franks, Drake & Froend 2007). Here we have shown that it indeed is better able to explain number of features of stomatal control than previous formulations.

In conditions that lead to sugar accumulation in leaves, the numerical solution of the model very easily leads to rapid viscosity build-up and unstable behaviour. More stable behaviour was obtained when down-regulation of photosynthetic rate was included as a response to sugar concentration. Such down-regulation of photosynthesis would be expected with viscosity increase due to sugar concentration as that would directly influence the mesophyll conductance for CO2. Down-regulation has also been observed in a number of studies (e.g. Goldschmidt & Huber 1992, Ainsworth & Bush 2011; Nebauer et al. 2011) but the signal to which down-regulation responds is not yet understood. Nebauer et al. (2011) showed that the down-regulation occurred prior to any sugar build-up in the leaf once the end-product usage started to slow down. Such a mechanism would guarantee that no harmful accumulation of carbohydrates would take place in the phloem. For apoplastic loaders, Vaughn et al. (2002) suggested that lowered loading rate with sugar accumulation in phloem would lead to such behaviour. The numerical sensitivity of the model to viscosity build-up even if the stomata were responsive to it, suggests that it might be a problem to real plants that are growing in conditions of high evaporative demand and photosynthetic rate such as those studied by Nebauer et al. (2011). However, as Goldschmidt & Huber (1992) showed, the down-regulation response to carbohydrates may be very different depending on species. Different loading mechanisms [e.g. apoplastic as in many herbaceous species versus symplastic down the concentration gradient as in many trees (Turgeon 2010; Liesche et al. 2011)] would yield most likely quantitatively a different outcome as the route of getting the sugars to the vascular tissue is different. However, the overall behaviour should not be very different, as with all mechanisms, high viscosity in sieve cells or elements is not beneficial. Suggested tight coordination between sink and source activities (Ainsworth & Bush 2011) would also point out to this direction.

When phloem transport is used as a criterion in stomatal control, the leaf productivity becomes sensitive also to starch storing capacity. Similarly to actual loading mechanism, this influences the connection between the photosynthetic production and sugar loading into transporting phloem. In our base case, we assumed direct linking between the rate of photosynthetic production and the loading rate, which resembles most closely the loading mechanism of many trees, where sugars are transported from mesophyll to phloem down the sugar concentration gradient. The simulations where we added a separate storage pool that was linked to sugar concentration, reproduced observed enhancement of photosynthetic production in high productivity conditions when the starch storing capacity is high (Ludewig et al. 1998). When the leaf sugar concentration is high, turnover to starch allows the stomata to be more open as phloem sap viscosity remains smaller and there is less down-regulation of photosynthesis.

At constant soil water availability, the modelled results were almost identical to the ones obtained optimizing carbon uptake with respect to water (e.g. Mäkeläet al. 1996; Hari & Mäkelä 2003). This approach and the semi-empirical approach of, for example, Ball et al. (1987), are currently the most used approaches to model stomatal behaviour, and have been recently shown to produce virtually identical results (Medlyn et al. 2011). These approaches agree with observations but, unlike our model, they do not suggest a functional linkage with tree structure or soil properties. Also, Tuzet, Perrier & Leuning (2003) suggested explicit linking between the soil–plant–water continuum and leaf water potential, but they did not include explicit linking with photosynthesis. Instead, they used leaf internal CO2 concentration. Different from any previous approach, our new approach explains influences of soil drying and tree structure on carbon uptake and it does not require a priori determination of the marginal cost of water as the optimal stomatal control approaches do. The regulation of turgor strength at the source that drives phloem transport combines both the hydraulic influences and the source–sink interaction. Thus, the stomata need to regulate the water loss more tightly the lower the soil water potential is. Because of this, our approach also coincides with the suggested drought-induced accelerated mortality of carbohydrate-starved trees (Sala 2009) as in such conditions trees are faced with problems of maintaining the phloem transport because they are lacking the capacity to build up sufficient cell turgor in leaves.

Plants are often divided into groups of isohydric or anisohydric behaviour depending if the minimum leaf water potential remains constant or decreases as soil dries (Franks et al. 2007). Qualitatively, our model would bring out both behaviours (results not shown) due to the physical boundary conditions that parallel xylem and phloem transport set on assimilate transport. When the soil is drying, the sugar concentration in the root phloem needs to grow to match the lower water potential in the root xylem and soil in order to maintain turgor pressure. This means that things being equal in the leaves, the maximum turgor pressure gradient between the leaves and the roots decreases, which slows down the sugar transport and leads to gradual sugar build-up in the leaves. If the transpiration rate remains unaltered, the water potential of the leaves drops as soil gets drier. If this drop is larger than the increase in sugar concentration, leaves need to close the stomata to maintain the hydraulic balance between xylem and phloem. However, if the photosynthetic rate is large enough, then the osmotic strength of phloem solution in leaves can match the drop in hydrostatic pressure and the leaf water potential can drop until the viscosity limit is reached. As Franks et al. (2007) stated, anisohydric plants tend to be quite small with high photosynthetic capacity. Such plants in our simulations would indeed behave in anisohydric way as the soil drying has much stronger influence on the driving turgor gradient in small than in large trees, leading, together with high assimilation rate, to rapid increase in leaf sugar concentration. To take anisohydry to extremes, our model also suggests that these plants should be able to store carbohydrates as starch in the leaves to avoid the viscosity build-up. Indeed, there seems to be also an overall trend from herbs to trees in their loading mechanism that suggests greater degree of decoupling between photosynthetic rate and loading rate in herbs (Turgeon 2010). The benefit of anisohydric behaviour in contrast to isohydric behaviour is that the plant can maintain carbohydrate translocation to other parts of the plant, such as roots, even during drought.

For the sake of simplicity, we considered that assimilates are transported as sucrose, although in reality a number of other compounds are transported, in particular in the phloem of trees. Although the concentration–viscosity relationship of various transported sugars is different from that of sucrose, in high concentration they all will cause lowered transport (e.g. Lang 1978) and qualitatively similar behaviour as sucrose. In our simulations with Scots pine, stomatal closure was due to maintaining sufficient turgor pressure gradient in phloem with relatively low solute concentrations where viscosity does not play a big role. In addition, our model did not consider potassium which has been suggested to have important role in phloem transport (Lang 1983; Vreugdenhil 1985; Thompson & Holbrook 2003) as it does not increase sap viscosity even in high concentration. However, the viscosity problem cannot be completely avoided as sugars need to be transported from the source to the sink in any case. Potassium merely increases the turgor pressure and its gradient by drawing in water to the phloem without increasing the solution viscosity. In our approach, this would mean more open stomata in high productivity conditions. Potassium increases the turgor pressure gradient in the phloem by the same amount as its fraction is in the phloem sap. If this varies, it could have an influence on the simulated response patterns. Particularly, with low sucrose concentrations, the role of potassium has suggested to be large (Smith & Milburn 1980b; Thompson & Zwieniecki 2005). Increasing potassium loading in our approach would again allow the stomata to remain more open also in low productivity conditions.

Closing of stomata is shown to be linked to abscisic acid (ABA) concentration and production of reactive oxygen species (ROS) in the guard cells (Wang & Song 2008). It is suggested that leaf-produced ABA would have an important role in leaf response to water deficit and that some of the leaf-produced ABA is actually moving to roots (Ikegami et al. 2009). ABA and photoassimilates have been observed to be transported in a similar manner from leaves (Hoad 1995), their accumulation has provoked stomatal closure (Hoad 1995) and ABA has also been suggested to influence unloading rates at carbohydrate sinks (Hoad 1995). Combination of bulk flow, leaf-produced ABA and dependence of stomatal responsiveness to ABA and ROS in guard cells could thus offer a mechanistic explanation for the modelled linkage between stomatal control and assimilate transport.

Our model suggests that the water use efficiency of leaves is related to whole tree assimilate transport. Many known features of stomatal behaviour follow from the requirement that assimilate transport rate is maintained as high as possible under the physical boundary conditions of xylem and phloem transport. The numerical results were also almost identical to those with optimal stomatal control principle. The wide usability of the latter or similar approaches (Medlyn et al. 2011) suggests that our approach has potential in further improving our understanding on tree water relations. It links the leaf level behaviour to tree structure and is based on directly measurable tree properties and process rates, and it connects the known responses of leaf gas exchange to above- and below-ground environmental variables. Due to these features, the approach can be readily improved by including more accurate description of physiology, particularly that of carbohydrate storage and loading, and leaf and whole tree structure. The numerical test against observations showed that quite good correspondence can be achieved. In that sense, it is a promising new hypothesis for studies of whole-tree behaviour and forest–atmosphere interactions.


The work used the data acquired by the FCoE ‘Physics, Chemistry, Biology and Meteorology of Atmospheric Composition and Climate Change’, Integrated Carbon Observation System (ICOS), Instrumentation for Measuring European Carbon Cycle (IMECC) and Greenhouse gas management in European land use systems (NitroEurope). We acknowledge the support received to this work from Nordic Centres of Excellence CRAICC and Defrost, Vulnerability assessment of ecosystem services for climate change impacts and adaptation (VACCIA), and Finnish Academy projects #1132561 and 124531.



Xylem and phloem transport model

Axial water flow rate for each xylem and phloem element was calculated using Darcy's law


where Qi,j,in,ax is the axial inflow of water [m3 s−1] from element i-1 to i, k is permeability [m2], µ is viscosity [Pa−1 s−1], P is pressure [Pa], l is the distance between the element midpoints [m], ρ is density of water (1000 kg m−3), g is the gravitational constant (9.81 ms−1) and A is the cross-sectional area [m2]. i refers to axial elements and j refers to radial compartment (j = 1 is the xylem, j = 2 is the cambium, and j = 3 is the phloem). The subscript x/p refers to either xylem or phloem. Viscosity was made a function of sucrose concentration using the formula as in Hölttäet al. (2006). The axial outflow rate from an element is


The radial water exchange between adjacent components is


where L is the radial hydraulic conductance [mPa−1 s−1], C is sugar concentration, R is the molar gas constant (8.314 J mol−1 K−1), T is temperature [K] and Arad is the cross-sectional area in radial direction [m2]. The radial outflow rate from an element is


A mass balance equation is then written for each element


where m is water mass [kg] and t is time [s]. Finally, Hooke's law relates the change of pressure to change in water content (Irvine & Grace 1997)


where E is the elastic modulus [Pa], the subscript x/c/p refers to either xylem, cambium or phloem, and V is volume [m3].

In the phloem, sugars are transported by advection along with the phloem sap so that the change in the sugar concentration in a phloem element is written as


where S and U denote a source or sink of sugar. Outflow from the topmost xylem element equals the transpiration rate and the bottommost xylem element exchanges water with the soil, that is, Pi−1, is equal to soil water potential. Sugar loading occurs at the top-most phloem numerical element representing the leaves, and unloading at the bottommost numerical element representing the roots. Sugar–starch dynamics was only considered for the top-most phloem element. No radial movement of sugar from one component (xylem, phloem, cambium) to another was allowed.


List of parameters and variables used in the equations in alphabetical order:

A Photosynthetic rateg CO2 m−2 s−1
A 0 Photosynthetic rate (in the absence of down-regulation)g CO2 m−2 s−1
A c Cambium cross-sectional aream2
A leaf Leaf aream2
A p Phloem cross-sectional aream2
A rad Radial cross-sectional area between adjacent components (i.e. xylem to cambium to phloem)m2
A x Xylem sapwood cross-sectional aream2
b Empirical coefficient related to the clay content of the soilUnitless
C a Ambient CO2 concentrationppm
C leaf Sugar concentration in leafmol m−3
C sink Sugar concentration at sinkmol m−3
E c Cambium elastic modulusPa
E p Phloem elastic modulusPa
E x Xylem elastic modulusPa
g Gravitational constantm2 s−1
h Tree height hm
I Photosynthetically active radiation (PAR) µmol m−2 s−1
i Axial numerical element number 
J Transpiration rateg m−2 s−1
j Radial compartment (j = 1 is the xylem, j = 2 is the cambium, and j = 3 is the phloem) 
K s Soil hydraulic conductancemol m−1 s−1 Pa−1
K sat Saturated soil hydraulic conductancemol m−1 s−1 Pa−1
k p Phloem axial permeability,m2
k s The effective hydraulic conductance for water uptake from rootskg s−1 Pa−1
k x Xylem axial permeabilitym2
L Radial hydraulic conductancemPa−1 s−1
l Distance between the element midpointsm
M H2O Molar mass of waterkg mol−1
M 1 Constant in sugar to starch conversions−1
M 2 Constant in sugar to starch conversionmol m−3
M 3 Constant in sugar to starch conversions−1
m Water masskg
P Water pressurePa
Q i,j,in,ax Axial inflow of water from element i − 1 to i in radial element jm3 s−1
Q i,j,out,ax Axial outflow rate from an element i to i + 1 in radial element jm3 s−1
Q i,j,in,rad Radial inflow of water from element j − 1 to j in axial element im3 s−1
Q i,j,out,rad Radial outflow of water from element j to j + 1 in axial element im3 s−1
R Molar gas constantkg mol−1 K−1
R l Root length indexm−1
R leaf Leaf respiration rateg CO2 m−2 s−1
r root Root radiusmm
r cyl Radius of a cylinder of soil to which the root has access tomm
S Phloem loading ratemol s−1
S starch Sugar to starch conversion ratemol s−1
S sugar Starch to sugar conversion ratemol s−1
T TemperatureK
t Times
t iter Period of iteration for stomatal conductances
U Phloem unloading ratemol s−1
V Volume of the elementm3
Ψ s Soil water potentialPa
Ψ e Air entry water potential for soilPa
Ψ Water potentialPa
α Photosynthesis down-regulation parametermol m−3
β Photosynthesis parameterms−1
γ Photosynthesis parameter µmol m−2 s−1
δ Photosynthesis down-regulation parametermol m−3
µ Dynamic viscosity of waterPa−1 s−1
ρ Density of waterkg m−3
ω Sugar unloading coefficientmol−1