 © 2007 American Society of Plant Biologists
Abstract
In tree trunks, the motor of gravitropism involves radial growth and differentiation of reaction wood (Archer, 1986). The first aim of this study was to quantify the kinematics of gravitropic response in young poplar (Populus nigra x Populus deltoides, ‘I4551’) by measuring the kinematics of curvature fields along trunks. Three phases were identified, including latency, upward curving, and an anticipative autotropic decurving, which has been overlooked in research on trees. The biological and mechanical bases of these processes were investigated by assessing the biomechanical model of Fournier et al. (1994). Its application at two different time spans of integration made it possible to test hypotheses on maturation, separating the effects of radial growth and cross section size from those of wood prestressing. A significant correlation between trunk curvature and Fournier's model integrated over the growing season was found, but only explained 32% of the total variance. Moreover, over a week's time period, the model failed due to a clear out phasing of the kinetics of radial growth and curvature that the model does not take into account. This demonstrates a key role of the relative kinetics of radial growth and the maturation process during gravitropism. Moreover, the degree of maturation strains appears to differ in the tension woods produced during the upward curving and decurving phases. Cell wall maturation seems to be regulated to achieve control over the degree of prestressing of tension wood, providing effective control of trunk shape.
Gravitropic movements have been observed on a large range of herbaceous species as well as woody plants. Two different motors enabling axis curvature exist: (1) in zones of primary elongation growth, the motor of reorientation is differential growth; the growth is stimulated, preferentially on one side of the organ (Barlow et al., 1989; Cosgrove, 1997); (2) in zones where elongation is completed but radial growth still active, stem curvature occurs by differential maturation between the two sides of the organ.
Maturation strains appear because of dimensional changes of the cells during the maturation of the secondary cell wall (Fournier et al., 1994). The dimensional changes can be either shrinkage or swelling, depending on the direction, wood type, and species. In woody angiosperm species such as poplar (Populus spp.), maturation strains involve shrinkage in the longitudinal and radial directions, and swelling in the tangential direction (Archer, 1986). Maturation strains in the longitudinal direction are about 10 times higher than in the other directions (Archer, 1986). Tree axes are generally slender structures and can be considered as beams. From a mechanical point of view, tropic curvature is mainly due to the effect of longitudinal maturation strains. However, these strains are constrained within the trunk. Indeed, inner layers of older wood to which differentiating cells are stuck resist this deformation because of their mechanical rigidity. A mechanical equilibrium in which maturation stresses are produced is achieved (Fournier et al., 1994). Peripheral, differentiating wood cells undergo longitudinal tension because their maturation strains are resisted and compress the inner tissues. If the amount of inner tissue is large enough compared with the thickness of the maturation zone, the rigidity of the inner core resists, developing lockedin maturation strains. Large maturation stresses are generated near the periphery.
The maturation stresses can be measured indirectly: The internal mechanical state is locally disrupted by making cuts or drilling a hole in the wood after removal of the bark. Disruption of the internal mechanical equilibrium enables the lockedin maturation strains to be released. The wood that was under tension within the standing tree subsequently shrinks. The longitudinal shrinkage is then measured. It has been be observed that the measured residual longitudinal maturation strains (rlms) underestimate the original maturation strain for two reasons: (1) The piece of wood on which shrinkage is measured is not completely disconnected from the rest of the trunk (Coutand et al., 2004); (2) green wood is known to be a viscoelastic material so that the release of some of the initial strains is delayed and can only be released after a long time or after an accelerating hygrothermal treatment (Gril and Thibaut, 1994). Maturation strains are generally higher (in absolute values) in reaction wood than in opposite or normal wood so that any asymmetric production of reaction wood induces a reorientation of the stem (Archer, 1986).
Two types of reaction wood have been observed: compression wood in most softwood species (Timell, 1986) and tension wood in most hardwood species (Sinnott, 1952). In many angiosperms, including poplars (Jourez et al., 2001), the formation of reaction wood is characterized by the differentiation of a specific cellulose cell wall layer referred to as the G layer (Scurfield, 1973). Reaction wood formation is often accompanied by eccentric radial growth (Sinnott, 1952). Mechanical analysis revealed that the efficiency of the righting process through reaction wood formation depends on a combination of at least three factors: (1) the radial growth rate, (2) the difference in maturation strains from reaction wood to opposite wood, and (3) the diameter of the trunk at the onset of the righting movement (Fournier et al., 1994).
Although two motors appear to exist for gravitropic reactions, most studies have been conducted on organs displaying only primary growth, such as hypocotyls (Cosgrove, 1997), epicotyls (Firn and Digby, 1980), coleoptiles (Tarui and Iino, 1997; Meskauskas et al., 1999a), roots (Selker and Sievers, 1987; Perbal and DrissEcole, 1994; Wolverton et al., 2002; LaMotte and Pickard, 2004), the stipe of Coprinus (Meskauskas et al., 1998), and algae (Kiss, 1997). Studies of gravitropism in aerial organs with secondary radial growth such as tree trunks or branches are less frequent (e.g. Sinnott, 1952; Archer and Wilson, 1973; Yoshisawa and Okamoto, 1986; Wilson and Gartner, 1996). Moreover, quantification and mechanical analyses of the tropic movement of lignified axes over time have been little considered (except for a few studies on conifers; Archer and Wilson 1973, 1982; Fournier et al., 1994). On young organs displaying primary growth, the gravitropic response consists of an upward curving response followed by a decurving phase, enabling the axis to avoid overshooting the vertical. This phase cannot be controlled only by the perception of inclination since it starts before passing the vertical. It requires the perception of variables other than the inclination angle. As a result, it has been referred to as autotropic straightening (Firn and Digby, 1979; Tarui and Iino, 1997), automorphosis (Hoson et al., 1995), or autotropism (Meskauskas et al., 1999b), depending on the author (see Stankovic and Volkmann, 1998, for a discussion about the use of these terms). These biphasic responses have been rarely mentioned for trees (Archer and Wilson, 1973; Fournier et al., 1994, on conifers), and the question of whether or not it is autotropism has not been addressed. The first question to be raised is: Does the gravitropic movement in radially growing parts of axes exhibit the same pattern as the one in elongating organs? We especially hoped to find out whether or not the autotropic phase also exists in woody plants (including angiosperms) by determining whether decurving begins when the stem overshoots the vertical or before.
Another gap in knowledge about gravitropism concerns the mechanistic links between changes in axis shape and the motor(s) producing it. Biomechanical models, based on explicit mechanical relationships between the motor processes and the gravitropic response have only been developed for tropic movements, due to secondary growth (Archer and Wilson, 1973; Archer, 1986; Fournier et al., 1994; Almeras et al., 2005). Following the above mentioned works of Archer and coworkers, Fournier et al. (1994) proposed a simple model to simulate the changes in shape of a lignified stem during its gravitropic movement. It is based on explicit mechanical relationships between growth and the gravitropic response. It uses input data concerning: (1) the addition of new material due to cambial growth, and (2) the asymmetry of maturation strains due to the formation of reaction wood on one side of the trunk. Its output is the change of curvature of the stem. The model uses beam theory principles within the framework of linear elasticity, aiming to simulate the tropic response as simply as possible (i.e. including the most significant processes with the lowest number of parameters). For example, in the simplest version of the model, radial growth eccentricity has been neglected because it is supposed to be a secondorder motor variable.
The rationale of the model is illustrated in Figure 1 . The model is defined at the crosssection scale for an infinitesimal step of growth. The maturation process is assumed to be instantaneous, i.e. cell maturation occurs as soon as a cell leaves the cambium and its maturation strain (along the cell axis) is instantaneously locked in. Thus, each newly formed cell immediately contributes to the curving process and no longer develops or differentiates after the instantaneous step of growth and maturation. Two other hypotheses, a circular cross section and a sinusoidal variation of maturation strains from the reaction wood side to the opposite one, lead to a simple closedform equation (Fournier et al., 1994). This predicts the variation of curvature using: (1) the difference between maturation strain of compression wood (positive expansion) and opposite wood (low shrinkage) in the axis bending plane, referred to as α, and (2) the increment of radius (dr) divided by the square radius of the cross section (r^{2}):(1)This model has been used by other authors to simulate the reorientations of a whole tree during its entire life, using finite elements or transfer matrices (Fourcaud and Blaise, 2003; Fourcaud and Lac, 2003; Ancelin et al., 2004). However, these works are mainly modeling approaches. Up until now, no complete and independent assessment of kinematic data dealing with both maturation strains and radial growth has been conducted to test the validity of Fournier's model. Thus, the second question to be raised was: Can such a simple biomechanical model predict the observed changes in curvature during the gravitropic movement of the trunk?
To answer the first question, we studied the movement in the part of the trunk where only radial growth takes place, i.e. the trunk and not the annual shoot (Fig. 2 ). Poplar cuttings bearing a 1yearold stem grown from a lateral bud were planted. They were artificially tilted and the successive shapes of each 1yearold stem were measured weekly during its second year of growth. In the following, trunk or stem refers to the 1yearold stem grown from the lateral bud of the cutting. These data were then used to quantify the spatiotemporal behavior of the gravitropic response. In some studies of gravitropism, changes in axis shape have been quantified by the change in stem tip angle (Bagshaw and Cleland, 1993; Perbal and DrissEcole, 1994). As stated by Firn and Digby (1980), this can lead to artifacts in the interpretation of axis shape changes because the tip angle is dependent on an integrated value, i.e. the sum of curvatures all along the axis and the angle at the stem base. As a result, different patterns of reorientation can lead to the same value of the change in tip angle, as illustrated in Figure 3 . To avoid this bias, the gravitropic response was quantified via a kinematic analysis of curvature fields. Curvature at point i measures the rate of change in angle dθ between two points separated by a distance ds (Fig. 4 ):(2)Curvature is the most consistent variable that can be used to describe local changes in axis shape (Silk, 1984). However, quantitative kinematic analyses of curvature fields during tropic responses are rare (Silk and Erickson, 1978; Meskauskas et al., 1998), and none have been applied to gravitropic responses due to secondary growth.
To answer the second question, the experimental data and Fournier's model were used to check for quantitative links between the motors and the reorientation movement. Equation 1 could not be directly used because it required measurement of growth and variation of curvature at infinitesimal time steps, which was not the case with these experimental data. Therefore, a timeintegrated expression of the model was used to assess the major assumptions of the model: (1) the homogeneity of distributions of maturation strains, and (2) the relevance of neglecting the time course of cell wall maturation with respect to growth. Although the model significantly correlated the radial growth with curving over the whole growing season, it failed to explain observed variations of curvature at short time scales. Two hypotheses were suggested: (1) the maturation time cannot be neglected at short time scales, and (2) maturation strains are not constant over time and/or space (along the trunk). Finally, we checked these hypotheses by: (1) studying the relative kinetics of radial growth and curvature, and (2) measuring the peripheral rlms along the trunks.
RESULTS
Kinematics of the Gravitropic Response
Two typical examples of the changes of trunk shape over time are given in Figure 5 . During the first week after tilting, trunk shape remained unchanged in 75% of the plants (Fig. 5A), whereas 25% of plants sagged (Fig. 5B). Two weeks after tilting, the reorientation process became clearly visible in the apical region of the trunk in both cases. It then increased and very quickly progressed toward the trunk base (within 2 weeks). At the end of the growing season, most of the trunk was straight and vertical. It should be observed that once a part of the trunk reached the vertical position, it maintained this position with no oscillation around the vertical. Figure 5, C and D, shows the successive shapes of the stem base (cutting + base of the stem grown from the lateral bud). This figure shows that the gravitropic movement was not limited to the 2yearold stem grown from the lateral bud since even the basal cutting exhibited a tropic movement. Figure 6 shows the curvature fields of a representative trunk of the experimental set of trees (the trunk presented in Fig. 5A). In Figure 6A, the fields of curvature are plotted versus time and position from the stem base (which refers to position 0). Figure 6B represents the changes of curvature versus time for different cross sections along the trunk. Three main phases were distinguished: During the first week after tilting, the curvature remained constant or sometimes became more negative. After the two first weeks, the curvature of all parts of the trunk increased and became positive. Finally, the curvature decreased toward zero with decurving of the trunk. It can be observed during the latter phase that no part of the trunk had overshot the vertical. The decurving process started in the most distal cross sections and progressively extended toward the stem base. We refer to these three phases as the latent phase, the gravitropic phase, and the autotropic phase, respectively.
During the autotropic phase, the decurving was much higher than the curving at the stem base. This should have led to a downward movement of the tip of the stem. However, this was not observed in Figure 5, A and B, where the tip remained at a constant angle. This is due to the gravitropic reaction of the cutting that changed the leaning angle of the stem base.
Tension Wood Cartography
A typical pattern of tension wood development along a trunk at the end of the growing season is given in Figure 7 . During the gravitropic process (upward curving), sectors of reaction wood were produced all along the trunk on the upper side (all cross sections of the figure). In the most distal cross sections, reaction wood was no longer produced on the upper side, but a second segment of reaction wood was produced on the initially lower side (Fig. 7). This was consistent with the two phases of gravitropic and autotropic curvature observed in Figure 6. For each cross section, the position of the cross section along the trunk from the stem base is given in centimeters.
Radial Growth
Figure 8 shows the average radial growth rates of the 10 plants from the experiment conducted in 2001, over time. Average increases in diameter at three heights along the trunk for the two sets of plants are given in Table I . Stems exhibited typical continuous radial growth. No stoppage of growth was observed during the growth season and the radial growth kinetics was highly nonlinear, with the highest peak of radial growth in mid June. For most of the stems, two other peaks of radial growth were present. Error bars demonstrate a relatively large heterogeneity of radial growth rate between trees.
Adaptation of Fournier's Model to Our Data
Sign of the Parameter α in Fournier's Model
To adapt Fournier's model to angiosperms, α was defined as the difference of maturation shrinkage between the normal wood and the tension wood. Therefore, if tension wood of higher maturation shrinkage was formed on the upper side, as expected in true gravitropic stages, α would take a positive value with a positive variation of curvature dC (upward curving movement). However, if tension wood was formed on the opposite side, as expected in the autotropic stages at the end of the response, α would take a positive value with a negative value of variation of curvature dC (autotropic decurving movement).
Integration of Fournier's Model over Time to Test Its Fit with Experimental Data
Integration between two times, t_{i} and t_{f}, can be computed as:(3)To evaluate the integral, an additional assumption must be made concerning α, i.e. the variation of longitudinal maturation strain asymmetry. As a first approximation, we assume a constant value of α. Then, for a cross section between two times, t_{i} and t_{f}, Equation 3 finally results in:(4)During a time span when the autotropic response follows the gravitropic response, e.g. a gravitropic phase from time t_{1} and radius r_{1} to time t_{2} and radius r_{2} with tension wood formed on the upper left side, followed by an autotropic phase from time t_{2} and radius r_{2} to time t_{3} and radius r_{3}, with tension wood formed on the opposite right side, the response might integrate these two successive stages:
(1) during the first gravitropic curving stage:(5)
(2) during the following autotropic decurving stage:(6)
The sum of the curving and decurving actions of the two opposite sectors of tension wood, alternately produced on opposite sides, then resulted in:(7)To simplify the text, the sum of curvatures including gravitropic and autotropic phases was referred to as sum(dC).
Using different time spans for the integration, the above forms (Eqs. 5, 6, and 7) of the timeintegrated model (Eq. 3) could be used to investigate several hypotheses concerning the distribution of α, i.e. the distribution of the differential of maturation strains. With a time span equal to the whole growing season, it was possible to test if the gravitropic response intensity is explicable as a firstorder approximation by radial growth alone during the season (the thicker the ring of wood formed, the greater the gravitropic curvature produced), modulated by the cross section size (the thicker the stem, the less it is liable to curve), with a constant value of α over the duration of the growing season and a spatial homogeneity of α for any cross sections. The plot of curvature variations as a function of (1/r_{i} − 1/r_{f}) should be fitted by a linear regression, the slope of which would give the value of 4α, with the intercept equal to 0. If this is true, the radial growth rate should be a more relevant parameter for explaining curvature variation than the modulation of tension wood quality (from strong tension wood [high α and high gravitropic efficiency] to a mild one [lower α and lower gravitropic efficiency]).
This could also be assessed with a shorter integration time step. With a 1week time step, it was also possible to check if α was constant over time. If so, all the regressions for the successive weeks should have yielded similar parameters.
Relationship between Gravitropic Curvature Changes and Radial Growth
Results of statistical analyses on the timeintegrated results of Fournier's model (Eq. 7) over the growing season are given in Table II . Despite clear scattering of the points in Figure 9 , there were significant correlations between sum(dC) and 1/r_{i} − 1/r_{f} for both years. The regression parameters exhibited no statistically significant difference between experiments. The merged data sets fitted a regression with a slope of 5.08 10^{−3} and an intercept equal to 3.96 10^{−3} m^{−1} (Fig. 9). The slope of 5.08 10^{−3} corresponded to a value of α equal to 1.27 10^{−3}. The intercept was significantly different from zero and positive. Under the assumption that α was spatially homogenous, this would have meant that trees could right themselves without any growth increment. This result was not consistent with the basic assumption of Fournier's model that the maturation of a new layer of cells can be considered as instantaneous and only applies to new growth increments. Some cross sections were circular in the data set but with an obvious eccentricity of the pith. Since the model does not take eccentricity into consideration, cross sections exhibiting a pronounced eccentric radial growth were removed from Figure 9. However, this did not significantly affect the R^{2} value or regression parameters. Statistical tests of the application of Fournier's model at shorter time steps revealed that the model could not explain the variability of curvature changes, quantified by sum(dC) and by (1/r_{i} − 1/r_{f}), either when considering a time effect for a given location along the tree (Table III ), or a spatial effect of section position at different times (data not shown).
Rlms
Typical values of rlms measured at the end of the growing season and at the periphery of both sides of the trunk are shown in Figure 10A . These pairs of measurements allowed us to compute the value of the differential of maturation strains (α) as the difference of rlms between normal wood and tension wood. As shown in Figure 10B, the value of α was highly variable along the trunk, and the highest values were found in the part of the stem where tension wood was differentiated within the autotropic phase. However, even in the autotropic tension wood, the values of α were not constant.
Relative Kinetics of Radial Growth and Curvature
To assess the hypothesis that the lag between growth and wood differentiation could be neglected, we studied the relative kinetics of the two variables developed from the model analysis (see Eq. 7): the sum of curvature variations, sum(dC), and the geometrical variable (1/r_{i} − 1/r_{f}) that integrates the effects of both radial growth and initial size for each cross section of each plant stem. Two main patterns of relative kinetics were identified. Figure 11A (pattern A) gives a representative example of the relative kinetics found in 96% of the cross sections; Figure 11B (pattern B) gives a representative example of the relative kinetics found in the other 4% of the cross sections. In pattern A, the kinetics of curvature variations and the radial growth or (1/r_{i} − 1/r_{f}) were clearly out of phase: Radial growth and variation of curvature did not vary linearly with time, and two peaks could be observed for most of the cross sections. It is interesting to note that peaks of radial growth preceded peaks of curvature change. The time lag between the first peaks was about 7 to 14 d, whereas the time lag between the second peaks ranged from 7 to 20 d. In pattern B, both curvature and radial growth were in phase, corresponding to an almost linear kinetics of radial growth.
DISCUSSION
Kinematical Analysis of the Righting Process
Our results demonstrate that a spatiotemporal response is developed in the gravitropism of young trunks of poplar trees, with three clear phases: a latency phase, an upward gravitropic curving, and finally, an autotropic decurving phase. These three phases have not been acknowledged in the literature about gravitropism in trees. In the gymnosperm stems of Pinus strobus, Archer and Wilson (1973) also reported that shifts in the location of compression wood from one side of the stem to the other started before the overshooting of the vertical. Although they did not recognize it as autotropism and did not emphasize this phenomenon in their subsequent synthetic reviews (e.g. Wilson and Archer, 1979), it is very likely that this threephase dynamic, including an autotropic phase, is general in woody plants.
The results also demonstrate the importance of quantifying the gravitropic response using curvature fields rather than the inclination of the stem tip. In our study, if inclination of the tip angle had been used, the autotropic phase would not have been identified; indeed during this phase, the inclination of the tip angle remained constant, whereas the curvature field along the trunk considerably changed with opposite trends between lower and upper parts of the plant stem. The constancy of the vertical tip angle is not due to a lack of response, but to the result of a complex regulation process. Analyses of curvature fields have generally been little used in attempts to quantify shape changes of plant aerial organs (Silk and Erickson, 1978; Moulia et al., 1994; Coutand and Moulia, 2000) and even more seldom used in the area of aerial organ tropisms, except for studies on the gravitropic response of the fruit of Coprinus (Meskauskas et al., 1998). The phenomenology of righting kinematics characterized here for gravitropism due to secondary growth is very similar to that found for the fruit body of Coprinus (Meskauskas et al., 1999b). Even if the mechanism of reorientation is different in the two cases (production of tension wood during radial growth rather than asymmetrical growth of cells) and despite different time courses (vertical position obtained in 12 h for Coprinus and in approximately 100 d for trees), the similarities in the phenomenology of the righting process are striking. This suggests either: (1) a common mechanism of control for the balance between gravitropic curving and autotropic decurving that could have been conserved during evolution, or (2) a strong selective pressure for efficient straightening processes that would have promoted analogous mechanisms (Moulia et al., 2006).
Link between Stem Curvature Changes, Radial Growth, and Tension Wood Maturation
The latency phase during the first week after tilting (when curvature remained constant or became more negative) may be explained in two ways: (1) a purely physical sagging due to the viscoelasticity of wood resulting in creep without any growth or compensating initial gravitropic curving or (2) a sagging due to the increase of loading resulting from growth, combined with a delay in tension wood differentiation and maturation. The latter hypothesis finds some support in previous reports on the time necessary for a fiber to acquire its matured state. Although this time seems to be plant age and species dependent, it was always of the order of magnitude of weeks, ranging from 6 to 10 d in Aesculus hippocastaneum epicotyls (Casperson, 1966), and from 7 to 39 d for other species (Scurfield and Wardrop, 1962). This is consistent with an initial latency phase due to the time required for new tension wood to differentiate and mature. It should, however, be observed that these two hypotheses are not mutually exclusive and that viscoelastic sagging could occur concomitantly with tension wood development.
At the time scale of a single growth season, we found a significant correlation between curvature variation and radial growth, taking mechanical effects of stem size into account (1/r_{i} − 1/r_{f}), in accordance with the model prediction. Thus, as a first approximation, (1/r_{i} − 1/r_{f}) is a pertinent variable for predicting curvature changes. The value of the slope fitted to the entire data set gives a mean value of the difference (α) between maturation strain of tension and normal wood that is in the range of magnitude generally measured but nevertheless a bit high. Actually, a previous study (Coutand et al., 2004) gave the rlms for normal wood and tension wood for a subsample of the experimental trees discussed in this article, attributing a value of about −0.0005 to α, i.e. onehalf the one found here. Moreover, radial growth variations and size (1/r_{i} − 1/r_{f}) explained only 32% (R^{2} = 0.32) of the variability of curvature variations. This suggests that a hypothesis of a constant tension wood quality (constant absolute value of α) is overstated and that important mechanisms of tension wood quality and modulation over time and spatial localization along stems are still unknown in this model of development. Lastly, the intercept differed very significantly from 0, which suggests that a tree can right itself without any radial growth. This contradicts the basic assumption of Fournier's model that the maturation of a new layer of cells is instantaneous and only applies to each new growth increment. In fact, the exact developmental process might be quite complex; for example, wood development and differentiation may well have started prior to tilting, so that the G layers of tension wood cells may only have been differentiated rather late, following the gravitropic stimulus. This possibility is supported by observations at the end of the vegetative season. In this case, the anatomy of the first cells of the tree ring on the upper side, produced before tilting, exhibit a G layer that was thinner than in tension wood fibers produced after tilting (data not shown), in accordance with a later shift to Glayer differentiation. Such a possibility was also observed by Jourez and AvellaShaw (2003) in their study of tension wood formation in poplar. On the contrary, in the case of temporal integration with a time step of a week, the model no longer fit the data. Two hypotheses might explain this: (1) spatio and/or temporal changes of tension wood quality (variations of α) during the gravitropic response are large and become more significant than geometrical effects, and (2) the time lag between radial growth and differentiation of a cell into tension wood fibers cannot be neglected at short time scales. Our measurements of rlms clearly demonstrated spatial changes of α along the poplar stems. However, since measurements of rlms in this case are destructive to the plants and only carried out on peripheral wood at the end of the response observations, they did not permit an unequivocal interpretation of the temporal versus spatial changes of α, because changes occurred in part of the trunk where autotropism was under way. Therefore, changes could be equally related to a spatial effect (changes along the trunk) or to a temporal effect (changes between the true gravitropic and autotropic phases) or even to both. Measurements of kinetics of α at different locations along the trunk would be necessary to reach a final conclusion. Reports of the kinetics of longitudinal maturation strains in upright trees were described by Okuyama et al. (1981). Indeed, they found an increase of rlms between May and September, but the time sampling is too long (compared with the time scales of this study) to fully compare the results. The second hypothesis that the time between radial growth and the differentiation of a cell into tension wood fibers cannot be neglected at short time scales is supported by several observations:
The relative kinetics of curvature and radial growth clearly showed time lags between the variation of curvature and radial growth in most cross sections. It is interesting to observe that few cross sections showed changes in curvature that were in phase with the radial growth increment and only in cases where the radial growth is established at a constant rate (permanent regime). In this case, Fournier's model predicted the change of curvature relatively well.
Despite the fact that tilting occurred after the onset of radial growth, even the very first cells produced on the upper side of the trunk displayed a clear G layer throughout the trunk.
The range of time lags between peaks of curvature and the radial growth corrected from stem size (1/r_{i} − 1/r_{f}) is within the same range of magnitude as the time of cell maturation, as reported by Scurfield and Wardrop (1962) and Casperson (1966). Despite a lack of precise knowledge about the kinetics of the cell maturation process and more precisely the time necessary for a cell (fiber) to acquire its mature state, maturation time is in the range of 1 or 2 weeks. For a cell wall maturation time of this order of magnitude, the observed change in curvature is not likely to be due to increments of radial growth during the lapse of time but, instead, to maturation of the cells produced earlier. This highlights the problem of how to associate the observed change of curvature with the active increment of radius in the righting process. Because of a nonsteady growth regime (i.e. radial growth rate is not constant) and a difference in the two time lags observed through the righting process, the duration of the differentiation phase may not be simply taken into account by introducing a simple constant time lag between the curvature variation (dC) and growth increment (dr) within the model. A more detailed analysis of the kinematics of cell wall maturation and the design of a new biomechanical model, including the time course of the differentiation process, should be developed to solve this question.
CONCLUSION
This study gives a quantitative description of the reorientation process in a hardwood species. It provides evidence for distinct phases of the gravitropic response in a tree species, including a phenomenon of anticipative curvature or autotropism, which has been overlooked in the literature on tree gravitropism. Moreover, it provides a conceptual framework, methods, and tools for quantifying local changes in the shape of plant axes, which can be used for analyzing shape changes in any plant axis (i.e. due to primary or secondary growth) and data analysis.
The results demonstrate the usefulness of simple and simplified biomechanical models with as few parameters as possible, to study the relationship between radial growth and variations of tree shape. The use of Fournier's model with two different integration time steps made it possible to test hypotheses on maturation, separating the geometrical effects (radial growth and size) from those of wood mechanical characteristics (α). However, for short time steps (about a week), the results demonstrated that the model failed to explain variations of curvature. At the time scale of an entire growth season, the model explains 32% of the changes in curvature. It is thus likely that Fournier's model, mainly designed for studies of longterm straightening in adult trees with a time scale of years, is only usable at a large time scale, greater than cell wall maturation (weeks). Hence, in its present form, it is not adequate for gravitropic studies at short time scales.
The inability of the model to predict observed curvature can be explained by two main hypotheses that are supported by data analysis. First, measurements of residual maturation strains revealed large differences in levels of tension wood responsible for the righting process and in tension wood responsible for the autotropic process. This means that a view of a single type of reaction wood within a tree is an oversimplification. Experimental results clearly demonstrate a modulation of prestress levels during the gravitropic movement. From a modeling point of view, this means that the hypothesis of a single value of wood quality asymmetry (α) is insufficient to properly describe a quantitative tropic response. Second, analyses of relative kinematics of shape and radial growth revealed a lag in time between variations of curvature and radial growth. This suggests a key role in the time of cell maturation when short time responses are considered. To be used for short time steps, Fournier's model should be extended to include an application for cell wall maturation. This study demonstrates a crucial need for further exploration of the relationships between radial growth, maturation process, and maturation prestress generation for a complete understanding and modeling of the reorientation process at annual or intraannual time scales.
MATERIALS AND METHODS
Experimental Design
The experimental work was conducted under field conditions at Institut National de la Recherche Agronomique (INRA) ClermontFerrand, France (45°47′ N, 3°10′ W, 329 m elevation), and repeated for 2 years (1999 and 2001). Poplar (Populus spp.) was chosen because of its rapid radial growth and potentially rapid gravitropic movement. The hybrid (Populus deltoides × Populus nigra) ‘I4551’ was chosen for its low sensitivity to phototropism (http://www.bariteau.org) to limit the potentially complicating effects on gravitropism. In January, cuttings bearing a 1yearold shoot grown from a lateral bud were planted in individual 60L containers filled with Limagne soil. The bottoms of the pots were drilled to enable roots to develop within the soil outside of the pots. In each experiment, 10 trees were artificially tilted at the end of May. Since the trunks were not perfectly straight and because of the effect of self weight, the inclination was adjusted on each trunk so that the mean angle between the first quarter of the trunk and the vertical was 35° degrees. All pots were then buried in the soil to limit their movements and to maintain a normal soil temperature around the roots (see Fig. 2). Trees were watered continuously by an individual irrigation device throughout the experiments. Initial dimensions of trees are summarized in Table IV .
Recording of Tree Shape
From day one of tilting until August, the shape of each tree was measured weekly by threedimensional digitizing (Polhemus and POL software, see Sinoquet et al. [1991] for details). Ink marks, approximately 2 to 3 cm apart, were made on each trunk to facilitate digitizing; these marks ensured the trace of identified material points on each trunk during the righting process (Fig. 2), enabling an accurate match of local curvature with corresponding crosssection anatomy.
Definition of Curvilinear Abscissa for Curvature Analysis
The position of a given material point i along a curved line is given by its curvilinear abscissa s_{i} and was computed using the following equation:(8)The curvilinear abscissa field is used to plot the curvature against the position along the trunk for each digitized point.
Computation of Fields of Curvature with Digitizing Data
The field of curvature of each tree shape was computed using digitized data. Computation of curvature requires first and second derivatives of a given curve. Since digitization gives only a discrete series of points, each set of points must first be fitted by a polynomial. Since plants can produce intricate shapes, a global fitting with a single polynomial is generally not appropriate. The method of locally fitting polynomials developed by Moulia et al. (1994) was not suitable in our case because of equally spaced digitizing points (in Moulia's approach, the density of points cannot be fixed and constant: the higher the curvature, the higher the density of points).
We used a method based on a smoothing spline function (spaps procedure available in the Matlab spline toolbox). In this method, the spatial step for fitting is automatically optimized by the algorithm. Moreover, it has two other main advantages: (1) it automatically ensures the continuity of the polynomials describing the spline, and (2) since the spline belongs to the nonparametrically fitted curves based on a set of local polynomials in varying amounts, they are much less likely to induce biases compared with other fitting methods.
In the spline procedure, the fitting is controlled by two parameters: the maximal degree of polynomials of the spline and a parameter referred to as tolerance. This parameter is defined as the square of the se of the distance between two successive points. It was experimentally estimated by digitizing the same trunk 10 times and by computing the average distance between two points, which was always less than 4 cm. A tolerance of 16 cm^{2} was thus retained.
The spline requires monotonic (continuously increasing) abscissas. The points coordinates (x,y) obtained were thus recomputed in the principal axes of the set of points (Dagnélie, 1973; Moulia et al., 1994) before spline fitting.
The first and second derivatives (dP[x_{i}]/dx and d^{2}P[x_{i}]/dx^{2}) of the equation of the fitting curve (P[x]) were computed and used to compute the value of curvature at each digitized point (C_{i}) along the trunk for each tree and at each date, by the classical equation:(9)The equations of the derivatives of the fitting curve were computed using the fnder function of Matlab and the values of the first and second derivatives were computed using the fnval function. The classical convention for curvature sign with counterclockwise set positive was retained. The consistency of the method developed in this study to compute curvature fields was checked by two controls. First, we computed the mean field of curvatures from the 10 curvature fields obtained with the 10 sets of points digitized on the same tree at one date. The maximal sd obtained was equal to 9.66 10^{−5} cm^{−1}, i.e. 2% of the full range of curvature variation. Second, as explained in the introduction, the variation of angle between the tip and the base of a curve is the integral of the curvature field. We thus compared the variations of angle between the stem base and the base of the annual shoot measured on trunk shapes with the variations of angle computed by the integral of curvature fields along the trunks. This comparison was done for each tree of the 2001 experiment and for all digitized samplings. If the fitting procedure is adequate, then plotting the two variables should give a straight line with a slope of 1 and an intercept of 0. We proceeded with an orthogonal regression (Dagnélie, 1973). It resulted in a linear relationship with equation y = 1.1x − 1.4 and a correlation coefficient R^{2} = 0.94. Statistical tests were performed to test if the slope differed from 1 and if the value of the intercept differed from 0 (Dagnélie, 1975). At the 5% level of confidence, the slope did not significantly differ from 1 (t_{obs} < t = 0.43). The intercept did significantly differ from 0 but was very small: −1.4°. In fact, if we take the average length of trunk as being equal to 180 cm (Table IV), then a bias of 1.4° (0.024 rad) on total angle change will introduce a curvature bias of approximately 0.024/180 cm = 0.00014 cm^{−1}, i.e. 4% of the full range of curvature. Moreover, it is not clear whether the bias comes from computation or from direct measurements. The method had a sd of about 0.0001 cm^{−1} and a mean bias of, at the most, 0.00014 cm^{−1}. These values had to be compared to the changes in curvature observed between dates: during the gravitropic phase when curvature increased from 0 to 0.003 cm^{−1} in 3 months, i.e. about 30 times the accuracy and 22 times the bias. The average change of curvature per week during the gravitropic phase was about 0.001 cm^{−1}, i.e. 10 times the sd and 7 times the bias. The range of curvature during the autotropic phase was of the same order of magnitude for the uppermost cross sections. Thus, although curvature estimation always remains a difficult issue, the original method proposed in this work is accurate.
Since the trunk shape is time and space dependent, the changes in curvature were studied spatially and temporally by plotting curvature versus time and/or position along the trunk (this latter is given by a curvilinear abscissa).
Growth Measurements
The radial growth of each tree within the tilt plane was measured weekly by a Vernier caliper at three locations along the trunk: the trunk base, 1 m from the base, and 1.5 m from the base.
Elimination of Sagging Phases for Testing Fournier's Model
During the growing season, the trunk may undergo several processes: sagging, true gravitropic upward curving, and decurving (also referred to as autotropism). Sagging results in a negative variation of curvature and a downward curving; the true gravitropic phase shows a positive variation of curvature and an upward curving of the trunk; the decurving phase shows a negative variation of curvature and a straightening of the trunk. Thus, both sagging and autotropism lead to a negative variation of curvature, but they can be distinguished using trunk shapes. The model does not integrate trunk sagging, so only data showing no sagging or insignificant sagging were retained for the analysis with equations.
Statistical Analyses
We tested the model by linear regression and variancecovariance analysis. We looked at the ability of the variability of 1/r_{i} − 1/r_{f} to explain the variability of sum(dC) in the case of an integration step equal to the growing season and, in the case of an integration time, equal to a week. Data sets consisted of repeated measurements with two repetitions: experiments of 1999 and 2001. Because of repeated data, a mixed model was used for statistical tests. Statistical analyses were processed with SASX8.1 (SAS Institute). Since measurements were repeated spatially along the trunks and over time, statistical analyses used the SAS mixed procedure, as described in the study of Christophe et al. (2003). This analysis involves possible variations in residual variances and possible covariances between repeated measurements.
Test of Fournier's Model on a Time Span Corresponding to the Whole Vegetative Season
The data sets consisted of spatially repeated measurements (three cross sections per tree). In this case, the fixed effects are the growth measured by the variable (1/r_{i} − 1/r_{f}) and the year of experiment (1999 or 2001); the random effect is due to the individuals, the trees in this case.
Test of Fournier's Model with a Time Span of About a Week
In this case, data sets consisted of spatially and temporally repeated measurements. To our knowledge, none of the statistical tests enabled us to take the time and spatial repetitions into account at the same time. Statistical tests were thus performed in two steps: (1) step one to test time effects and (2) step two to test spatial effects, as detailed below.
To test time effects, the data set must be separated into three subsets corresponding to the cross sections located at the base of the trunk, 1 m from the stem base and 1.5 m from the stem base. For each data set, we used the following programming procedure: Proc mixed; Class tree, time; Model sumdC = (1/r_{i} − 1/r_{f}) time (1/r_{i} − 1/r_{f}) × time; Repeated Sub = tree.
To test the spatial effects, the data set must be separated into subsets corresponding to the intervals between digitizing dates. For each data set, we used the same procedure as above but replaced time by section within the programming procedure. Since we did not know the structure of the covariance, three covariance models were used for each test: VC, CS, and UN (Littell et al., 1996). VC considers variable variances (heterocedasticity) and no covariance, CS considers a compound symmetry of the variance and covariance, and UN makes the hypothesis that there is no mathematical structure. The best covariance model was chosen according to the maximal value of the Akaike's Information Criterion and Schwarz Bayesian Criterion criteria (Littell et al., 1996). Both criteria indicated CS as the best covariance model. It was thus selected for statistical tests.
Measurements of rlms
Rlms were measured using strain gauges (Vishay Micro Measurements EP08 250BG 120 Ohm) and an extensometry bridge (P3500, Vishay Micro Measurements; see Coutand et al., 2004 for details). Measurements were made in pairs at the periphery of two trees of the 2001 experiment: one measurement on the upper side and the second on the lower side, at several locations along each trunk, as explained in Figure 12 .
Cartography of Tension Wood Sectors
At the end of each experiment, trees were felled and cut into segments and cross sections. The cross sections were used to make a cartography of tension wood sectors. Mapping of tension wood sectors was performed using the naked eye or by the method developed by Grzeskowiak et al. (1996): An iodinezinc chloride solution was applied with a paint brush on a 1cm thick, freshly cut cross section. The dye gives a red color to Glayer tension wood while normal wood remains unstained or turns slightly yellow. The coloration is temporary (several minutes) so that immediately after application of the dye, the contours of tension wood sectors were manually drawn on a transparent sheet of paper placed on the cross sections.
Acknowledgments
We would like to thank Stéphane Ploquin (INRA, Unité Mixte de Recherche [UMR] Physiologie Intégrée de l'Arbre Fruitier et Forestier [PIAF], ClermontFerrand) for his work on both experiments and his technical support for digitizing, Gaelle Jaouen (INRA, UMR PIAF, ClermontFerrand) for her participation during digitizing and growth measurements of the experiment in 2001, and Patrice Chaleil (INRA, UMR PIAF, ClermontFerrand) for tree planting and maintenance of watering devices during the experiment. We would also like to thank Pr. AnneMarie Catesson (ENS, Paris) and Dr. Brigitte Chabbert (INRA, Reims) for the discussions about cell wall maturation processes. We are very grateful to Nick Rowe (CNRS, Montpellier) and George Jeronimidis (University of Reading, Reading, UK) for improving the English and for their helpful comments.
Footnotes

The author responsible for distribution of materials integral to the findings presented in this article in accordance with the policy described in the Instructions for Authors (www.plantphysiol.org) is: Catherine Coutand (coutand{at}clermont.inra.fr).

↵[C] Some figures in this article are displayed in color online but in black and white in print.

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 Received August 11, 2006.
 Accepted April 16, 2007.
 Published April 27, 2007.