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It has been known for some time that small but easily detectable changes in tree stem and branch diameter reflect variations in plant water status (Klepper et al., 1971; Simonneau et al., 1993; Zweifel et al., 2001; Steppe et al., 2006), as the volumes of living cells (Nobel, 1991) and xylem conduits (Irvine & Grace, 1997) vary according to the pressures to which they are subjected. Cambial growth can also be clearly distinguished in the stem diameter measurements (Génard et al., 2001; Steppe et al., 2006). The changes in cell and xylem conduit pressure arise mainly from changes in the transpiration rate, which determine the water potential of the xylem (Perämäki et al., 2001). The water potentials in the extra-xylary tissues largely follow the water potential variation of the xylem (Sevanto et al., 2002), as water potentials tend to equilibrate within the tree with time scales of minutes to several hours. The elastic portion of the bark, the functional phloem and the cambium are the main extra-xylary tissues that contribute to the whole-tree stem diameter variations (De Schepper et al., 2012).
By assuming a linear relationship between tissue volume and pressure, that is, Hooke's law (Nobel, 1991), the changes in inner bark (cambium and phloem) and xylem pressure can be calculated from the measured xylem diameter changes (Perämäki et al., 2001). If thickness variations of the inner bark were driven only by changes in xylem water potential, the inner bark thickness, discarding cambial growth, would be expected to follow changes in xylem diameter in a predictable manner (Sevanto et al., 2003). We should then be able to predict its movements on the basis of the hydraulic coupling between the xylem and the inner bark and the elastic properties of the tissues (Sevanto et al., 2011). This prediction is largely confirmed by measurements (Zweifel et al., 2001; Sevanto et al., 2003; Steppe et al., 2006), but changes in xylem water potential alone cannot explain completely the diurnal changes in stem diameter (Sevanto et al., 2002, 2003, 2011). This has led to the hypothesis that phloem sugar dynamics might also influence inner bark thickness variations (Sevanto et al., 2003). However, no theoretical framework exists to infer sugar dynamics in the phloem region from thickness change measurements of the inner bark.
In this study, we develop a model that allows the determination of this putative signal generated in the phloem region, and use this model to extract the turgor/osmotic signal from simultaneous measurements of xylem and inner bark thickness variations. We link the inferred changes in this signal throughout one growing season with the measured gross primary production (GPP) determined by concurrent eddy covariance measurements. Finally, we test whether a link between GPP, tree radial growth, sugar loading and unloading, and phloem translocation can be established in our boreal pine forest.
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Figure 1. Diagram illustrating the main geometrical features of our measurement system. (a) Representation of a stylized tree, illustrating the two heights at which the two pairs of LVDTs (linear variable displacement transducers) were placed. (b) Cross-sectional view of the geometrical arrangement of the system. The metal frame (depicted in grey) is held around the trunk by means of a metal plate (not depicted), screwed in position above the section shown in the figure (see figure in Sevanto et al. (2005) for a diagram illustrating this arrangement). The system coordinates are defined by the black screw opposite to the position of the LVDT (left-hand side of figure), which provides a fixed starting point for the measurements by being fixed directly into the tree xylem. A similar screw is also employed below one of the LVDTs on the right-hand side to obtain direct contact with the xylem tissue without any further debarking damage. On its side, a second LVDT is placed in direct contact with the inner bark after carefully cutting away the dead bark. The bark diameter is obtained by difference between the two readings of xylem and xylem plus bark diameter made by the two LVDTs. (c) Readings are carried out for subsequent intervals. In this case, at time t = 2, both bark and xylem have shrunk to some degree. The LVDTs provide directly the reading of the diameter and thickness changes dDx and dDb. The final quantities employed in the analysis (cf. main text), however, are ΔDx and ΔDb. They are obtained by subtracting the diameter changes measured on the midnight of the first day of measurement (i.e. and ) from all subsequent measurements in the time series. See text for further details.
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Under conditions of slow, elastic changes of small magnitude, the changes in xylem diameter dDx over a time interval dt reflect changes in xylem water pressure dPx (MPa) according to Hooke's law (Irvine & Grace, 1997; Perämäki et al., 2001):
- (Eqn 1)
(, xylem diameter (m) at a reference pressure; Er,x (MPa), radial elastic modulus of the xylem tissue). The same can be written for the inner bark thickness Db:
- (Eqn 2)
(dPb (MPa), change in the pressure (tissue-averaged turgor pressure) of the live inner bark; (m), inner bark thickness at a reference pressure; Er,b (MPa), radial elastic modulus of the inner bark tissue). The inner bark tends towards water potential equilibrium with the xylem with a time scale dependent on the radial resistance between the two (Sevanto et al., 2011). Here, we treated and as two model parameters, measured at a reference pressure, although, strictly speaking, Hooke's law requires the use of the actual diameter/thickness at each time interval in the numerator of Eqns 1 and 2. In practice, this is not an issue, as the absolute changes caused by the shrinkage and swelling are extremely small relative to the xylem diameter and bark thickness (cf. Fig. 1), and the approximation of constant values holds.
Assuming a negligible osmotic content in the xylem, the water flux J (m3 s−1) between the xylem and phloem can be written as:
- (Eqn 3)
(L (m MPa−1 s−1), hydraulic conductance of the cross-sectional area A (m2) of contact between the xylem and phloem through which radial water exchange takes place; Π (MPa), osmotic pressure of the inner bark).
We modelled the inner bark tissue as if it acted only as a dynamic water reservoir for the xylem with constant osmotic content, and then used the residuals from the predictions of this model against the observed bark thickness changes to obtain an estimate of the variations in the osmotic content. In other words, we calculated how the inner bark thickness would behave if no changes in bark osmotic concentration (which are primarily caused by phloem transport) took place over time. The difference between the inner bark thickness measured and that which was predicted on this basis should contain a turgor signal caused by the changes in osmotic concentrations taking place in the inner bark. This approach avoids the problem of having to formulate hypotheses on the unknown patterns of phloem osmotic transport and focuses on modelling the known processes.
An additional potential problem is that the bark will also change its thickness passively, simply as a result of the geometrical stretching caused by xylem diameter changes. This effect will occur independently and simultaneously to the changes in turgor and osmotic relationships of the bark, as a consequence of the changes in xylem diameter. For example, if the xylem swells by a certain amount, the bark thickness will be expected to shrink correspondingly to keep the bark volume constant (cf. diagram in Fig. S1). However, the magnitude of this geometrical effect is small (in our case, it is between 2% and 8% of the measured xylem diameter changes) and can therefore be discarded. The demonstration is given in Supporting Information Notes S1.
The expression of the bark thickness changes as a function of the pressure of the xylem and the turgor and osmotic pressures of bark leads to the following expression for (see Notes S2 for derivation):
- (Eqn 4)
γ is a constant representing the bark thickness change that occurs at the reference time (which is taken to be the first midnight of measurements) as a consequence of water potential disequilibrium between bark and xylem, Π* and are the reference osmotic pressure and the reference bark volume (also taken at the first midnight of measurement), respectively, and ΔDx and ΔDb are the changes in xylem diameter and bark thickness, respectively, relative to the reference measurement at the first midnight, as defined above. The reference state at the first midnight of the time series is used for practical convenience and has no bearing on model fitting or on the interpretation of the parameters from Eqn 4, and its choice has only a minute effect on the calculated values of the biophysical parameters (cf. Notes S2). In addition, τ is a time constant (i.e. α is measured in s−1), which reflects the velocity of the radial pressure propagation between xylem and inner bark. By contrast, β can be interpreted as the ratio of the changes in bark thickness to xylem diameter for a given change in xylem water potential. Parameters α and β are positive, because, in a realistic physiological range, > 0. In addition, the value of parameter β should be much higher at the top of a tree, because it depends on the ratio of the inner bark thickness to the sapwood xylem diameter, which is much larger there.
In practice, the model fitting is performed using measurements collected over a finite time period (in our case, 30 min), which means that the fitting estimates parameters and , averages over 30-min periods and with corresponding different units. The equivalent parameters from Eqn 4 can be easily derived. The bark thickness at time (t + dt) is then estimated from the previous thickness at time (t) and the predicted thickness change (cf. Notes S3 for more details on the fitting procedure):
- (Eqn 5)
where the caret is employed to emphasize the nature of the predicted variable for the bark thickness. Therefore, one predicts , at the end of the first interval after midnight, from the measured ΔDb(t = 0) and ΔDx(t = 0) (both set to zero at t = 0) using the predicted values of , β and . Once has been predicted, predictions for the subsequent 30-min intervals are obtained by iteration of the same procedure using the predicted bark thickness, and β. The difference between the measured (ΔDb) and the predicted () change in inner bark thickness gives the change in the inner bark thickness caused by the change in turgor of the bark region that is not explained by the xylem water potential:
- (Eqn 6)
By differentiating with respect to the measured values in Eqn 6, one confounds the estimate of the turgor term with the residual error in the time series. Assuming the time series contains only uncorrelated random errors (i.e. white noise), Eqn 6 can be used to obtain the changes in the osmotic pressure of the inner bark which are caused only by the change in the number of moles of solutes (i.e. ΔΠsol):
- (Eqn 7)
(ns, number of moles of solute that are being diluted into volume V; R and T, molar gas constant (8.314 J K−1 mol−1) and temperature (K), respectively). If α is obtained by fitting the model to the data, the radial hydraulic conductance L can be calculated:
- (Eqn 8)
Assuming that information is available on the xylem elastic modulus and diameters, one can invert Eqn 4 for β and substitute into Eqn 8, that is:
The confidence intervals in the estimates of radial hydraulic conductance can be obtained by propagating the error terms contained in all the measured variables and coefficients employed in the calculations using standard derivatives calculus and appropriate variance estimates (Hughes & Hase, 2010).
Empirical field measurements
Simultaneous xylem and inner bark thickness measurements were conducted from 28 June to 4 October 2006 at the Helsinki University research station in Hyytiälä, southern Finland on three 47-yr old, 15-m-tall, Scots pine (Pinus sylvestris L.) trees. A general analysis of the behaviour of earlier time series from the same forest can be found in Sevanto et al. (2002, 2003). Here, we concentrate on testing the proposed theory.
The measurement system on each tree consisted of two pairs of pen-like linear displacement differential transducers (LVDT; Solartron AX/5.0/S, Solartron Inc., Bognor Regis, West Sussex, UK) attached next to each other onto two rectangular metal frames mounted around the stem on the opposite side from the transducers at two different heights (1.5 and 10 m; cf. Sevanto et al., 2002, 2005 for further details and additional drawings of the system). For each pair, one of the sensors was mounted onto the smooth inner live bark, whereas the other was mounted onto xylem tissue, each pair providing diameter data every 30 min. The xylem sensor yielded direct measurements of xylem shrinkage and swelling, whereas the net inner bark thickness change was calculated by subtracting the xylem diameter changes from the stem diameter changes for each time interval (cf. Fig. 1). Air, sapwood (at 1 cm depth into the wood) and metal frame temperatures were available from copper-constantan thermocouples, and the thermal expansion properties of wood and frame were considered (cf. Sevanto et al., 2005). The data were collected once per minute and averaged over 30 min. Stem diameters were c. 15 cm (at 1.5 m height) and c. 5 cm (at 10 m height), and the inner bark thickness was c. 2 mm at both heights for the three trees.
Before subsequent analyses, the time series of inner bark thickness was de-trended to subtract the seasonal signal of cambial growth (Zweifel et al., 2001) using piecewise linear regression between visually identified break points in growth. This procedure gave acceptable results for our time series, although more sophisticated approaches are possible (Zweifel et al., 2001).
Evapotranspiration (ET) was measured and ecosystem GPP (an estimator for canopy photosynthesis) and total ecosystem respiration (TER) were inferred from half-hourly measurements of net ecosystem exchange (NEE) measured with a three-dimensional sonic anemometer (R3IA; Gill Instruments Ltd, Lymington, UK) and a closed-path CO2/H2O infrared gas analyser (LI6262; Li-Cor Inc., Lincoln, NE, USA) installed above the stand at a height of 23 m. The instrumentation is documented in more detail in Vesala et al. (2005). Gap filling, quality control of ET, NEE values and calculation of GPP followed standardized protocols (Reichstein et al., 2005; Papale et al., 2006).
Equation 4 was modelled using nonlinear mixed effects modelling (Pinheiro & Bates, 2000; Gelman & Hill, 2007) employing the NLME library (version 3.1-96; Pinheiro & Bates, 2000) in R 2.9.0. This allowed the representation of parameter α of Eqn 4 as a random variable estimated separately for each day of the year, which allowed the examination of the possibility that the radial hydraulic conductance varied from day to day (Steppe et al., 2011) (see Notes S3 for further details on model structure and results of preliminary analyses).
As no alternative empirical method is available to determine the turgor properties in the field at the spatial and temporal scales of our measurements, we validated our approach using model simulations. We employed the finite-element model of Hölttä et al. (2006) to perform an initial validation of the predictions of the model presented here. The more complex finite-element model was initialized from assumed daily cycles of canopy photosynthesis and transpiration. It predicted the magnitude of the changes occurring in xylem and bark thickness and the likely timing of the transfer of sucrose and of pressure–concentration waves (sensu Thompson & Holbrook, 2004) from the canopy to the two points of measurement down the trunk. In the finite-element model, we also varied the numerical values of L, the radial hydraulic conductance between xylem and bark, and of Er,b, the elastic modulus of the bark. We then ran this model on the subsequent time series of bark and xylem thicknesses and verified whether we could retrieve the original parameter values.
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It is generally accepted that diurnal variations in inner bark thickness can be understood on the basis of tissue elasticity and water exchange with the xylem, as outlined in our modelling section (Irvine & Grace, 1997; Perämäki et al., 2001; Zweifel et al., 2001; Steppe et al., 2006; De Schepper & Steppe, 2010). Our approach shows that the bark thickness changes are not entirely attributable to changes in xylem tension, and that an additional component must be invoked to explain the observations. We interpreted this additional signal as arising from the changes in inner bark osmotic concentration that could be related to phloem loading, transport and unloading of sugars.
In the absence of an alternative method to determine changes in phloem turgor and osmotic concentration at the time and spatial scale at which we worked, we employed a modelling validation approach. Our validation provided an important confirmation that the model presented here could retrieve the original parameter values. We also demonstrated that the discrepancy of c. 10–15% between the imposed and retrieved parameter values occurred because of fast-travelling turgor waves in the phloem. In other words, this model strictly yields the part of the bark thickness change that is not explained by changes in xylem water potential. However, phloem turgor pressure (and therefore partially also bark thickness) also responds to fast-travelling pressure waves caused by a change in sugar loading (Thompson & Holbrook, 2004; Mencuccini & Hölttä, 2010a).
Interpretation of model parameters
The parameter α (the inverse of the time constant for pressure propagation between the xylem and phloem) varied significantly over the course of the season, particularly in response to changes in air temperature (Fig. 5). It also appeared to decline significantly following the sudden rainfall events that broke the drought periods (Fig. 4). Its absolute value varied between 3.61 × 10−5 and 3.31 × 10−4 s−1, suggesting a time constant for the pressure transfer from xylem to phloem on the order of 1 to c. 5 h, much slower than the speed of sound, but as expected from previously reported values of xylem–phloem resistance (Génard et al., 2001; Sevanto et al., 2011). The radial conductance between the xylem and phloem, the main variable controlling α, is at least partly controlled by aquaporin activity (Martre et al., 2001), which might be expected to vary over space and time (Sevanto et al., 2011; Steppe et al., 2011), particularly in response to the need to regulate turgor in phloem cells following sudden perturbations of plant water potential. Our analysis, however, accounted for these dynamics, at least at the daily time scale, because α was allowed to vary freely from day to day. Conversely, we found no evidence of short-term (night-time vs mornings and afternoons) variability in α. We also failed to detect a significant temperature effect on α within a day (i.e. the coefficient for temperature was either nonsignificant or had a very small numerical value). However, when we calculated the temperature dependence of α based on its day-to-day variability in relation to the mean daily temperature, we obtained Q10 values varying between 1.15 and 1.86, with the value at the top of the tree being very close to the temperature dependence of the water viscosity.
Our measurements at the two heights were also consistent with other theoretical predictions, that is, values of β were much higher at the upper position (Table 1), by a factor of c. 3.6 on average, which corresponded well with the ratio of the xylem diameter to phloem thickness at the two heights, which averaged 3.0 (cf. Eqn 4). In addition, the values estimated for the radial hydraulic conductance L and for the bark elastic modulus for the three trees using Eqn 4 were of the same order of magnitude as found in previous studies, that is (4.65 ± 0.41) × 10−8 and (3.64 ± 1.99) × 10−8 m MPa−1 s−1 for the hydraulic conductance at the upper and lower positions, respectively (Génard et al., 2001; Sevanto et al., 2011).
Daily patterns in bark osmotic concentration
Inner bark thickness decreased during the day, primarily because of the transpirational pull from the needles, and increased during the afternoons and evenings, when water potential recovered. However, in addition to changes in xylem water potential, another signal with a diurnal rhythm could be detected (Figs 2, 3), which we interpreted as arising from the changes in the osmotic concentration of the phloem region. These phloem-induced turgor changes occur in addition to the water potential changes causing the fluctuations in bark thickness. Actual phloem turgor is given by the integration of the two signals. During the summer, the diurnal changes in the modelled inner bark osmotic concentration were found to be closely related to changes in GPP (Figs S2, S3). This is in accordance with the observed trends of diurnal variations in phloem sugar concentrations, which typically peak during the day and decrease during the night (Sharkey & Pate, 1976; Hocking 1980; Pate & Arthur, 2000; Cernusak et al., 2002). It is possible that the turgor increase detected in the mornings down the stem may have resulted from fast pressure propagation in the phloem caused by direct sucrose loading at the top. Similarly, the early decline in bark turgor at midday may have been a consequence of reduced phloem loading, perhaps caused by reductions in leaf water potential.
When the leaf water potential drops, phloem turgor also drops if phloem loading does not increase (Hölttä et al., 2006). Nikinmaa et al. (2012) showed that the observed stomatal regulation of leaf gas exchange on these very same trees acted to maximize the assimilate transport from leaves. Under such a scheme, the changes in water potential caused by transpiration and the changes in phloem loading caused by assimilation control stomatal aperture. If transpiration causes the water potential to drop, whereas assimilation does not increase sufficiently to compensate for the turgor loss via increased phloem loading, stomata should close until a new equilibrium is reached. This closure would prevent further leaf water potential declines, but would also result in decreased phloem loading and decreased turgor, as observed here.
The exact timing of this turgor peak varied during the season. In particular, peaked before GPP during the period of active above-ground growth, which, in Scots pine at Hyytiälä, takes place between May and early July (see Sevanto et al., 2003). Instead, typically followed GPP by c. 0.5–1.5 h after the cessation of above-ground growth and whilst root activity continued below ground during August and September (Iivonen et al., 2001; Niinistö et al., 2011).
At this daily time scale, the time lags observed in the late summer between the CO2 assimilation rate and changes in the osmotic concentration are in accordance with the time lags predicted theoretically (Thompson & Holbrook, 2004; Hölttä et al., 2006) and observed experimentally (Zimmermann, 1969; Mencuccini & Hölttä, 2010a; Heinemeyer et al., 2011; Vargas et al., 2011) for pressure concentration waves.
Phloem transport during drought
July and August 2006 were exceptionally dry at our measurement site, and clear reductions in GPP, TER and ET, resulting from stomatal closure, were observed for canopy fluxes (Kolari et al., 2009). Interesting dynamics of the inner bark sugar concentration were observed during and immediately after the drought period. Our analysis indicated that sugars tended to accumulate somewhat at the bottom of the stem during the drought, whereas the sugar concentration (and turgor) of the phloem region simultaneously decreased at the tree top (Fig. 6). We can hypothesize two reasons for this behaviour. First, osmoregulation may have played a role. As the drought progressed, sugar accumulation was probably required to prevent turgor loss, particularly in the living cells in the cambium and phloem of the roots. Closer to the top of the tree, such a strategy is required to a much lower degree, because Scots pine behaves isohydrically and stomatal closure allows the maintenance of stable water relations under most circumstances (Magnani et al., 2002). Second, the reduction in GPP and water potential during the drought must also have reduced the phloem loading in the leaf, leading to smaller osmotic driving gradients in the phloem and lower rates of translocation. This behaviour is also in accordance with the proposal that stomata act to maximize assimilate transport from leaves under drought conditions (Nikinmaa et al., 2012). If the drought-induced drop in leaf water potential cannot be matched osmotically by increased phloem loading, the stomata will close to avoid further declines in assimilate transport because of the hydrodynamic pressure drop caused by transpiration, especially so in isohydric plants (Nikinmaa et al., 2012). The increase in at the base of the stem is also in accordance with the often noticed reduction in soil autotrophic respiration during drought (Tang et al., 2005; Carbone et al., 2011).
The response of the term to rainfall was very rapid and may also have been caused by changes in the plant water potential and canopy GPP (in this case, mediated by the rapid propagation of the water potential wave caused by the sudden increase in soil water content). Interestingly, at the bottom of the tree, the rainfall events were immediately followed by two sudden episodes of rapid reduction in the bark term, which could only have been caused by reductions in osmotic concentrations, suggesting a rapid release of solutes from the tree base following re-watering. In other words, when the rain ended the drought, sugars at the bottom of the phloem transport pathway were probably unloaded. A rapid increase in soil heterotrophic respiration after a precipitation event has been noticed in numerous studies (Law et al., 2001), with more recent studies showing that plant autotrophic respiration also responds rapidly (within 3 d) to rain pulses (Carbone et al., 2011). From the point of view of the optimization of phloem translocation, it also seems necessary for the sugars accumulated at the stem base during the drought to be disposed of externally to increase the phloem axial osmotic gradient to match the phloem sugar transport rate with the increased sugar production rate after the drought, and to avoid osmotic bottlenecks that would prevent a recovery in the sugar production rate (Nikinmaa et al., 2012). Conversely, the equally rapid increase in the term at the top of the tree could have been caused by two, not mutually exclusive, processes. First, the rapid increase in leaf water potential may have elicited rapid changes in leaf phloem loading and turgor, which propagated down the phloem to the point of measurement. Second, stomatal re-opening and consequent increases in photosynthesis may also have increased phloem loading and turgor.
Seasonal patterns in phloem transport and relationships with GPP
At the seasonal scale, the osmotic signal was very well coupled with canopy photosynthesis (Fig. 7). The significant relationship between the daily totals of GPP and the daily totals of (Fig. 8) is remarkable, because it suggests that the absolute value of changed over time depending on the levels of GPP attained in the canopy. The fact that the first maximum correlation between these two variables was obtained with lags of 1 and 9 d at the top and bottom of the tree, respectively (as suggested by the significance level of t values for the correlation coefficient), suggests that it may take more than a week for the concentration profile of the tree to adjust to changes in canopy GPP, again supporting the distinction between fast information transmission and slow solution transport (Hölttä et al., 2006; Mencuccini & Hölttä, 2010b). The presence of a second peak of even higher significance of the correlation coefficient at longer lags (10 and 31 d, respectively) supports the existence of multiple temporal scales over which canopy photosynthesis, phloem transport and below-ground processes may be coupled (Kuzyakov & Gavrichkova, 2010; Heinemeyer et al., 2011; Vargas et al., 2011). Our model predicts that about a week is necessary for the concentration profile to achieve steady state after a sudden perturbation, but it cannot reproduce the time scale of 30 d for the peak in correlation coefficients at the bottom of the trees. It is therefore likely that additional processes not accounted for in our model, such as sucrose loading and unloading in the transport zone and dynamic changes in sucrose storage along the pathway, caused these longer delays in the equilibration of the phloem concentration profile.
The slope of the relationship between the cumulative daily canopy photosynthesis and daily totals of was steeper in the upper part relative to the lower part of the tree. This may reflect the downward dampening of the sugar concentration signal from canopy GPP (Thompson & Holbrook, 2004; Mencuccini & Hölttä, 2010a). During the periods of cambial growth and drought, phloem loading and transport seemed to be less coupled to photosynthesis. This result also appears logical, as the location of the major sinks varies spatially during the season, and the sugar to starch conversion dynamics in the needles and along the transport pathway also varies (Rühr et al., 2009).
The model presented here appears to be capable of capturing one important dimension of bark thickness change which had been overlooked previously, that is, the role of phloem transport. Because the majority of inner bark thickness variations are caused by water exchange with the xylem, we were able to model the thickness variations by omitting the changes in osmotic content. Comparing our model results with the measured inner bark thickness variations revealed a diurnal signal that correlated strongly with photosynthetic production. We interpreted this additional signal in relation to phloem transport, and several lines of evidence supported this interpretation. Critically, the model assumes that all the relevant processes affecting changes in the xylem diameter and bark thickness are correctly represented in a mechanistic fashion, that is, that the discrepancies between the observations and model predictions can be attributed entirely to an osmotic-mediated process taking place in the phloem region. This seems to be justified on the basis of our analyses. The model presented here can be further improved to provide additional insights into the behaviour of the phloem transport system of trees. In particular, empirical validation can be obtained on small seedlings where continuous independent measurements are possible using nuclear magnetic imaging under laboratory conditions (Windt et al., 2006; De Schepper et al., 2012). Second, additional manipulations, such as canopy shading or phloem girdling (cf. Sevanto et al., 2011), either in the field or on small plants, should also be very informative, and should allow the determination of the extent to which bark thickness shrinkage reflects the processes of osmotic adjustment in the bark vs the purely passive adjustment to water potential changes in the xylem.