Numerical modelling of shape regulation and growth stresses in trees

This paper is the first of a series focusing on the biomechanical analysis of live trees. The finite element method (fem) is the most common method used for the analysis of complex mechanical structures. Several fem industrial codes exist, but they need to be adapted to calculate the mechanical behaviour of growing trees. A general incremental model has been developed for this specific application. In this model, time was discretised and for any developmental stage, a new equilibrium was written considering the increment of weight due to the mass of new wood layers and new vegetative elements being added. Maturation strains of new-formed cells were also considered for the simulation of the shoot reorientation process. This model was intended for use at the whole plant level. A multi-layer beam finite element is presented, which is well adapted to discretise tree limbs. The shape evolution of the structure was represented at each time step by the nodal displacement vector. The mechanical stresses induced as a result of growth were determined within the stem using a cumulative process taking into account the past history of each growth ring. The first basic results of growth stresses and shape evolution were compared with already published results at the branch level.     

Numerical study of how creep and progressive stiffening affect the growth stress formation in trees

It is not fully understood how much growth stresses affect the final quality of solid timber products in terms of, e.g. shape stability. It is, for example, difficult to predict the internal growth stress field within the tree stem. Growth stresses are progressively generated during the tree growth and they are highly influenced by climate, biologic and material-related factors. To increase the knowledge of the stress formation, a finite element model was created to study how the growth stresses develop during the tree growth. The model is an axisymmetric general plane strain model where material for all new annual rings is progressively added to the tree during the analysis. The material model used is based on the theory of small strains (where strains refer to the undeformed configuration which is good approximation for strains less than 4%) where so-called biological maturation strains (growth-related strains that form in the wood fibres during their maturation) are used as a driver for the stress generation. It is formulated as an incremental material model that takes into account elastic strain, maturation strain, viscoelastic strain and progressive stiffening of the wood material. The results clearly show how the growth stresses are progressively generated during the tree growth. The inner core becomes more and more compressed, whereas the outer sapwood is subjected to slightly increased tension. The parametric study shows that the growth stresses are highly influenced by the creep behaviour and evolution of parameters such as modulus of elasticity, micro-fibril angle and maturation strain.     

Identification of biomechanical factors involved in stem shape variability between apricot tree varieties

BACKGROUND AND AIMS: Stem shape in angiosperms depends on several growth traits such as elongation direction, amount and position of axillary loads, stem dimensions, wood elasticity, radial growth dynamics and active re-orientation due to tension wood. This paper analyses the relationship between these biomechanical factors and stem shape variability. METHODS: Three apricot tree varieties with contrasting stem shape were studied. Growth and bending dynamics, mechanical properties and amount of tension wood were measured on 40 1-year-old stems of each variety during one growth season. Formulae derived from simple biomechanical models are proposed to quantify the relationship between biomechanical factors and re-orientation of the stems. The effect of biomechanical factors is quantified combining their mechanical sensitivity and their actual variability. RESULTS: Re-orientations happened in three main periods, involving distinct biomechanical phenomena: (a) passive bending due to the increase of shoot and fruit load at the start of the season; (b) passive uprighting at the fall of fruits; (c) active uprighting due tension wood production at the end of the season. Differences between varieties mainly happened during periods (a) and (b). CONCLUSIONS: The main factors causing differences between varieties are the length/diameter and the load/cross-sectional area ratios during period (a). Wood elasticity does not play an important role because of its low inter-variety variability. Differences during period (b) are related to the dynamics of radial growth: varieties with early radial growth bend weakly upward because the new wood layers tend to set them in a bent position. The action of tension wood during period (c) is low when compared with passive phenomena involved in periods (a) and (b).     

The Biomechanics of Tree Fork Design

Two biomechanically different types of tree fork are described: the “compression fork” where the two jointed stems are pressed against each other at the contact face by the action of reaction wood, and the “tension fork” where the two connected stems are bent away from each other by gravity or wind action leading to tensile stresses in the connective zone. It is well known that trees permanently try to improve their own designs by adaptive growth in order to maintain a state of constant mechanical stress at the tree surface. In the case of these two different types of tree fork, adaptive growth also takes different ways in order to avoid high localized stress peaks which could lead to failure of the tree under wind loading. In this paper only the tension fork is assessed with respect to its shape optimization by computer simulation of adaptive growth. It is shown that the tensile fork is shape optimized in a very perfect way in order to avoid any dangerous localized stress peaks (notch stresses) which could lead to failure of the tree.     

Biomechanical design and long-term stability of trees: Morphological and wood traits involved in the balance between weight increase and the gravitropic reaction

Studies on tree biomechanical design usually focus on stem stiffness, resistance to breakage or uprooting, and elastic stability. Here we consider another biomechanical constraint related to the interaction between growth and gravity. Because stems are slender structures and are never perfectly symmetric, the increase in tree mass always causes bending movements. Given the current mechanical design of trees, integration of these movements over time would ultimately lead to a weeping habit unless some gravitropic correction occurs. This correction is achieved by asymmetric internal forces induced during the maturation of new wood.The long-term stability of a growing stem therefore depends on how the gravitropic correction that is generated by diameter growth balances the disturbance due to increasing self weight. General mechanical formulations based on beam theory are proposed to model these phenomena. The rates of disturbance and correction associated with a growth increment are deduced and expressed as a function of elementary traits of stem morphology, cross-section anatomy and wood properties. Evaluation of these traits using previously published data shows that the balance between the correction and the disturbance strongly depends on the efficiency of the gravitropic correction, which depends on the asymmetry of wood maturation strain, eccentric growth, and gradients in wood stiffness. By combining disturbance and correction rates, the gravitropic performance indicates the dynamics of stem bending during growth. It depends on stem biomechanical traits and dimensions. By analyzing dimensional effects, we show that the necessity for gravitropic correction might constrain stem allometric growth in the long-term. This constraint is compared to the requirement for elastic stability, showing that gravitropic performance limits the increase in height of tilted stem and branches. The performance of this function may thus limit the slenderness and lean of stems, and therefore the ability of the tree to capture light in a heterogeneous environment.     

Effects of bending stress on taper and growth of stems of youngEucalyptus regnans trees

A glasshouse experiment investigated the effect of bending stress on stem radial and height growth and stem taper ofEucalyptus regnans seedlings. Eighteen-week-old, potted seedlings were bent continuously for 8 weeks with a static bending stress. The bending treatment was then removed and the seedlings grown for another 12 weeks. Other seedlings were stayed vertically throughout the experiment whilst control seedlings were neither bent nor stayed. Seedlings were rotated every 2 days to prevent reaction wood developing asymmetrically in the stems of bent trees. Bent trees had higher radial growth rates, developed more tapered stems and had higher safety factors (the ratio of stem radius to the minimum radius required to prevent the tree toppling over) than unbent seedlings. They produced a band of tension wood in their stems and ceased height growth whilst bent. When bending ceased, they resumed normal radial and height growth. Unbent trees developed more cylindrical stems. There were no differences in growth behaviour between stayed and control trees. Bent and unbent trees all developed a butt swell, the taper of which was not affected by treatment. It was concluded that bending stress has substantial effects on both the size and taper of tree stems. However, the development of butt swell is independent of the bending stress applied. The results were considered in relation to biomechanical theories of tree stem development.     

Effect of tree size and competition on tension wood production over time in beech plantations and assessing relative gravitropic response with a biomechanical model

? Premise of the study: Gravitropic movements are unexpected mechanical processes that could disturb tree design allometries derived from the physics of nonliving bodies. We investigated whether the scaling law of gravitropic performance (power of -2 of stem diameter) derived from integrative biomechanical modeling is disturbed by ontogeny or environment, then discuss the silvicultural and dendroecological consequences. ? Methods: In a beech (Fagus sylvatica) plantation, four plots with different initial planting densities evolved without any intervention for 26 yr. Regular tree inventories and a silvicultural model were used to monitor competition over time in each plot. The radial production of tension wood was quantified using a cross-section of the stems at 1.30-m height, and an integrative biomechanical model computed the tree gravitropic performance over time. ? Key results: All trees developed tension wood over the whole period, with higher amounts at the youngest age, resulting in theoretical lean corrections of ca. 20-30° on the first 4 m of the stem over the whole period. The scaling law of gravitropic performance is slightly larger than the power of -2 of stem diameter. ? Conclusions: Gravitropic performance in forest ecosystems is mainly limited by size (diameter). Ontogenic acclimation of tension wood formation allows the youngest trees to be more reactive. No additional effect of spacing was found. However, silviculture influences size and, therefore, tree reactivity at a given age. Such results will be helpful for dendroecological approaches that use wood as a marker of environmental disturbances or a trait linked to plant strategies.     

Wind-induced stresses in cherry trees: evidence against the hypothesis of constant stress levels

We calculated the wind-induced bending moments and stresses generated in the stems of five Prunus serotina conspecifics differing in height and canopy shape and size (based on detailed measurements of stem projected area and location with respect to ground level) to test the hypothesis that wind-loads generate uniform and constant stress levels along the lengths of tree twigs, branches, and trunks. These calculations were performed using five different wind speed profiles to evaluate the relative importance of the shape of wind speed profiles versus the ’geometry’ of tree shape on stem stress distributions and magnitudes. Additionally, we evaluated the effect of absolute tree size and stem taper on wind- induced stresses by scaling the size of smaller conspecifics to the absolute height of the largest of the five trees yet retaining the original stem proportions (i.e., diameter relative to stem length) for each plant. Finally, we also determined how the factor of safety for wind-loading (i.e., the quotient of stem yield stress and wind-load stress) changed as a function of tree size (and, presumably, age). Our results indicate that wind-load stress levels (1) vary along stem length even for the same wind speed profile and the same maximum wind speed; (2) would increase to dangerous levels with increasing tree height if it were not for ontogenetic changes in stem taper and canopy shape that reduce stress intensities to manageable levels; (3) tend to be more dependent on stem taper and canopy shape and size than on the shape of the wind speed profile; and (4) the factor of safety against wind-induced mechanical failure decreases as trees get larger, but varies along the length of large trees such that preferential stem failure is likely and functionally adaptive. We thus (1) reject the hypothesis of constant wind-induced stress levels; (2) support the view that size-dependent changes in stem taper are required to maintain wind-load mechanical reliability; and (3) suggest that certain portions of mature trees are ’designed’ to fail under high winds speeds, thereby reducing drag and the bending moments and stresses experienced by trunks. Received: 24 May 1999 / Accepted: 8 October 1999     

Transgenic poplars with reduced lignin show impaired xylem conductivity, growth efficiency and survival

We studied xylem anatomy and hydraulic architecture in 14 transgenic insertion events and a control line of hybrid poplar (Populus spp.) that varied in lignin content. Transgenic events had different levels of down-regulation of two genes encoding 4-coumarate:coenzyme A ligase (4CL). Two-year-old trees were characterized after growing either as free-standing trees in the field or as supported by stakes in a greenhouse. In free-standing trees, a 20 to 40% reduction in lignin content was associated with increased xylem vulnerability to embolism, shoot dieback and mortality. In staked trees, the decreased biomechanical demands on the xylem was associated with increases in the leaf area to sapwood area ratio and wood specific conductivity (k(s)), and with decreased leaf-specific conductivity (k(l)). These shifts in hydraulic architecture suggest that the bending stresses perceived during growth can affect traits important for xylem water transport. Severe 4CL-downregulation resulted in the patchy formation of discoloured, brown wood with irregular vessels in which water transport was strongly impeded. These severely 4CL-downregulated trees had significantly lower growth efficiency (biomass/leaf area). These results underscore the necessity of adequate lignification for mechanical support of the stem, water transport, tree growth and survival.     

Growth stress controls negative gravitropism in woody plant stems

In the shoots of woody plant species, reaction-wood fibers are formed on the upper or lower side of the secondary xylem of a leaning trunk or branch wherever large, internal growth stress is generated. Negative gravitropic movement in woody plant stems is proposed to be the result of growth stress generated in the reaction-wood tissue. This study examines the interaction between bending moment due to increasing self-weight and recovery moment resulting from asymmetric growth stress, and tests a hypothesis that describes the relationship based on the structural mechanics "beam theory". Simulations of observed tree branch morphology of Magnolia kobus DC., Juniperus chinensis L., Abies saccharinensis Fr. Schum., and Prunus spachiana Kitamura f. spachiana cv. Plenarosea showed that (i) the growth stress generated in the reaction wood is sufficient to counteract the gravitropic response to increasing self-weight, and (ii) the specific directional angle of the shoot apex or preferred angle of the elongation zone plays an important role in controlling the spatial shape of the branch stem that is peculiar to plant species with large growth stress generated in the reaction-wood tissue.     

Main trends in the genetic transformation of <Emphasis Type="Italic">Populus</Emphasis> species

The main advantages that could be obtained by poplar plantation production were described in this review. We also described the significance of poplars for industry and for solutions to ecological problems. Taking into consideration the results obtained by genetic engineering methods, we analyzed the trends in the improvement of the Populus phenotypes related to the resistance to biotic and abiotic stresses and herbicides, as well as to the modification of the wood quality (decreasing or modifing the lignin content), phytoremediation, plant growth acceleration, and changes in the plant morphology.     

Measuring and modelling stem growth and wood formation: An overview

The immediate environment of a cambial initial (weather and nutritional factors, growth regulators, physical stresses) varies continuously over time. Consequently local conditions in the cambium influencing wood formation at any given instant are unique. The distribution of these conditions can be influenced by longitudinal gradients (stem base to apex), circumferentially or by local factors, such as proximity to branches. Not surprisingly, therefore, the variation in wood properties within a stem is large and in seasonal climates, the greatest variation is typically found within an annual ring.A great advantage for the study of wood is that the net product of seasonal processes is recorded in the wood structure across the stem radius. Thus by studying the pattern of wood property variation, within the context of its growth history, we can gain insight into cause and effect relationships between the drivers of wood variability. Combining this with temporal, high-resolution measurements of stem growth, weather, and process modelling enables us to better understand and test hypotheses of wood formation and the causes of variability in wood properties.Over recent years and in partnership with industry and other research providers, we have been attempting to model tree growth (Cabala) and cambial activity (TreeRing and CAMBIUM) at a daily time step to explain radial variability in wood properties. CAMBIUM is the latest development of this effort, modelling a population of eucalypt cambial cells, accounting for fibre and vessel formation using physiologically meaningful relationships.     

林木蛋白质组学研究进展

近年来,应用新的基因组学技术来研究林木生长发育以及林木对生物与非生物胁迫的反应已使得人们对林木生物学有了相当大的了解。蛋白质组学是林木生物学的重要内容。本文综述了林木蛋白质组学在群体遗传、遗传作图、逆境生理、组织器官以及木材形成等方面的研究进展,并简要介绍了林木蛋白质组数据库。最后展望了林木蛋白质组学的发展前景。     

A model to simulate the gravitropic response and internal stresses in trees,considering the progressive maturation of wood

Key message

The developed model of gravitropism takes non-instantaneous maturation of wood into account which enabled to correctly simulate different gravitropic phases and realistic internal stress profiles.

Abstract

A new biomechanical model of tree movement in relation to gravity (gravitropism) is proposed in this study. The modelling of the progressive maturation of wood cells is taken into account, as well as spatio-temporal variations in maturation strains (MS) and mechanical properties. MS were identified using an inverse method that allows the model to fit the gravitropic reaction observed experimentally. For this purpose, the curvature during righting movement, the geometry and the mass distribution of a two-year-old poplar tree was measured. The identified MS are higher than expected, which shows the underestimation of MS by usual measurements. By using the same mechanical parameters and MS as an input, the model gives satisfying results in terms of shape modelling for different trees up to 32 days after tree tilting. The model is able to simulate the latency phase observed in the tree righting movement, and the internal stress profile in the trunk is realistic (low compressive value in the central part of the trunk and zero stress in newly formed cells). The next development of the model will aim to simulate the end of the gravitropic phase in relation with the regulation of MS by the tree.     

Biomechanical Plasticity Facilitates Invasion of Maritime Forests in the southern USA by Brazilian pepper (Schinus terebinthifolius)

Biomechanical plasticity and within-species growth form diversity are traits that can facilitate invasion by non-native plant species. We support this argument with evidence from the invasion of coastal habitats in northern Florida, USA, by Schinus terebinthifolius and describe some of the consequences of this invasion for overtopped saltmarsh plants. In crowded stands, Schinus grows more like a vine than a tree, with stem height : diameter ratios nearly twice than those observed in open-grown individuals but with no changes in wood density or the modulus of elasticity of stem material. When extracted from the surrounding vegetation, the formerly crowded Schinus stems buckle under their own weight. Schinus crowns also extend much further over adjacent saltmarsh than crowns of Juniperus virginiana, the only other tree species abundant in the study site. Along forest edges, the above-ground biomass of saltmarsh plants overtopped by Schinus crowns was reduced by more than an order of magnitude. The biomechanical plasticity of Schinus allows it to adapt its growth form to suit habitat conditions and can dominate the edges of salt marshes as a sprawling shrub and maritime forests as either a free-standing tree or a woody vine, depending on stand crowding     

Linking stem cell function and growth pattern of intestinal organoids

Intestinal stem cells (ISCs) require well-defined signals from their environment in order to carry out their specific functions. Most of these signals are provided by neighboring cells that form a stem cell niche, whose shape and cellular composition self-organize. Major features of this self-organization can be studied in ISC-derived organoid culture. In this system, manipulation of essential pathways of stem cell maintenance and differentiation results in well-described growth phenotypes.We here provide an individual cell-based model of intestinal organoids that enables a mechanistic explanation of the observed growth phenotypes. In simulation studies of the 3D structure of expanding organoids, we investigate interdependences between Wnt- and Notch-signaling which control the shape of the stem cell niche and, thus, the growth pattern of the organoids. Similar to in vitro experiments, changes of pathway activities alter the cellular composition of the organoids and, thereby, affect their shape. Exogenous Wnt enforces transitions from branched into a cyst-like growth pattern; known to occur spontaneously during long term organoid expansion. Based on our simulation results, we predict that the cyst-like pattern is associated with biomechanical changes of the cells which assign them a growth advantage. The results suggest ongoing stem cell adaptation to in vitro conditions during long term expansion by stabilizing Wnt-activity.Our study exemplifies the potential of individual cell-based modeling in unraveling links between molecular stem cell regulation and 3D growth of tissues. This kind of modeling combines experimental results in the fields of stem cell biology and cell biomechanics constituting a prerequisite for a better understanding of tissue regeneration as well as developmental processes.     

Source-driven remobilizations of nutrients within stem wood in Eucalyptus grandis plantations

Nutrient remobilizations in tree ligneous components have been little studied in tropical forests. A complete randomized block design was installed in Brazilian eucalypt plantations to quantify the remobilizations of phosphorus (P), potassium (K), calcium (Ca), magnesium (Mg), and sodium (Na) within stem wood. Three treatments were studied: control with neither K nor Na addition (C), 3 kmol ha^?1 K applied (+K), and 3 kmol ha^?1 Na applied (+Na). Biomass and nutrient contents were measured in the stem wood of eight trees destructively sampled at 1, 2, 3 and 4 years after planting in each treatment and annual rings were localized on discs of wood sampled every 3 m in half of the trees. Chemical analyses and wood density measurements were performed individually for each ring per level and per tree sampled. Nutrient remobilizations in annual rings were calculated through mass balance between two successive ages. Our results show that nutrient remobilizations within stem wood were mainly source-driven. Potassium and Na additions largely increased their concentration in the outer rings as well as the amounts remobilized in the first 2 years after the wood formation. The amount of Na remobilized in annual rings was 15 % higher in +Na than in +K the fourth year after planting despite a 34 % higher production of stem wood in +K leading to a much higher nutrient sink. A partial substitution of K by Na in the remobilizations within stem wood might contribute to enhancing Eucalyptus grandis growth in K-depleted soils.     

Stem biomechanics of three columnar cacti from the Sonoran Desert

The allometric relationship of stem length L with respect to mean stem diameter D was determined for 80 shoots of each of three columnar cactus species (Stenocereus thurberi, Lophocereus schottii, and S. gummosus) to determine whether this relationship accords with that predicted by each of three contending models purporting to describe the mechanical architecture of vertical shoots (i.e., geometric, stress, and elastic similitude, which predict L proportional to D(alpha), with alpha = 1/1, 1/2, and 2/3, respectively). In addition, anatomical, physical, and biomechanical stem properties were measured to determine how the stems of these three species maintain their elastic stability as they increase in size. Reduced major axis regression of L with respect to D showed that alpha = 2.82 ± 0.14 for S. thurberi, 2.32 ± 0.19 for L. schottii, and 4.21 ± 0.31 for S. gummosus. Thus, the scaling exponents for the allometry of L differed significantly from that predicted by each of the three biomechanical models. In contrast, these exponents were similar to that for the allometry previously reported for saguaro. Analyses of biomechanical data derived from bending tests performed on 30 stems selected from each of the three species indicated that the bulk stem tissue stiffness was roughly proportional to L2, while stem flexural rigidity (i.e., the ability to resist a bending force) scaled roughly as L3. Stem length was significantly and positively correlated with the volume fraction of wood, while regression analysis of the pooled data from the three species (i.e., 90 stems) indicated that bulk tissue stiffness scaled roughly as the 5/3-power of the volume fraction of wood in stems. These data were interpreted to indicate that wood served as the major stiffening agent in stems and that this tissue accumulates at a sufficient rate to afford unusually high scaling exponents tot stem length with respect to stem diameter (i.e., disproportionately large increments of stem length with respect to increments in stem diameter). Nevertheless, the safety factor against the elastic failure of stems (computed on the basis of the critical buckling height divided by actual stem length) decreased with increasing stem size tot each species, even though each species maintained an average safety factor equal to two. We speculate that the apparent upper limit to plant height calculated for each species may serve as a biomechanical mechanism for vegetative propagation and the establishment of dense plant colonies by means of extreme stem flexure and ultimate breakage, especially for S. gummosus.     

Variations in maturation strains and root shape in root systems of Maritime pine (Pinus pinaster Ait.)

 In order to determine if different types of wood were being laid down in the root system of Maritime pine (Pinus pinaster Ait), in response to wind loading, longitudinal residual maturation strains (LRMS), indicating the existence of mechanical stress in developing wood cells, were measured in the trunk and lateral roots. Two age groups of trees (5- and 13-year- old) were compared. LRMS were greater in the trunk and roots of 13-year-old trees than in 5-year-old trees. This phenomenon may be due to increased competition between older trees. LRMS in leeward roots of both age-groups were positive i.e. the wood cells had developed under compression, as also occurs in reaction wood of gymnosperms. As leeward roots are placed under compression during tree sway, an abnormal type of wood may form in the roots in order to counteract the increased stress. In other roots, the strains were negative i.e. the cells had developed under tension, as occurs in normal wood. In the roots of younger trees, LRMS were also positive nearer the stem, thus indicating that wood formation may also be influenced by bending stresses experienced in this zone. In addition to LRMS measurements, radial growth in roots was examined in order to determine the influence of mechanical loading on secondary growth. In older trees, there was a significant increase of 34% in woody growth below the biological centre, compared to that above. This eccentricity is unlike that found in most other tree species, where secondary growth is usually greater on the upper side of the root. However, Maritime pine has a tap root, which will alter the pattern of stress within the root system. Under wind loading, a concentration of mechanical stress will develop at the bases of the stem, lateral roots and tap root. Received: 7 July 1997 / Accepted: 11 December 1997     

Woody plant and gravity]

In this review, we attempted to summarize the effect of gravity on growth of woody plants, broad leaved trees, on earth. It is well known that in tilted broad leaved trees, tension wood formed in the secondary xylem causes negative gravitropism. Gibberellin has been shown to induce tension wood in weeping branch, causing its upright growth. Recent study has shown that seedling of Japanese cherry tree grown on three dimensional clinostat, a device that simulates microgravity, grew at random angles, and that the formation of secondary xylem, as supporting tissue for upright growth, decreased. In the decreased xylem formation, the inhibition of the differentiation and development of fiber cell was clearly observed. These results suggest that in attitude control and morphogenesis of stem in woody plant, secondary xylem formation seriously relates to gravity on earth. In woody plant, the mechanism of gravity perception and the following signal transduction have not yet been elucidated, although the recent study reported the possibility that endodermal starch sheath cells and plant hormones may play some role in the mechanism. Space experiment for woody plant is expected to study these problem.