<figure><img src="image_219.png" /><figcaption>Fig. 4. Transverse section of mung bean hypocotyl in the growing zone; epidermis and parenchyma with primary cell wall. ×450.</figcaption></figure>

<figure><img src="image_220.png" /><figcaption>Fig. 5. Same level, parenchyma; view of a cell corner with local ('rib') layered thickenings of the cell wall. G, intercellular gas space. ×1200.</figcaption></figure>
microstructural viewpoint. When the thickened parts of the growing cell wall are observed in ultrathin sections, the characteristic so-called arced layers are seen, especially in the youngest domains, i.e., in the upper levels and near the plasmalemma (Fig. 6).

It is now well-known that arced layering is frequent in microfibrous biological composites and is indicative of helicoidal architecture. It corresponds to a progressive rotation of the orientation of the successive cellulosic microfibrils which are secreted and deposited beyond the plasmalemma. The whole construction forms a twisted plywood (Fig. 7). The cell wall shows, for a while, a liquid crystal-like architecture (of the cholesteric or chiral nematic type). The

<figure><img src="image_221.png" /><figcaption>Fig. 6. Ultrastructural organization of the cell wall of epidermis in the course of growth; arced patterning near the plasmalemma (pl) and progressive dispersion in the former parts (arrows). cu, cuticle. ×20,000, reduced to 90% for reproduction.</figcaption></figure>
<figure><img src="image_222.png" /><figcaption>Fig. 7. Orientation of the cellulose microfibrils in a twisted plywood: (A) section for five successive positions; (B) face view.</figcaption></figure>

construction of the helicoidal architecture—from a single arc to complete arced layering—is briefly summarized in Fig. 8. Neville (1993) gives a full explanation of this characteristic mesophase and its specific properties. Each layer of cellulosic microfibrils is oriented in parallel sheets; the direction of the microfibrils changes through a small angle from sheet to sheet (helicoidal geometry is thus somewhat comparable to the change in direction of the grain of a wooden spiral staircase).

The spatial disposition of the microfibrils is multidirectional; it is efficient and well adapted, owing to the longitudinal intramolecular arrangement of the cellulose [close array of β-(1,4) glucan chains], to resist tension. The primary wall composite combines multidirectional microfibrils of high stiffness with a matrix of lower stiffness. It is specially suited to resist shearing forces coming from all directions; fracture energy is absorbed and dissipated through the laminated composite. The propagation of fractures will be prevented or limited and buckling reduced. From the mechanical point of view, it must be emphasized that the skeletal microfibrils are strategically oriented so as best to be able to cope with various tensile stresses and strains operating upon them, both from an internal source (weight, etc.) and from environmental origin (wind, etc.).

Therefore, on one hand, the twisted architecture optimizes the distribution of forces and has the advantage to secure the best multidirectional strengthening supported by young and slender organs, and, on the other hand, the liquid crystal-like phase remains fluid enough and the molecules are internally mobile when forces are acting (in the present case, tension due to osmotic pressure).
<figure><img src="image_223.png" /><figcaption>**Fig. 8.** Diagrammatic construction of the helicoidal pattern: (A) section of one arc with progressive rotation; (B) three-dimensional view of neighboring arcs; (C) nested arcs associated in a layer; (D) section of successive arcs assembled at the plasmalemma interface.</figcaption></figure>

Submitted to shearing forces, the helicoids are stretched and a dramatic new molecular arrangement occurs. The primary cell wall construction is transient: When moving toward the base of the growth gradient and/or the outer part of the cell wall, the helicoidal texture changes and progressively vanishes. Identical gradients of destruction are seen on sections for different tissues (arrows, Fig. 9A). In the older parts of the cell wall, the nested arcs become less clear and the layering disappears (Fig. 9B). In the course of growth, the cholesteric-like ordering is used and destroyed by the surface expansion (Fig. 9C). The cell wall becomes thinner, and when no more helical order is seen, growth ceases.
<figure><img src="image_224.png" /><figcaption>Fig. 9. Changes in epidermal cell wall ultrastructure along the growth gradient. (A) General view of the progressive change (arrows) from a helicoidal texture (recent) to a scattered texture (older and stretched). pl. plasmalemma; cu, cuticle. (B) The very end of the growth gradient (epidermis): thinning of the cell wall and dispersion of the last layer. (C) Complete dispersion after growth. (A, B) ×26,000 and (C) 32,000; reduced to 90% for reproduction.</figcaption></figure>
MECHANICAL PROPERTIES OF THE GROWING CELL WALL

It is usually difficult to study the mechanical properties of a growing cell wall of a living specimen. It is necessary to exclude cytoplasmic components to cancel out the action of the internal vacuolar pressure and use external tension as a substitute. Different procedures can be used to eliminate the cytoplasmic components and preserve cell wall microstructures. More often, the technique requires either hot methanol followed by the action of a pronase (Cleland, 1981), simple boiling in water, or freezing. According to the method used, the functional integrity of the cell wall microstructures is more or less preserved, especially the *in muro* proteins and enzymatic activities.

To study the rheological properties of biological materials, various methods can also be used (Prat and Paresys, 1987), for example, (1) measurement of the relaxation capacities of tissues under constant deformation [sequential stress/relaxation, developed mainly by the Japanese school (Yamamoto *et al.*, 1975)], (2) tension response of specimens under a cycle of tension/relaxation at a constant rate of deformation (Fig. 10) [improved by Cleland (1981, 1984)], and (3) simple measurement of deformation during a sequential tension/relaxation (Fig. 11). The latter, the easiest to perform, allows the distinction between an irreversible plastic (P) deformation and a reversible elastic (E) deformation, comparable to those observed in natural conditions for a living specimen (Fig. 3). In all cases, a decrease in the intrinsic wall plasticity is noticed when the cell walls are prepared from specimens whose growth has ceased (Fig. 10B).

From the deformation curves obtained under a constant load (Fig. 11A),

<figure><img src="image_225.png" /><figcaption>Fig. 10. Extensibility measurements at a constant rate of deformation of segments excised from the elongating zone (A) and nonelongating zone (B) of *Vigna radiata* hypocotyls. The segment is extended until a 40-g stress is reached, relaxed, then reextended until 50 g is reached. Arrows indicate the direction of the deformation.</figcaption></figure>
<figure><img src="image_226.png" /><figcaption>Fig. 11. Extensibility measurements under a constant load of a segment of the elongating zone of *Vigna radiata* hypocotyl. (A) Deformation after loading (+) then unloading (-). The force used is 40 g. P and E represent the plastic and elastic deformations. The creep represents the slow deformation under a constant load. (B) Creep plotted versus log time.</figcaption></figure>

an immediate elasticity and a progressive plasticity can be distinguished. This low deformation, or creep, is the most comparable to the actual growth of a living specimen. However, its exponential shape (Fig. 11B) is significantly different from a growth curve obtained when the rate is steadily maintained.

An important part of the immediate elasticity is due not to the wall structures but to the alveolar (cellular) organization of plant tissues (Fig. 12) and has as a consequence, to produce reversible volume changes in relationship with external pressure changes. It seems that only the "low plasticity," or creep, can be correlated with the intrinsic properties of the cell wall structure.
<figure><img src="image_227.png" /><figcaption>Fig. 12. Schematic behavior of a piece of tissue during longitudinal stress: A great part of the deformation is due to the elastic deformation of the cell wall network.</figcaption></figure>

Among the main physiological factors regulating expansion, the mechanism of action of a growth hormone, auxin, is now the best known. It is generally considered that auxin has a first and short action via secretion of protons. This produces a lowering of pH within the wall compartment, which can be experimentally mimicked by the addition of an acid buffer to the incubating medium. Figure 13A shows the response of a living specimen: A pH of 5 highly and rapidly stimulates elongation. No significant response is noticed with a killed specimen under tension, at the same pH (Fig. 13B) (Rayle and Cleland, 1972; Prat, 1987). Only a lower pH (pH 3) produces a stimulation. The latter result is explained by a purely chemical process (ruptures of weak linkages), whereas the pH 5 stimulation requires the integrity of the cell structure and is probably of a physiological and biochemical nature. It is possible to check the point using, in the same experiment, specimens killed in different ways (Fig. 14). Specimens killed by methanol show no reaction for a moderately acid pH (pH 4.7); conversely, specimens simply frozen and thawed (in which the enzymatic activities are preserved) react rapidly.
<figure><img src="image_228.png" /><figcaption>Fig. 13. Effect of low pH's on elongation of a living segment (A) and deformation under a constant load of a killed segment (B). The segments were excised from the elongating zone of *Vigna radiata* hypocotyls when cuticle was abraded with carborundum powder. In B, the segment was killed by boiling in methanol during 5 min. The segments were incubated with 5 mM phosphate buffer, pH 7.7. The medium was changed to phosphate buffer, pH 4.7 (first arrow), then to HCl solution, pH 3 (second arrow).</figcaption></figure>

Numerous diagrams have been proposed for the supramolecular architecture of the growing wall. One of the last published (Carpita and Gibeaut, 1993), for example, shows with great accuracy the intermolecular relationship which allows the assembly of the network. It is now widely accepted that specific hemicelluloses (xyloglucans) play a major role in the cellulosic microfibril connection. Transglucosidases seem to be good candidates for the regulation of cell wall plasticity (Nishitani and Tominaga, 1991). Their optimum pH (acidic) agrees well with the role of low pH in growth stimulation. Numerous factors modulate
<figure><img src="image_229.png" /><figcaption>Fig. 14. Effect of preparation of killed segments on responses to low pH. The segments excised from the elongating zone were abraded with carborundum powder, then boiled for 5 min in methanol (M) or freeze-thawed (FT). The segments were incubated in 5 mM phosphate buffer, pH 7.7, then transferred (arrow) to phosphate buffer, pH 4.7.</figcaption></figure>

the local intrawall pH, especially growth hormones, such as auxin, and specific wall enzymes, namely, pectinmethylesterases.

CONCLUSION

The mechanical properties of the primary cell wall are the key features for the regulation of plant growth (Cosgrove, 1993). Knowledge of them is essential to understanding the successive events which interact in the course of the surface expansion (exponential increase, progressive decrease, and final cessation) of the cell wall. However, experimental measurement of these properties indicates only the immediate growth potentialities or, more precisely, the "immediate past wall extensibility" (Cleland, 1984) when the tissue was excised. The later properties will depend on the intrinsic potentialities of the wall but also on the actual regulating processes controlled by the cell biology: (a) synthesis and assembly of the wall components and (b) biochemical and enzymatic reactions interacting within the previously secreted components, which continuously change the positioned network.

It is nearly impossible to deduce the properties of one cell from the study of the whole specimen, for various reasons: (a) the cell number is very high (about $10^5$ cells in a 10 mm-long segment); (b) the size of associated cells in the course of growth is heterogeneous (diameter from 5 µm for procambial cells to 100 µm for pith cells); and (c) in a given cell, the wall is a sort of puzzle of microdomains, each with a different thickness, composition, and structure.
Experiment and structure are found to agree well with the so-specific behavior of the growing cell wall, associating dramatic surface expansion and supporting function. The cholesteric mesophase, owing to its short-lived multidirectional architecture, appears to be outstandingly important in controlling the mechanical efficiency of growing plant organs. However, numerous questions still need to be answered if we are to understand the detail of dispersion and randomization of the cellulosic network (Figs. 15 and 16). The risk of modeling is an oversimple explanation: The primary wall cannot be considered

<figure><img src="image_230.png" /></figure>

Fig. 15. Diagram of scattering and change in the cellulose microfibril density according to surface extension; two hypothesis concerning the behavior of the fibrils during growth process. (A) Fibril arrangements before growth in a three-dimensional model. (B, C) Fibril arrangements after growth (×5 in length): (B) first hypothesis—sliding and change of orientation of the fibrils according to expansion; (C) second hypothesis—microbeaks without sliding or reorientation. The orientation of the fibrils is indicated as degrees. (A', B', C') Superimposed views of A, B, and C layers.
<figure><img src="image_231.png" /><figcaption>Fig. 16. Time change of the wall arced patterns during cell elongation. The fibrils of cellulose were synthesized by cellulose synthetase located in the plasmalemma (Pl). In the periplasmic interface, the fibrils were self-assembled in the inner layer of the cell wall. During the same time, the outer (former) layers of the cell wall were dispersed. An identical diagram is obtained according to the first or the second hypothesis described in the legend to Fig. 15.</figcaption></figure>
as an ordinary composite material with physical preestablished properties; it is
a specific biological compartment continuously renewed and locally modified.

REFERENCES

Cleland, R. E. (1981). Wall Extensibility: Hormones and wall extension. In Tanner, W., and Loewus, F. A. (eds.), *Plant Carbohydrates. II. Extracellular Carbohydrates*, Springer Verlag, Berlin, pp. 255-269.

Cleland, R. E. (1984). The Instron technique as a measure of immediate-past wall extensibility. *Planta* **160**:514-520.

Cosgrove, D. J. (1993). How do plant cell walls extend? *Plant Physiol.* **102**:1-6.

Frey-Wyssling, A., and Mühlethaler, K. (1963). *Ultrastructural Plant Cytology*, Elsevier, Amsterdam.

Carpita, N. C., and Gibeaut, D. M. (1993). Structural models of primary cell walls in flowering plants: Consistency of molecular structure with the physical properties of the walls during growth. *Plant J.* **3**:1-30.

Neville, A. C. (1993). *Biology of Fibrous Composites*. Development Beyond the Cell Membrane, Cambridge University Press, Cambridge.

Nishitani, K., and Tominaga, R. (1991). In vitro molecular weight increase in xyloglucans by an apoplastic enzyme preparation from epicotyls of *Vigna angularis*. *Physiol., Plant.* **82**:490-497.

Prat, R. (1985). A comparative study of some methods for measuring cell growth potentials in *Vigna radiata* hypocotyls, *in situ* and after excision. *J. Exp. Bot.* **36**:1150-1158.

Prat, R. (1987). Interrelations between IAA- or H⁺-induced cell elongation and cell wall extension in mung bean hypocotyl. *Plant Physiol. Biochem.* **25**:401-408.

Prat, R., and Paresys, G. (1987). Multiple use apparatus for cell wall extensibility and cell elongation studies. *Plant Physiol. Biochem.* **27**:955-962.

Prat, R., Vian, B., Reis, D., and Roland, J. C. (1977). Evolution of internal pressure, vacuolation and membrane flow, during cell growth in mung bean hypocotyl. *Biol. Cell.* **28**:269-280.

Rayle, D. L., and Cleland, R. E. (1972). The *in-vitro* acid-growth response: Relation to *in-vivo* growth responses and auxin action. *Planta* **104**:282-296.

Roland, J. D., Reis, D., and Vian, B. (1992). Liquid crystal order and turbulence in the planar twist of the growing plant cell wall. *Tissue Cell* **24**:335-345.

Silk, W. K. (1992). On the kinematics and dynamics of plant growth. In Karalis, T. K. (ed.), *Mechanics of Swelling*, Springer Verlag, Berlin Heidelberg, pp. 165-177.

Yamamoto, R., Kawamura, H., and Masuda, Y. (1975). Stress relaxation properties of the cell wall of growing intact plants. *Plant Cell Physiol.* **15**:1073-1082.
Volume 2, Number 3

Special Issue:
Plant Biomechanics Congress

September 1994

BIMIEL 2(3) 193-282 (1994)

ISSN 1059-0153

Biomimetics

PLENUM PRESS • New York and London
Biomimetics

Volume 2, Number 3

September 1994

CONTENTS

Special Issue: Plant Biomechanics Congress

Montpellier, France, September 5-9, 1994

Morphogenesis and Training of Trees: Questions from Physiology to Biomechanics

Jacques Crabbé

193

Kinematics and Dynamics of Primary Growth

Wendy Kuhn Silk

199

Plant and Fungal Cell Growth: Governing Equations for Cell Wall Extension and Water Transport

Joseph K. E. Ortega

215

Tree Biomechanics: Growth, Cumulative Prestresses, and Reorientations

Meriem Fournier, Henri Bailleres, and Bernard Chanson

229

Stem and Root Tree Architecture: Questions for Plant Biomechanics
C. Edelin and C. Atger

253

The Biomechanics of Leaf Rolling

Bruno Moulia

267
# Morphogenesis and Training of Trees: Questions from Physiology to Biomechanics<fnref n="1" />

Jacques Crabbé<fnref n="2" />

A review is presented with comments on the various effects on plants—partic-
ularly on trees—exerted by gravity, with or without other mechanical stress of
different origin. Questions are raised about these phenomena, which would
require collaboration with a biomechanical approach in order to improve the
physiological understanding of the sequence of events from perception of the
stimulus to growth response.

KEY WORDS: trees; gravity stimulus; gravitropism; reaction wood; gravimorphism; plagiotropy.

Among the factors acting mechanically on self-sustaining, tall plants, such as
trees, **gravity** is certainly the most prevalent and ubiquitous. While growing
and ramifying, the tree constantly has to resist—through its growth processes—
the constraints generated by the weight of its limbs. Other external factors (wind,
rain, snow), of course, intervene, but mostly because they amplify the strain or
change the direction of the resultant force. Incidentally a gust of wind may
increase the lateral component force by bending a trunk or a branch—perfectly
self-supporting at the start—and lead to its sudden buckling. Nevertheless, an
analysis of the problems to bring together biomechanics and physiology must
first consider the effects of gravity.

The different growth manifestations wherein a perception of gravity can be
suspected are presented in Fig. 1 and its legend. These phenomena are essential
to describe the architectural models of the trees (Troll, 1937; Rauh, 1939; Hallé
et al., 1978) and to understand the changes that they undergo in various cir-
cumstances. For example, many horticultural practices, such as staking of trunks,

<footnote n="1">¹This is the published version of a paper presented at the Plant Biomechanics Congress, Mont-
pellier, France, September 5-9, 1994.</footnote>

<footnote n="2">²Faculté des Sciences agronomiques, Morphogenèse végétale appliquée, Avenue de la Faculté 2, B-5030 Gembloux, Belgium.</footnote>
<table><thead><tr><th></th><th>HERBACEOUS SHOOTS</th><th>WOODY SHOOTS</th></tr></thead><tbody><tr><td>TRUNKS OR MAIN STEMS</td><td>1. GRAVITROPISM</td><td>2. REACTION WOOD</td></tr><tr><td>BRANCHES OR LATERAL STEMS</td><td>4. CROTCH-ANGLE FORMATION<br>EPINASTY</td><td>5. ORTHOTROPY<br>5bis. HYPOTONY<br>5ter.</td></tr><tr><td></td><td>6. PLAGIOTROPY<br>4ter.<br>HYPONASTY</td><td>6bis. AMPHITONY<br>EPITONY (gravimorphism)<br>4c.</td></tr></tbody></table>
Fig. 1. Diagrammatic representation of the various reactions of trunks and branches implying gravity perception. Trunks or main stems. (1) *Gravitropism* is a primary growth reaction of herbaceous shoots, wherein the curvature results from an unequal distribution of growth factors (auxin), according to a still controversial theory (Fim and Digby. 1980; Wilkins, 1984). (2) *Reaction wood* contributes to putting the trunk upright again, when the tree is moderately or lowly tilted. The response occurs at the level of xylem differentiation (Wilson and Archer, 1977). (3) *Gravimorphism* consists of preferential outgrowth of upward-oriented lateral buds and happens in fast tilting (steep slope, river bank) or sharp bending (buckling through excess load, manual arching). This is discussed by Crabbé and Lakhoua (1978) and Crabbé (1987). *Branches or lateral stems*. (4) Young herbaceous shoots initially form a *crotch angle* due to intraangular pressure of tissues (Jankiewicz, 1964). Later they may curve upward—hyponasty (4cr), generally a gravitropic reaction—or downward—epinasty (4bis), often a prelude to plagiotropism. (5) *Orthotropy* is a nearly erect growth direction of an axis, except for the formerly established crotch angle. Its lateral buds or shoots sometimes show *hypotony* (5bis), a differential development favoring those at the lower side. The opposite reaction, *epitony* (5ter), appears in gravimorphism, i.e., when a branch is rapidly arched by hand or by its own weight. (6) *Plagiotropy* is a nearly horizontal growth direction of an axis, depending or not on a prevailing apical dominance. Often, lateral buds and shoots preferentially develop in a horizontal plane: *amphitony* (6bis). As far as the direction of growth is concerned, both orthotropy and plagiotropy require a balanced formation of reaction wood to maintain the direction (Fisher and Stevenson, 1981). Both also demand perception of gravity, only the translations of the stimulus in the proper response are different.
palisading, and bending of branches, disturb the plant's primary reactions to gravity and related factors.

From left to right, Fig. 1 shows an increasing gradient in complexity of the plant's response. In herbaceous shoots, the gravitropic reaction consists of a different rate of cell extension between the upper and the lower halves of the subapical growing zone of the organ. The current explanation, known as the Cholodny-Went hypothesis, requires a regulated asymmetrical distribution of growth factors, to a sufficient extent, through that zone: a situation which is thought doubtful in dicotyledonous shoots (Firn and Digby, 1980). In woody shoots anyway, such a direct growth curvature is no longer possible and different mechanisms are involved. As far as reaction wood formation is implied (upright bending of a trunk, maintenance of ortho- or plagiotropy in branches), the sequence of events from the perception of the gravity stimulus to the response may be short. On the other hand, when the reaction consists of differential bud formation and outgrowth, as in hypo- and epitony and gravimorphism, correlative influences between plant parts clearly play an essential role in its expression.

Our knowledge is still deficient and controversial about all of these phenomena. Superficial morphological descriptions and premature hormonal interpretations seem to be the main shortcomings. A correct approach would conform to the classical scheme used in the study of irritability phenomena and neglect none of its steps.

**PERCEPTION of the stimulus (Where? How? Intensity?)**

<figure><img src="image_232.png" /></figure>

**TRANSDUCTION (Nature of the signal? Modes of production and transmission?)**

<figure><img src="image_233.png" /></figure>

**RECEPTION of the signal (Where? When? How is it translated?)**

<figure><img src="image_234.png" /></figure>

**RESPONSE (Where? When? How? Intensity?)**

And somewhere a step of "memorization" or "fixation" would be inserted in the sequence, when a long time lapse occurs between perception and reaction, as demonstrated in gravimorphism (Crabbé and Lakhoua, 1978).

The level of **perception of the stimulus** is probably the one that needs a close collaboration between biomechanics and physiology. Most experimental approaches indicate the stem itself—and almost certainly living tissues within the stem—as the locus of perception. In woody shoots, the cambial zone and the ray parenchymas thus appear to be the possible candidates. It seems reasonable to point to the former, when reaction wood formation is the response, as this process is linked to xylem cell differentiation. But what about hypotony and gravimorphism, where differential bud formation and development are involved?
The very first questions to the biomechanician would consequently be: What are the strains appearing in normal and modified positions of the stem? and How do both candidates undergo these constraints? Further, according to the stiffness of the stem, its actual elongation and diameter growth, and the changes in the distribution of mass subsequent to growth, foliage development, and branching, how are these constraints modified by the mechanical stresses which may be added by spontaneous or artificial bending of the axis? In this respect, is the different response induced by slow or rapid tilting, i.e., reaction wood or gravimorphism, due only to a different intensity of the same stimuli or to a combination of gravity and flexion stresses, whose perceiving structures might be as different as their transduction and response mechanisms evidently are?

A correct and precise evaluation of the constraints and a precise identification of the affected anatomical structures represent the inescapable requirements for further progress in the physiology of these phenomena. First, the physiologist should be able properly to appreciate, in mechanical terms, the treatments that he is applying in his experiments. Further, on the basis of the strains laid upon possible sensor tissues, the next steps toward the response could be identified and described.

Finally, attention should be paid to the effects of staking or, generally speaking, of the suppression of tree swaying, perhaps the simplest of mechanical situations. Foresters (Larson, 1965) and horticulturists (Leiser et al., 1972) have demonstrated that the absence of mechanical stress changes elongation and diameter growth and the shape of the trunk, and causes a significant decrease in flexion and buckling resistance (Barnola and Crabbé, 1993). It is thus evident that gravity and mechanical stress combined, however subtle their effects may be, are an essential but ill-recognized part of the plant's environment. They shape its architecture through many varied stimulus-response processes. Moreover, over many years, man has empirically learned to use them in order to tailor the plant to his needs. This would suffice to unite our skills to understand them better.

## REFERENCES

* Barnola, P., and Crabbé, J. (1993). L'activité cambiale, composante active ou passive dans les réactions de croissance de l'arbre? *Acta Bot. Gallica* **140**(4):403-412.

* Crabbé, J. (1987). *Aspects Particuliers de la Morphogenèse Caulinaire des Végétaux Ligneux et Introduction à Leur Étude Quantitative*, IRSIA, Bruxelles.

* Crabbé, J., and Lakhoua, H. (1978). Arcure et gravimorphisme chez le pommier. Mise en évidence d'effets gravimorphiques sur bourgeons isolés. *Ann. Sci. Nat. Bot.* **19**(2):125-140.

* Firn, R. D., and Digby, J. (1980). The establishment of tropic curvatures in plants. *Annu. Rev. Plant Physiol.* **31**:131-148.

* Fisher, J. B., and Stevenson, J. W. (1981). Occurrence of reaction wood in branches of dicotyledons and its role in tree architecture. *Bot. Gaz.* **142**(1):82-95.

* Hallé, F., Oldeman, R. A. A., and Tomlinson, B. P. (1978). *Tropical Trees and Forests. An Architectural Analysis*, Springer Verlag, Berlin.
Jankiewicz, L. S. (1964). Mechanism of the crotch angle formation in apple trees, *I. Acta Agrobot.* **15**:21-49.

Larson, P. R. (1965). Stem form of young *Larix* as influenced by wind and pruning. *Forest Sci.* **11**(4):412-424.

Leiser, A. T., Harris, R. W., Neel, P. L., Long, D., Stice, N. W., and Maire, R. G. (1972). Staking and pruning influence trunk development of young trees. *J. Am. Soc. Hort. Sci.* **97**(4):498-503.

Rauh, W. (1939). Ueber die Gesetzmässigkeit der Verzweigung und deren Bedeutung für die Wuchsförmen der Pflanzen. *Mitt. Deutsch. Dendrol. Gesell.* **52**:81-111.

Troll, W. (1937). *Vergleichende Morphologie der höheren Pflanzen*, Bornträger, Berlin.

Wilkins, M. B. (1984). Gravitropism. In Wilkins, M. B. (ed.), *Advanced Plant Physiology*, Pitman, London, pp. 163-185.

Wilson, B. F., and Archer, R. R. (1977). Reaction wood: Induction and mechanical action. *Annu. Rev. Plant Physiol.* **28**:23-43.
# Kinematics and Dynamics of Primary Growth<fnref n="1" />

Wendy Kuhn Silk<fnref n="2" />

The spatial distribution of growth is reviewed in terms of growth trajectories and fields of displacement velocity and growth rate. A one-dimensional analysis has been suitable for many studies of roots, stems, and monocot leaves. To analyze growth in two dimensions, growth rate tensors and finite element methods have been used to show the direction and magnitude of the local growth rates. Natural coordinate systems have helped to analyze the growth of curved surfaces such as twining stems. A decade ago, thermodynamic considerations suggested the existence of a nonuniform water potential field in growing tissues. In recent years the water potential field for primary growth has been characterized by comparing the local growth rate to intracellular pressure and osmotic potential. For maize roots in solution culture, when growth is steady, the growth-sustaining influx of water appears to be driven by a radial gradient in osmotic potential.

**KEY WORDS:** plant growth; kinematics; spatial patterns; water potential; root; *Zea mays*.

## INTRODUCTION: PLANT GROWTH AS FLUID FLOW

When seeds of broad-leaved plants germinate, the seedlings grow through the soil with a curved shape, a hook, which is thought to protect the apex from damage. After emergence from the soil, if the plant grows in dim light, the hook is maintained for many hours. As the seedling grows the hook can be said to migrate from the hypocotyl into the first and then the second internode. But from another perspective the shape of the hook can be seen to be steady, i.e., unchanging in time. The steady form is apparent in a moving reference frame

<footnote n="1">¹This is the published version of a paper presented at the Plant Biomechanics Congress, Montpellier, France, September 5-9, 1994.</footnote>

<footnote n="2">²Department of Land, Air, and Water Resources, Hoagland Hall, University of California, Davis, California 95616.</footnote>
attached to the plant apex: A plot of stem curvature vs distance from the plant apex does not change (Fig. 1). The steady form, however, is composed of changing elements. An epidermal hair appears to flow through the hook. Initially found near the shoot meristem, the hair will be observed later at the hook summit and then on the basal straight side of the hook. This observation tells us that the seedling hook is composed of a parade of tissue elements, each of which first curves and then straightens as it is displaced from the apical meristem during growth.

The situation of a form that is unchanging (or slowly changing) in time, and that is composed of elements that are experiencing rapid change, is reminiscent of fluid structures. And indeed, allusions to rivers, fountains, and the wakes of boats appear often in classical botanical literature. The seedling hook is a paradigm for primary growth in plants.

Another example is the root apex, which can be described in terms of regions of cell division, rapid cell expansion, and cell differentiation. These regions are formed by a procession of cellular tissue elements, which first grow slowly while producing new cell walls, then elongate rapidly and then differentiate. If one observes root development from an origin attached to the (moving) apex, tissue elements appear to flow through the standing pattern. In an analogous fashion, the shoot bears a succession of leaves of different developmental stages; during a plastochron (time to make a leaf), a leaf may grow to have the same size, shape, and biochemical composition as its distal neighbor as observed

<figure><img src="image_235.png" /><figcaption>Fig. 1. Seedling hook, a steady structure composed of changing elements. The form (curvature distribution, bottom figures) is maintained over time, while constituent tissue elements first curve and then straighten as they are displaced from the seedling apex during growth. Thus a surface mark seems to flow through the hook. [From Silk (1984); reproduced with permission.]</figcaption></figure>
at the beginning of the plastochron. Shoot form also is produced by a parade of
changing tissue elements.

The analogy between primary growth and fluid flow suggests the use of concepts and numerical methods from fluid dynamics to solve problems in plant development. The distinction between Eulerian (spatial or site-specific) and Lagrangian (material or cell particle-specific) specifications of developmental variables is important for understanding plant development (Green, 1976; Silk and Erickson, 1979; Gandar, 1980). And evaluation of local, convective and material derivatives is necessary for calculating rates of change in growing tissue (Silk and Erickson, 1979; Gandar, 1980, 1983). An especially important application is the characterization of growth (volumetric expansion) itself.

SPECIFICATION OF THE GROWTH FIELD IN ONE DIMENSION

Historically, botanists have observed that primary growth is confined to apical regions of roots and shoots. Marks placed near an apex move away from the soil surface and also separate from each other. Marks placed far from the apex do not separate from each other, although they are found progressively farther from the growing apex. One way of specifying growth is to plot the position of cellular particles, or applied marks, versus time (Fig. 2, top). The resulting "pathline" (Gandar, 1983) or "growth trajectory" (Silk and Wagner, 1980; Hejnowicz, 1984) is a material specification of growth, because a real or material particle is followed. If distance is measured from the apex, the slope of the growth trajectory increases monotonically from small values near the apex to a constant value at the base of the growth zone. To characterize the growth pattern, a family of curves can be obtained to show growth trajectories of particles at different initial positions.

Another useful way to describe growth is with a velocity field, $v(z)$ (Erickson and Sax, 1956). This is a spatial, or Eulerian, specification of growth because it involves the movements of many particles found instantaneously at different locations. The velocity field increases monotonically with position, because progressively more expanding tissue is found between the origin and the point of interest (Fig. 2, center). At the base of the growth zone, where expansion is no longer occurring, the growth velocity acquires a uniform value equal to the elongation rate of the organ. In the maize root, individual particles accelerate to a growth velocity (rate of displacement from the apex) of $3.1 \text{ mm h}^{-1}$. The velocity field can be calculated from short-term marking experiments. The velocity field is used in some important physiological applications, including calculation of biosynthesis rates and material derivatives in growing tissue (Silk and Erickson, 1979; Gandar, 1980).

For the physiologist, the most useful growth descriptor is probably the relative elemental growth (REG) rate (Erickson and Sax, 1956), which engineers
<figure><img src="image_236.png" /><figcaption>**Fig. 2.** Kinematics of growth of the primary root of *Zea mays*. These figures were obtained by computer-assisted analysis of time-lapse marking experiments. Top: Growth trajectories. The growth trajectory is a fate map and can be used with data on spatial distributions of developmental variables to find the time course of these variables. Middle: Growth velocity field. Growth velocity is used to calculate rates of change in expanding tissue. Bottom: Relative elemental growth rate. For the physiologist, this curve is the best way to localize growth.</figcaption></figure>

call the growth strain rate. In one dimension, the REG rate, *r* is the velocity gradient; mathematically, $r = \partial v/\partial z$. If care is taken to choose appropriate spatial and temporal scales, the REG rate field can be evaluated from measurement of the separation of applied marks (Erickson and Silk, 1980). This field
shows quantitatively the location and magnitude of growth within the organ. In the maize root, for example, growth can occur over a length of 11 mm (Fig. 2, bottom). The REG rate maximum, $0.42 h^{-1}$, is found 4 mm from the apex. In contrast, a vine stem may have a smaller REG rate of $0.02 h^{-1}$ distributed over a longer, 200-mm region to give a similar axis extension rate of 2 mm $h^{-1}$ (Silk and Haidar, 1986).

The effect of environmental variation on growth can be quantitatively described with the REG rate field. We know, for instance, that a decline in temperature causes a decrease in the growth rate almost uniformly throughout the growth zone of the maize root (Pahlavanian and Silk, 1988). In contrast, water stress does not affect the REG rate in the apical 3 mm of the maize root but causes elongation to cease in the basal half of the growth zone (Sharp et al., 1988).

A one-dimensional analysis is also suitable for the growth of some gra-mineous leaves. In recent years, studies of monocot leaves have revealed growth patterns in important crop and pasture species and the effects of environmental and genetic variation on the spatial and temporal distribution of growth rate (Schnyder et al., 1987; Schnyder and Nelson, 1989; Paolillo and Sorrels, 1992; Meiri et al., 1992; Bernstein et al., 1993).

## SPECIFICATION OF THE GROWTH FIELD IN TWO DIMENSIONS

### Strain Rate and Vorticity Tensors

Expansion patterns of many leaves and apices can be described with a two-dimensional analysis (Richards and Kavanagh, 1943; Erickson, 1966). The early leaf growth analyses have been shown to be closely related to strain rate tensors of classical fluid dynamics (Silk and Erickson, 1979). The magnitude and direction of growth can be quantified by growth rate tensors, and change in element orientation can be characterized by the vorticity vector. Both growth rate and vorticity are obtained from the velocity field in two dimensions. The growth rate tensor is visualized as the rate at which circles placed on the leaf are deforming into slightly larger ellipses. The length of the axis of the ellipse in a particular direction gives the relative growth rate in that direction; the maximum and minimum relative growth rates are represented by the major and minor axes of the ellipse. Relative area expansion is given by the difference in area between the ellipse and the circle. The vorticity is the rate of rotation (as radians per unit time) and can be considered a measure of the change of orientation experienced by the elements.
Finite-Element Methods

In the past decade, finite-element computational methods have been de-
veloped to reveal the expansion pattern from "landmarks" or material points
on the surface of the growing tissue. In analyzing growth of grape leaves, Wolf
et al. (1986) identified vein intersections as material elements on the leaf surface.
Cartesian coordinates for each element were digitized in a sequence of photo-
graphs of the growing leaf. Particle pathlines were plotted in two dimensions.
Using the vein intersections as vertices, Wolf et al. also triangulated the leaf
surface and used finite-element formulae to find the local area growth rates; the
deformation of each element was completely specified. Their analysis considered
some numerical problems that arise in the measurement of large deformations.
In particular, the "relative growth rate," k1 = ∂A/A ∂t, becomes larger than
the "exponential growth rate," k2 = ∂ ln A/∂t, when ∂A/A becomes larger than
0.1, i.e., when the element area increases by more than 10% during the obser-
vation period. Wolf et al. point out that in the literature the methods giving the
"divergence of velocity" or "relative elemental growth rate" give numbers
close in value to k2. However, trends in growth rate and effects of environmental
variation are more easily detected with the more sensitive measure, k1.

Goodall and Green (1986) introduced several improvements for character-
izing the REG rate tensor. They decomposed growth strains into components
that yield canonical parameters to characterize the local growth pattern. Like
Wolf et al., they avoided Eulerian (site-specific) velocity fields and worked
directly with elements bounded by identifiable landmarks. Their method is a
regression technique and has the advantage of giving error estimates as well as
best-fit values for directionality of the local expansion. The computation meth-
ods of Goodall and Green are proving powerful tools for studies of mechanics
of morphogenesis. Used with nondestructive cellular imaging techniques in-
volving SEM of casts of growing apices (Williams, 1991), the numerical meth-
ods gave quantitative descriptions of leaf formation (Tiwari and Green, 1991)
and the morphogenesis of floral apices (Hernandez et al., 1991). These kine-
matic analyses are also providing the foundation for analysis of the mechanics
of apical morphogenesis (Selker et al., 1992).

Natural Coordinates

Hejnowicz and colleagues (e.g., Hejnowicz and Karczewski, 1993) have
introduced two-dimensional analyses using natural coordinates. The periclines,
lines formed by cell files in apices of roots and shoots, are hypothesized to
correspond to the particle pathlines during growth. The periclines are also as-
sumed to lie along the principal directions of growth. Hejnowicz and colleagues
fit analytical functions to the periclines and use tensorial expressions to calculate
the fields of velocity and growth rate. These studies have shown some interesting
properties of apical growth. For instance, as a consequence of minimizing shear
during growth, the lines of equal displacement velocity prove to be orthogonal
to the particle pathlines.

The analysis of twining growth was also facilitated by the use of a natural
coordinate system (Silk, 1989). Twining plants have a growth rate pattern that
maintains a corkscrew-shaped form. The twining stem was modeled as a tube
surrounding a helix α. Position vectors to points in the stem were given in terms
of α, the Frenet vectors, and the natural coordinates s representing arc length
along the helix; φ, angular displacement from the unit normal; and r, distance
along the radius of the tube (Fig. 3, top). Using growth velocities measured
along the helix (Silk and Haidar, 1986), a form of the strain rate tensor was
calculated for the curvilinear, nonorthogonal, natural coordinate system, and the
elongation rates were found for line elements in the twining stem (Fig. 3,
bottom). For stems such as *Pharbitis nil*, which twine clockwise and simulta-
neously twist counterclockwise about their stem axis, the growth rate maximum
occurs on the bottom surface of the stem.

DYNAMICS OF GROWTH

While the REG rate field, described above, provides the kinematic basis
for growth analysis, an understanding of growth in biomechanical terms must
also involve growth dynamics, i.e., the study of the forces and energies involved
in producing the observed spatial and temporal expansion patterns. The plant
cell is typically 85% water; and, to a first approximation, local volume increase
is caused by a local water influx. The driving force for growth is thought to be
the gradient in water potential, i.e., the gradient in the chemical potential energy
of water in the tissue.

Empirically, the cell growth rate has been shown to depend on the exten-
sibility of the wall and the difference between the turgor pressure and some
yield stress of the wall. The mechanics of cell wall expansion is an extensively
studied topic. Other contributors to this symposium are reviewing work on
mechanical properties of cell walls, on the development of new analytical chem-
ical techniques to understand the structure of the growing wall, and on recent
mechanical models for tissue expansion. An understanding of the role of water
potential is also important for growth dynamics. Here I review the theory and
empirical measurements of growth-sustaining water potentials in multicellular
tissues.

Equation for Growth-Sustaining Water Potential

To find the relationship between REG rate and water potential, $\psi_i$, one
must consider the growth velocity, $\mathbf{v}$ of the moving tissue element as well as
the flux $j$ (volumetric flow rate per unit cross-sectional area) of the water move-
<figure><img src="image_237.png" /></figure>

<figure><img src="image_238.png" /></figure>

tapered, rotating

Fig. 3. Top: Natural coordinate system for twining growth. The curvilinear coordinate *s* is distance along the central helix *α* (left). The Frenet vectors *t*, *n*, and *b* vary with *s*; and the coordinates *φ* and *r* are defined in terms of the Frenet trihedron (middle). Surfaces can be visualized corresponding to constant values of the natural coordinates (right). [Redrawn from Silk (1989); with permission.] Bottom: Growth rate patterns that produce twining in a tapered stem that is twisting counterclockwise about *t*. The numbers give the elongation rates of lines, initially 0.5 mm, long drawn on the surface in a longitudinal (left) or circumferential (middle) directions or along the radius in the normal plane (right). Rates are microns per hour. The growth rate maximum occurs on the lower surface. [Drawings from Silk (1989); reproduced with permission.]
ment, for only water that is moving faster than the growing element will cross the element boundary to enlarge the cell (Silk and Wagner, 1980). In symbols,

$$ \int_S (j - u) \cdot \eta \, dS = \int_S - (K_z \cdot \nabla \psi) \cdot \eta \, dS \quad (1) $$

Flux across surface = conductivity · driving force

where $\eta$ represents the unit normal to the surface, $S$, $\psi$ is the water potential, and $K_z$ is the hydraulic conductivity tensor. Then by the divergence theorem,

$$ \int_V \nabla \cdot (j - u) \, dV = \int_V \nabla \cdot (-K_z \cdot \nabla \psi) \, dV \quad (2) $$

Since water is largely incompressible,

$$ \nabla \cdot j \approx 0 \quad (3) $$

And we find that the fundamental relationship between growth at a point and water potential at a point in an expanding continuum is given by

$$ L = \nabla \cdot (K_z \cdot \nabla \psi) \quad (4) $$

where $L = (\nabla \cdot u)$ is recognized as the REG rate.

## Simplified Versions of Eq. (4)

Some simplifying assumptions make Eq. (4) more tractable. For the maize root, one can assume that the tissue is cylindrical, with radius $r$, and growing only in the direction of its long axis. The distribution of $\psi$ is axially symmetric. The growth pattern is steady, so that the equation may be treated as a time-independent problem with coordinate origin at the root tip. It is recognized that the frame of reference is moving at constant velocity and that one is solving for spatial values of $\psi$. Conductivities in the radial ($K_r$) and longitudinal ($K_z$) directions are independent so that radial flow is not modified by longitudinal flow. The boundary condition is to assume that the root is not transpiring and is growing in pure water or saturated air, i.e., $\psi = 0$ in the nongrowing region and outside the root. If these assumptions are made, then Eq. (4) becomes

$$ K_z \frac{\partial^2 \psi}{\partial z^2} + K_r \frac{\partial}{\partial r} \left( r \frac{\partial \psi}{\partial r} \right) + \frac{\partial K_z}{\partial z} + \frac{\partial \psi}{\partial z} + \frac{\partial K_r}{\partial r} \frac{\partial \psi}{\partial r} = L(z) \quad (4A) $$

Equation (4A) implies that if the spatial distributions of REG rate and hydraulic conductivity are known, the growth-sustaining distributions of $\psi$ can, in theory, be calculated. As implied in the section on growth kinematics, the REG rate distribution has long been known (e.g., Erickson and Sax, 1956). Good empirical data on radial and longitudinal hydraulic conductivities are es-
sential for solution of Eq. (4) and evaluation of the growth-sustaining $\psi$ field. However, measurements of hydraulic conductivity in the growing regions of maize roots have taken longer to achieve.

## Estimates for Hydraulic Conductivity

A mathematically rigorous analysis of water transport in a growing root, properly accounting for tissue geometry, is lacking in the literature. Nevertheless, several different experimental and numerical approaches seem to be yielding estimates of the same order of magnitude for hydraulic conductivity through cortical cells. Radial transport of water through the nongrowing cortex of maize roots (30–70 mm from the tip) was measured by Ginzburg and Ginsburg (1970). Correcting their value of conductivity ($1.2 \times 10^{-7} \text{ m MPa}^{-1}\text{s}^{-1}$) for the radial geometry of the root, one obtains $7.3 \times 10^{-11} \text{ m}^2 \text{ MPa}^{-1}\text{s}^{-1}$. This value is just a bit lower than the estimate of $1-5 \times 10^{-10} \text{ m}^2 \text{ MPa}^{-1}\text{s}^{-1}$ calculated by Bret-Harte and Silk (1994) assuming a modified Hagen-Poiseuille law with recent estimates for size and frequency of plasmodesmatal pores. From timing of turgor transients in growing regions of maize roots subjected to osmotic shock, Frensch and Hsiao (1994a) have calculated $1.3 \times 10^{-10} \text{ m}^2 \text{ MPa}^{-1}\text{s}^{-1}$. It is interesting that these estimates for hydraulic conductivity of maize root cortex, expressed per unit of cell length, have the same order of magnitude as the hydraulic conductivities reported for growing pea stem cortical cells (Cosgrove and Steudle, 1981) and *Kalanchoe* leaf mesophyll cells (Steudle *et al.*, 1980) from the timing of transient water influx or efflux induced by pressure perturbations.

## Solutions to the Simplified Water Potential Equation

It is instructive to consider two different sets of assumptions regarding hydraulic conductivity. Silk and Wagner (1980) assumed that longitudinal and radial conductivities are of a similar magnitude. The geometry of the maize root then determines that most of the growth-sustaining water flux is radial.

The growth-sustaining $\psi$ field, calculated from Eq. (4A) and assuming isotropic, uniform hydraulic conductivity of $1.3 \times 10^{-10} \text{ m}^2 \text{ MPa}^{-1}\text{s}^{-1}$ and the REG rate distribution of Erickson and Sax, is shown in Fig. 4A. The magnitude of $\psi$ increases with distance from the bounding water source and also parallels the strain rate to produce egg-shaped water potential shells. Individual tissue elements are displaced through the standing pattern and experience a decrease and then an increase in $\psi$ as they approach the nongrowing region. The solutions imply a small water potential differential, of the order of 0.04 MPa, sustains the maximum REG rate.

It is also interesting to consider the possibility that growth-sustaining water flux is longitudinal, with water flowing acropetally from the nongrowing region
<figure><img src="image_239.png" /></figure>

Fig. 4A. Contour map shows the distribution of $\psi$ in a longisection of the growth zone of the maize root. The root is presumed to be growing in pure water ($\psi = 0$) without transpiration. Hydraulic conductivity is assumed isotropic. The darkest region in the center has $-\psi = 4 \times 10^{-2}$ MPa, and water potentials become progressively less negative with distance from the root center and with decreasing growth rate. Current estimates for hydraulic activity are used with the solution (in terms of $K\psi$) shown by Silk and Wagner (1980).

of the root. This could happen if radial hydraulic conductivity is negligible. Then a one-dimensional version of Eq. (4) can be obtained by integration:

$$ \int v \, dz = A + Bz - K_z \psi \quad (4B) $$

One can solve for $\psi$ by numerical integration of the empirically determined velocity field. The growth-sustaining $\psi$ field under these assumptions is shown in Fig. 4B. Water potential becomes less negative with distance from the apex. The solutions are not physically reasonable. Even if longitudinal flow were
<figure><img src="image_240.png" /></figure>

**Fig. 4B.** Growth-sustaining $\psi$ field under the assumption that water flux is longitudinal. The graph shows the growth-sustaining water potential if the longitudinal hydraulic conductivity is high, $5 \times 10^{-10} m^2 MPa^{-1} s^{-1}$, and the radial conductivity is negligible. Water potential becomes less negative with distance from the apex. The solution is not physically reasonable, especially if flux is longitudinal throughout the growth zone (solid line), and even if flux is longitudinal only through the meristem (dashed line).

required only to sustain meristematic growth, the required differential water potential of -0.4 MPa is considerably greater than any observed pattern in the root tip. Thus growth-sustaining water flux must be mostly radial.

## Components of Growth-Sustaining $\psi$

Solutions of Eq. (4) displayed in Fig. 4A show the spatial pattern of growth-sustaining water potential in the maize root. For the physiologist the puzzle is then to find the components of water potential that are providing the driving force. According to conventional theory, the possible components are turgor pressure and osmotic potential.

In the past 5 years, improvements in technology—refinement of the pressure probe—have resolved spatial patterns of turgor pressure within growing regions. We now know the turgor pressure distribution within the growth zone of the maize root during steady growth (Spollen and Sharp, 1991; Rygol et al., 1993; Pritchard and Tomos, this symposium) and in response to perturbations in water
potential (Frensch and Hsiao, 1994a, b). The turgor pressure distribution appears to be fairly uniform during steady growth, although slightly higher pressures can sometimes be detected in the root center and at the tip of the growth zone. Thus the distribution of turgor potential would not drive a growth-sustaining influx of water.

There is no evidence for a growth-sustaining longitudinal gradient in solute potential (Silk et al., 1986; Sharp et al., 1990). Thus the prime candidate for the growth-sustaining water potential gradient is a radial gradient in solute potential. In recent years, improvement in technology—use of the nanoliter osmometer—has also permitted measurement of this putative gradient. Rygol et al. (1993) recently showed that, at least in roots of transpiring plants, the osmotic potential decreases with distance from the root surface. Frensch and Hsiao (1994a) also interpret their pressure probe studies in support of this hypothesis. They find that when a root is immersed in a medium of low solute potential, the half-time for turgor reequilibration increases with distance from the root center. They believe that the restoration of turgor depends on solute transport into the cells, and the solute concentration increases fastest in the region closest to the phloem. These empirical studies are at least consistent with the hypothesis that a radial gradient in solute potential drives the growth-sustaining flux of water from the bathing medium into the root center.

Not all empirical evidence is consistent with the hypothesis described above. In a nontranspiring plant, 7 mm from the root tip, solute potential was found to be uniform across the root cortex (Rygol et al., 1993). A likely explanation is that this region, especially in solution culture, was not growing rapidly. Or hydraulic conductivity may be at the high end of the estimated range, so that the growth-sustaining water potential is not empirically detectable. On the other hand, Nonami and Boyer (1993) found strong evidence for growth-sustaining solute gradients in the soybean hypocotyl. In this stem tissue, water is supplied from the xylem centripetally to the pith cells and centrifugally to the cortex. Using a nanoliter osmometer, Nonami and Boyer found that solute potential is least negative in the xylem and decreases in both directions from the water supply, as required by the theory. (The term "growth-sustaining water potential" used here is similar to the "growth-induced water potential" described by Boyer and colleagues. My choice of terminology is intended to beg the question of cause and effect.) The association of sucrose import with stem growth (Schmalstig and Cosgrove, 1990) is also consistent with an important role for solute potential in sustaining growth.

## CONCLUSION

More than a decade ago, theoretical studies predicted the shape of the field of water potential that sustains longitudinal growth (Molz and Boyer, 1978; Silk and Wagner, 1980). The tissue was considered to be a noncompartmented con-
tinuum in steady-state growth. The analyses considered the thermodynamic basis for growth and did not address the coupling to mechanical extensibility.

Measurement of the fields of turgor and solute potential in growing organs, at least for roots growing in aqueous solution, is now possible. Thus the water relations of root growth can empirically characterized with current technology, and a comprehensive theory seems achievable. Some mathematically rigorous analysis is needed to characterize rates of water transport in an appropriate geometry for the growing region. For the maize root in solution culture, growth-sustaining water flux appears to be mostly radial and driven by a radial gradient in solute potential.

## ACKNOWLEDGMENTS

The author's research is supported by Grant 92-37100-7625 from the United States Department of Agriculture.

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# Plant and Fungal Cell Growth: Governing Equations for Cell Wall Extension and Water Transport<fnref n="1" />

Joseph K. E. Ortega<fnref n="2" />

A set of governing equations for plant and fungal cell growth (enlargement) is reviewed, whose foundation is the augmented growth equations. These equations describe two interrelated and simultaneous physical processes that are involved in enlargement of plant and fungal cells: cell wall extension and water uptake. Special emphasis is put on the systematic development of the many governing equations that are used as theoretical foundations for pressure probe methods. Some new equations are presented which may provide the theoretical foundation for new measurement methods and/or better analyses.

**KEY WORDS:** augmented growth equations; governing equations; pressure probe; cell wall extension; water relations; plant cells; fungal cells.

## INTRODUCTION

Governing equations have played key roles in the evolution of physical sciences from a qualitative state to a quantitative state. They are fundamental to the physical sciences. Navier-Stokes equations in fluid mechanics and convective heat transfer and Maxwell's equations in electricity and magnetism, are but two examples of governing equations that have become the theoretical foundation of their respective field. Physical scientists, engineers, and, to some extent, biologists have come to rely on the use of governing equations to determine the

<footnote n="1">¹This is the published version of a paper presented at the Plant Biomechanics Congress, Montpellier, France, September 5-9, 1994.</footnote>

<footnote n="2">²Department of Mechanical Engineering, Bioengineering Laboratory, University of Colorado at Denver, Campus Box 112, P.O. Box 173364, Denver, Colorado 80217-3364.</footnote>
magnitude and behavior of a "parameter of interest" when several other parameters change simultaneously.

Growth is the result of biochemical and metabolic events in combination with physical processes. During the past decade, progress has been made in the development of governing equations that describe the relevant physical processes involved in plant and fungal cell enlargement, in terms of biophysical and bio-mechanical parameters. The magnitude and behavior of the biophysical and biomechanical parameters are controlled by relevant biological processes.

This paper attempts systematically to review governing equations relevant to plant and fungal cell enlargement. Emphasis is put on a systematic development of the governing equations used as theoretical foundations for "pressure probe" methods. This systematic approach might provide an effective method for investigators trained in related fields, and/or new investigators interested in using pressure probe methods, to review this topic. This paper does not attempt to provide a comprehensive review of all related work. Because the main objectives are to provide a systematic review of established governing equations for *in vivo* cell wall extension and water transport and to provide a systematic review of the theoretical foundations for common pressure probe methods, only selected references are provided, to give examples of applications and a deeper discussion of the method. A few new governing equations are derived and presented for the first time in this review. It is hoped that these new governing equations will provide theoretical foundations for new experimental methods and analyses.

# GOVERNING EQUATIONS

## The Augmented Growth Equations

The augmented growth equations (Ortega, 1985, 1990, 1993; Ortega et al., 1988a) provide the foundation for a set of governing equations that describe plant and fungal cell enlargement. These equations are extensions of the previously established growth equations (Lockhart, 1965; Taiz, 1984).

The first augmented growth equation describes the rate of cell wall extension and may be formulated to account for both irreversible and reversible extension (Fig. 1). This equation relates the relative rate of change in volume of the cell wall chamber, $v_c$, to the sum of the relative rate of irreversible (plastic) extension and the relative rate of reversible (elastic) extension (Ortega, 1985):

$$v_c = (dV_c/dt)/V_c = \phi(P - Y) + (dP/dt)/\epsilon \quad (1)$$

$$(\text{relative rate of extension}) = (\text{plastic}) + (\text{elastic})$$

where $V_c$ is the volume of the cell wall chamber, $t$ is time, $\phi$ is the relative
<figure><img src="image_241.png" /><figcaption>Fig. 1. A schematic illustration of the two simultaneous and interrelated physical processes that regulate the growth (enlargement) rate of plant and fungal cells: the rate of cell wall extension and the net rate of water uptake. The rate of wall extension is the sum of plastic (dashpot) and elastic (spring) extension rates. The net rate of water uptake is the difference between the rate of water uptake (flow in) and the rate of transpiration (flow out).</figcaption></figure>

irreversible wall extensibility, *P* is the turgor pressure (the pressure difference between the cell interior and the external medium), *Y* is the yield threshold (a magnitude of turgor pressure which must be exceeded before plastic wall extension occurs), and *ε* is the volumetric elastic modulus. The term *φ(P - Y)* represents the relative rate of plastic, or permanent, wall extension. The term (*dP/dt*)/*ε* represents the relative rate of elastic, or reversible, wall extension.

The second augmented growth equation is a mathematical statement describing the *net* rate of water uptake (Fig. 1) and may be obtained from the conservation of mass (water mass) (Ortega *et al.*, 1988a):

$$v_w = (dV_w/dt)/V_w = L(\sigma\Delta\pi - P) - T \quad (2)$$

(*net* rate of water uptake) = (flow in) - (flow out)

where $v_w$ is the relative rate of change in volume of the cell contents (mostly water), $V_w$ is the volume of the cell contents, $L$ is the relative hydraulic conductance of the cell membrane ($L = L_p A/V_w$), where $L_p$ is the hydraulic conductivity of the cell membrane and $A$ is the area of the cell membrane), $σ$ is the
solute reflection coefficient of the cell membrane (whose value can range from zero to unity; in most cases it is unity), $\Delta\pi$ is the difference in osmotic pressure between the cell sap, $\pi_i$, and the external medium, $\pi_e$, and $T$ is the relative transpiration rate. The term $L(\sigma\Delta\pi - P)$ represents the relative rate of water uptake, and the term $T$ represents the relative rate at which water is lost from the cell by transpiration. This term $T$ is zero for cells that are not exposed to the external environment and, thus, do not transpire. It should be noted that the volume of the cell contents must equal the volume of the cell wall chamber, $V_w = V_c$, and that the respective relative rates of change in volume must also be equal, $\nu_w = \nu_c$. Therefore, for the remainder of this paper, there is no distinction between $V_w$ and $V_c$, or between $\nu_w$ and $\nu_c$; they are simply represented by $V$ and $\nu$, respectively. Finally, observe that Eqs. (1) and (2) are coupled by the turgor pressure, $P$, and by the relative rate of change in volume, $\nu$.

## Turgor Pressure Behavior

The governing equation describing the turgor pressure behavior can be obtained by combining Eqs. (1) and (2) with the elimination of $\nu$:

$$ \frac{dP}{dt} + \epsilon\phi(P - Y) - \epsilon L(\sigma\Delta\pi - P) + \epsilon T = 0 \quad (3) $$

In the special case when the turgor pressure is constant (thus, $\frac{dP}{dt} = 0$) and the cell is not transpiring ($T = 0$), the following equation describing the turgor pressure is recovered: $P = (L\sigma\Delta\pi + \phi Y)/(L + \phi)$, which is the same equation derived by Lockhart (1965) for this same case.

## Pressure-Dependent Parameters

In the most general of cases, Eqs. (1), (2), and (3) are not linear differential equations, because the magnitude of some of the inclusive parameters is known to vary with the turgor pressure; $\epsilon$, $L$, and $\Delta\pi$ have all been shown to be functions of $P$. The functional relationship between $\epsilon$ and $P$, $\epsilon(P)$, may be described by the following equation (Ortega, 1990, 1993; Ortega et al., 1988a):

$$ \epsilon(P) = \epsilon_{\infty} - [(\epsilon_{\infty} - \epsilon_0) \exp(-kP)] \quad (4) $$

where $\epsilon = \epsilon_{\infty}$ when $P = P_{\infty}$ (for many cells, $P \approx P_{\infty}$, when $P > 0.5$ MPa), and $\epsilon = \epsilon_0$ when $P = 0$, and $k$ is a rate constant which is different for each species of cells.

Similarly, it has been reported that for some algal cells (Zimmermann and Steudle, 1974a, b) and higher plant cells (Steudle et al., 1982), the hydraulic conductivity of the cell membrane, $L_p$, increases at low turgor pressure (also see review by Steudle, 1993). Therefore, for these cells, there exists a functional relationship between $L_p$ and $P$. Based on the published results of Zimmermann
and Steudle (1974a, b) and Steudle et al. (1982), the following relationship is derived to describe $L_p(P)$:

$$L_p(P) = (L_{p0} - L_{p\infty}) \exp(-aP) + L_{p\infty} \quad (5)$$

where $L_p = L_{p\infty}$ when $P = P_\infty$, $L_p = L_{p0}$ when $P = 0$, and $a$ is a rate constant which is different for each species of cells. The functional relationship for $L$ and $P$ is obtained from the equation $L(P) = L_p(P)(A/V)$.

The dependence of $\Delta\pi$ on $P$, $\Delta\pi(P)$, can be described by the following equation, where the expression for $\pi_i(P)$, first derived by Dainty (1976), is employed:

$$\Delta\pi(P) = \pi_i(P) - \pi_e = (\pi_o - \pi_e) - \pi_o[P/\epsilon(P)] \quad (6)$$

where $\pi_i = \pi_o$ when $P = 0$, and Eq. (4) describes $\epsilon(P)$.

It is not known whether the parameters $\phi$, $Y$, and $T$ are dependent on the turgor pressure [although for some cells, there is evidence which suggests that $Y$ is "adjustable" or "nonadjustable," depending on the magnitude of $P$ (Green et al., 1971; Ortega et al., 1989)]. More investigations are needed to determine whether $\phi$, $Y$, and $T$ are dependent on $P$.

Equations (1)–(6) represent a "set of governing equations" for plant and fungal cell growth (enlargement). Importantly, the augmented growth equations and the original growth equations (the foundations of this set of the governing equations) have demonstrated utility in understanding and predicting the growth rate behavior of higher and lower plant cells (see reviews by Cosgrove, 1986, 1993a, b; Ortega, 1990; Steudle, 1993, Taiz, 1984; Tomos et al., 1989) and fungal cells (Money, 1990, 1992; Ortega, 1990, 1993; Ortega et al., 1988a, b, 1989, 1991, 1992).

## PRESSURE PROBE METHODS

The augmented growth equations have also proven useful in determining the magnitude and behavior of the inclusive parameters. Because the turgor pressure, $P$, is present in three of the four terms of the augmented growth equations, the cell pressure probe [which directly measures the turgor pressure of single cells (see review by Steudle, 1993)] has proven to be an important tool in determining the magnitude and behavior of the biophysical and biomechanical parameters in Eqs. (1) and (2) (Cosgrove, 1985, 1986, 1987, 1993a, b; Cosgrove et al., 1987; Hüsken et al., 1978; Money, 1990, 1992; Okamoto et al., 1989; Ortega, 1990, 1993; Ortega et al., 1988a, b, 1989, 1991, 1992; Steudle, 1993; Steudle et al., 1982; Tomos et al., 1989; Wendler and Zimmermann, 1982; Zimmermann and Steudle, 1974a, b, 1978). The governing equations for nearly all pressure probe methods can be obtained directly from
the augmented growth equations in their present form or by examining them in
limiting conditions.

Determination of ε

A typical method for determining ε utilizes a turgor pressure pulse-down and pulse-up produced with the pressure probe (Fig. 2a). After the cell is im-paled with the microcapillary tip of the pressure probe to measure the turgor pressure, cell sap is quickly withdrawn with the pressure probe to lower the turgor pressure, thus producing a step-down. Subsequently, a step-up of equal magnitude is produced by injecting the cell sap back into the cell, and if the time interval between the step-down and step-up is short, a pulse-down is pro-duced. If the change in pressure, ΔP, and the volume of the cell sap removed

<figure><img src="image_242.png" /><figcaption>Fig. 2. Examples of some turgor pressure traces pro-
duced or observed in pressure probe methods: (a) pulse-
down followed by pulse-up, (b) pressure relaxation from
a constant pressure, (c) step-up, (d) step-down, and (e)
step-down with subsequent relaxation, followed by a
step-up with subsequent relaxation.</figcaption></figure>
or injected, $\Delta V$, are measured, together with the initial volume of the cell, $V$, then $\epsilon$ can be calculated.

The governing equation that describes the “elastic” deformation of the cell wall chamber can be obtained from Eq. (1) for the special case when there is no plastic deformation, $\phi(P - Y) = 0$ (this would be the case for nongrowing cells):

$$v = (dV/dt)/V = (dP/dt)/\epsilon \quad (7)$$

Now multiplying Eq. (7) by $dt$, and solving for $\epsilon$, the following relationship is obtained (Philip, 1958):

$$\epsilon = V(dP/dV) \equiv \Delta P/(\Delta V/V) \quad (8)$$

To determine $\epsilon$ for growing cells, it is important to account for the plastic extension related to growth; $\phi(P - Y)\Delta t$. The relevant governing equation used to determine $\epsilon$ for growing cells is Eq. (1); by simply solving for $\epsilon$, and substituting in “finite-difference” operators, the following equation is obtained (Ortega, 1985):

$$\epsilon = \Delta P/[(\Delta V/V) - \phi(P - Y)\Delta t] \quad (9)$$

Recently, a new method to determine the *average* value of $\epsilon$ for cells in plant tissue was introduced (Ortega, 1993). In this method, the transpiration rate of the tissue and turgor pressure decay (relaxation) of the cells (Fig. 2b) are measured after the water source is removed from the tissue. The governing equation can be obtained from Eq. (3) by accounting for the following conditions; the water source is removed, $L(\sigma\Delta\pi - P) = 0$, and the cells are not growing, $\phi(P - Y) = 0$. Equation (10) is obtained by solving the resulting equation for $\epsilon$ (Ortega, 1993):

$$\epsilon_{av} = (-dP/dt)_{av}/T \quad (10)$$

## Determination of $\phi$ and $Y$

In Vivo *Stress Relaxation*. This method consists of measuring the pressure decay (relaxation) of cells (Fig. 2b) when the transpiration is suppressed and the water source is removed. The governing equation for the *in vivo* stress relaxation method can be obtained directly from Eq. (3) by accounting for the following conditions: water source is removed, $L(\sigma\Delta\pi - P) = 0$, and transpiration is suppressed, $T = 0$. Given these conditions, the following equation is obtained (Cosgrove, 1985; Ortega, 1985):

$$dP/dt + \epsilon\phi(P - Y) = 0 \quad (11)$$

When $\phi$, $\epsilon$, and $Y$ are assumed constant during the turgor pressure decay, this
differential equation can easily be solved to obtain the turgor pressure as a function of time, $P(t)$ (Cosgrove, 1985; Ortega, 1985):

$$P(t) = Y + (P_i - Y) \exp(-\epsilon\phi t) \quad (12)$$

where the initial condition $P = P_i$ at $t = 0$ is employed. It is apparent that at times greater than zero, the turgor pressure decays exponentially to a constant value, $Y$, with a time constant, $t_c = 1/\epsilon\phi$. Thus, if the magnitude of $\epsilon$ is known, theoretically $\phi$ and $Y$ can be obtained from the turgor pressure decay curve (Cosgrove, 1985, 1987; Ortega et al., 1989). A problem implicit to the *in vivo* stress relaxation method is that the pressure decay can take a significantly long period of time to complete (typically longer than an hour), and it cannot always be assumed that the parameters, $\phi$ and $Y$, remain constant during this long period of time.

Sometimes the turgor pressure decay occurs over the range where $\epsilon$ varies with $P$. In these cases, the following integral should be evaluated with the appropriate limits to obtain $P(t)$:

$$\int [(P - Y)\epsilon(P)]^{-1} dP = - \int \phi dt \quad (13)$$

where Eq. (4) is substituted for $\epsilon(P)$.

In Vivo Creep. This method consists of producing a step-up in turgor pressure, $\Delta P$ (Fig. 2c) and measuring the growth rate before and after the step-up to determine the difference. The governing equation for the *in vivo* creep method can be obtained from Eq. (1) by considering the effects of a step-up in turgor pressure on the plastic extension rate (growth rate) of the cell wall; the elastic extension that occurs immediately after the step-up is neglected in the analysis (Green et al., 1971; Okamoto et al., 1989; Ortega et al., 1989, 1991). The following equation is obtained when the total differential of modified Eq. (1) [(Lockhart, 1965) sometimes referred to as the "Lockhart equation," which is Eq. (1) without the elastic term] is taken and then solved for $\phi$ (it is assumed that $\phi$ and $Y$ remain constant for a short time interval before and after the $\Delta P$):

$$\phi = \Delta\nu/\Delta P = [\nu_{\text{ave(after)}} - \nu_{\text{ave(before)}}]/\Delta P \quad (14)$$

Once the magnitude of $\phi$ is determined, then the Lockhart equation may be used to solve for $Y$ using the obtained value of $\phi$ and the growth rate before $\Delta P$ (Ortega et al., 1989, 1991):

$$Y = P - \nu/\phi \quad (15)$$

A more direct method can sometimes be used to determine $Y$. In theory, a turgor pressure step-down (Fig. 2d) whose magnitude is equal to or greater than the difference, $P - Y$, will stop growth. When this $\Delta P$ can be determined experimentally, it can be used to determine $Y$ with the equation $Y = P - \Delta P$ (Green et al., 1971).
The *in vivo* creep method offers the important theoretical advantage (in comparison to the *in vivo* stress relaxation method) of not requiring $\epsilon$ in the analysis of the results. This eliminates both the need to measure $\epsilon$ for the determination of $\phi$ and the need to account for its dependence on $P$, if the pressure decay occurs in the range where $\epsilon$ varies with $P$.

## Determination of $L_p$

*Pressure Relaxation Method.* In this method, a rapid decrease in turgor pressure is produced with the pressure probe by withdrawing a small, fixed amount of cell sap in a short time (in less than a second) and recording the subsequent pressure relaxation (Fig. 2c). After the pressure relaxation is complete, the removed cell sap is injected back into the cell to produce a rapid increase in turgor pressure, and the pressure relaxation is again recorded. Typically, the half-times for the pressure relaxation curves are used to determine $L_p$.

The governing equation for the pressure relaxation method can be obtained from Eq. (3) for the following conditions: $T = 0$ (no transpiration), $\phi(P - Y) = 0$ (nongrowing cell) and $\epsilon = $ constant. The solution to the resulting differential equation (not shown here) is an exponential decay of turgor pressure with the time constant: $t_c = V/[AL_p(\epsilon + \sigma\pi_{io})]$, where $\pi_{io}$ is the equilibrium osmotic pressure of the cell sap. Typically, the $L_p$ is determined by measuring $\epsilon$, $\pi_{io}$, $V$, $A$, and the half-time of the exponential decay, $t_{1/2}$; in most cases, $\sigma$ can be assumed to be unity. The following equation is used to calculate $L_p$ (Hüsken et al., 1978; Zimmermann and Steudle, 1978):

$$L_p = V \ln 2/[t_{1/2}A(\epsilon + \sigma\pi_{io})] \quad (16)$$

In the case of a growing cell, when $\phi = $ constant (but not zero), the relevant governing equation can be obtained from Eq. (3) by applying the conditions $T = 0$ (no transpiration) and $\epsilon = $ constant (Ortega, 1985). Using Eq. (6) to describe $\Delta\pi(P)$, the solution to the resulting differential equation (not shown here) will be an exponential decay of turgor pressure with the following time constant (Cosgrove, 1986): $t_c = 1/[L(\epsilon + \sigma\pi_{io}) + \epsilon\phi]$.

*Pressure Clamp Method.* This method makes use of the turgor pressure step-down and/or step-up produced by the pressure probe (Figs. 2d and c). Using the pressure probe, a decrease in turgor pressure is produced and maintained (clamped) by continuously withdrawing cell sap into the microcapillary of the pressure probe; subsequently, the cell sap can be injected back into the cell to produce a step-up in turgor pressure. The rate at which the cell sap is removed from the cell, or injected into the cell, decreases as a function of time after the initiation of the step-down or step-up, i.e., a "volume relaxation." The governing equation for this volume relaxation can be obtained from Eq. (2) by
imposing the condition that $T = 0$ and by multiplying the equation by $V$. In this equation, the osmotic pressure of the cell sap is a function of time, $\pi_i(t)$, and is approximated with the relationship $\pi_i(t) = \pi_{io}[1 - V(t)/V_o]$, where $V_o$ is the initial cell volume (Wendler and Zimmermann, 1982). The resulting differential equation can be solved for $V(t)$. The solution (Wendler and Zimmermann, 1982) is an exponential decay with a time constant $t_c = V_o/(AL_p\sigma\pi_{io})$. It is apparent that $L_p$ can be determined when $t_c$, $V_o$, $A$, and $\pi_{io}$ are measured; typically, $\sigma$ is assumed to be unity. Importantly, Wendler and Zimmermann (1982) further demonstrated that $L_p$ can be determined from the initial slope of the volume relaxation curve:

$$L_p = -s_v/(A\Delta P) \qquad (17)$$

where $s_v$ is the magnitude of the initial slope. Thus, implicit in the pressure clamp method is the important advantage of being able to determine $L_p$ without knowing $V_o$, $\epsilon$, or $\pi_{io}$. On the other hand, unstirred layers adjacent to the cell membrane during the induced water flow can affect the accuracy of the results obtained with the pressure clamp method (Steudle, 1993).

## Determination of T

*Pressure Relaxation Method.* This method consists of removing the water source from nongrowing cell(s) and measuring the resulting decay in turgor pressure (Fig. 2b). The relative transpiration rate, and transpiration rate, can be calculated from the magnitude of the turgor pressure decay rate, provided that $\epsilon(P)$ is known, or measured independently. The relevant governing equation can be obtained from Eq. (3) by accounting for the following conditions: The water source is removed, $L(\sigma\Delta\pi - P) = 0$, and the cell is not growing, $\phi(P - Y) = 0$. The resulting equation can be solved for $T$ to obtain (Ortega, 1993; Ortega et al., 1988a, 1992)

$$T = -(dP/dt)/\epsilon(P) \qquad (18)$$

Since $\epsilon$ is a function of $P$, a series of governing equations can be obtained, each depending on the range in which the turgor pressure decay occurs (Ortega, 1993):

$$T = -(dP/dt)/\epsilon_{\infty} \qquad \{\text{high } P\} \qquad (18a)$$

$$T = -(dP/dt)/(\epsilon_o + bP) \qquad \{\text{low } P\} \qquad (18b)$$

$$T = -(dP/dt)/\{\epsilon_{\infty} - [(\epsilon_{\infty} - \epsilon_o) \exp(-kP)]\} \qquad \{\text{all } P\} \qquad (18c)$$

where $\epsilon_{\infty}$, $\epsilon_o$, and $k$ have been defined previously, and $b$ is the slope of the $\epsilon$ versus $P$ curve at low $P$.

*Pressure Clamp Method.* This method consists of removing the water
source from a growing or nongrowing cell and clamping the turgor pressure at its equilibrium value (the value of P that was measured before the water source is removed) by continuously injecting silicon oil into the cell vacuole. The transpiration rate, $[dV/dt]_T$, and relative transpiration rate, $T = [dV/dt]_T/V$, can be determined when the volumetric growth rate and the volumetric rate of oil injection are measured. The relevant governing equation can be obtained by multiplying Eq. (2) by $V$ and noting that, $Vv = [dV/dt]_g$ (volumetric growth rate), $VL(\sigma\Delta\pi - P) = [dV/dt]_i$ (volumetric rate of water uptake, which is equal to the volumetric rate of oil injection during the pressure clamp), and $VT = [dV/dt]_T$ (volumetric transpiration rate). The following equation is obtained by solving for the volumetric transpiration rate, $[dV/dt]_T$ (Ortega, 1993; Ortega et al., 1992):

$$[dV/dt]_T = [dV/dt]_i - [dV/dt]_g \quad (19)$$

Two theoretical advantages of the pressure clamp method (in comparison with the pressure relaxation method) are that (i) it can be used to measure transpiration rates from growing cells, as well as nongrowing cells, and (ii) $\epsilon$ is not used in the analyses. Two experimental disadvantages of the pressure clamp method are that (i) injection of silicon oil into the cell is required, although it is envisioned that water or cell sap can be substituted for the oil in the microcapillary tip and injected into the cell instead of oil, and (ii) this method cannot be easily adapted to measure transpiration rates from tissue.

## DISCUSSION

Analogous to the many sets of governing equations in the physical sciences, Eqs. (1)–(6) can be interpreted as a set of governing equations for plant and fungal cell growth (enlargement). This is the first time that all of these equations have been reviewed together, although some previous papers have reviewed a smaller and more limited set of equations (Cosgrove, 1986, 1993b; Lockhart, 1965; Ortega, 1990; Taiz, 1984). It is envisioned that this set of governing equations will grow as more research is conducted in this area. However, it is doubtful that this set of governing equations will ever become a “deterministic” set of equations, because of the implicit dependence of the inclusive biomechanical and biophysical parameters on the many biological processes that control their magnitude and behavior. Still, similar to other sets of governing equations, this set of governing equations [Eqs. (1)–(6)] provides an important feature that other mathematical models (such as correlations between parameters) cannot, that is, that the magnitude and behavior of a dependent parameter (of interest to the investigator) can be determined when several other parameters vary simultaneously. This important property of governing equations can provide more qualitative and quantitative insights and more in-depth knowledge of integrated
effects of many biomechanical and biophysical parameters (biological processes) on each other.

It is apparent that this set of governing equations [Eqs. (1)-(6)] also demonstrates utility in developing new methods to measure and study many biomechanical and biophysical parameters. In this review, it is demonstrated that nearly all the governing equations for pressure probe methods can be systematically derived from the initial set of six governing equations. It is reasonable to anticipate that these governing equations will provide the theoretical foundation for new and better methods to measure many biomechanical and biophysical parameters. It should be emphasized that these new methods are not restricted to pressure probe methods but can employ other related instruments (such as the pressure bomb and psychrometer) and may inspire the invention of new instruments such as the "pressure block" (Cosgrove, 1987).

## ACKNOWLEDGMENTS

The author gratefully acknowledges support by U.S. National Science Foundation Grants DCB-8514902, DCB-8801717, and IBN-9103760.

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# Tree Biomechanics: Growth, Cumulative Prestresses, and Reorientations<fnref n="1" />

Meriem Fournier,<fnref n="2" /> Henri Bailleres,<fnref n="3" /> and Bernard Chanson<fnref n="4" />

Three progressive steps are distinguished in the biomechanical study of tree growth and wood differentiation. For the first, tree mechanics, emphasis is put on the maturation prestress and on the surprising results of a time-differentiated analysis of radial growth. In particular, maximal bending stresses due to self-weight in leaning stems are usually not located in the peripheral youngest wood. The second step, biomechanical characterization, concerns a wide range of works. Here we present examples that deal with the relations between maturation strains and (i) tree architecture and morphology and (ii) histological and biochemical features of wood (microbiomechanics of wood). Finally, the state of the art concerning the biomechanical regulation of growth and differentiation is reviewed, through the examples of (i) the activation of radial growth to ensure a constant stress design at the stem periphery (Mattheck's model) and (ii) the righting movements of leaning stems due to asymmetric prestresses related to the differentiation of reaction wood.

**KEY WORDS:** tree mechanics; wood; growth stresses; reaction wood; cambial growth.

## INTRODUCTION

Tree stems are tall and slender mechanical structures that support other aerial organs, withstand winds, and allow the plant to explore its environment. The design and movement of such structures are related to some morphological, histological, and physiological aspects of growth. Moreover, they ask basic

<footnote n="1">¹This is the published version of a paper presented at the Plant Biomechanics Congress, Montpellier, France, September 5-9, 1994.</footnote>

<footnote n="2">²École Nationale du Génie Rural, des Eaux et Forêts, 14 rue Girardet, 54042 Nancy Cedex, France.</footnote>

<footnote n="3">³CIRAD Forêt, 45 bis Avenue de la Belle Gabrielle, 94 Nogent/Marne, France.</footnote>

<footnote n="4">⁴Laboratoire de Rhéologie du Bois de Bordeaux, BP 10, 33610 Cestas Gazinet, France.</footnote>
biomechanical questions as some phenomena are explainable by a physical necessity while others involve regulation or adaptation of the living plant. Hence, our approach will be organized in three steps, that state progressively the interaction between mechanics and biology.

## THE THREE STEPS OF OUR BIOMECHANICAL APPROACH

Tree mechanics is based on the mechanics of solids. The inputs are supported loads, the outputs are fields of local variables such as displacements, strains (variations of form and movements), and stresses (local distributions of loads). Global characteristics of the structure such as safety factors (Tateno, 1991; Tateno and Bae, 1990), buckling loads (Greenhill, 1881; McMahon, 1975), and frequencies of sways (Milne, 1991; Fournier et al., 1993) can also be included.

In addition to these inputs and outputs, a mechanical problem involves the geometry of the structure and the properties of materials: the critical stresses or strains that lead to failure and the local relation between stresses and strains, for example, the Young's modulus. The characterization of these properties requires the description of heterogeneities and anisotropies. Finally, note that a mechanical problem can be stated at different levels: The analyzed structure can be the whole tree (a truss of stems), the organ (a stem), the tissue (an elementary piece of wood), the isolated cell, or the cell wall.

Tree mechanics can be used to analyze the consequences of biological factors (genotype, environment, etc.) on the mechanical properties and response of the plant. This is biomechanical characterization. It aims a defining sets of data (inputs and parameters) for the mechanical problem, of typical biological situations, and can use data collected for other purposes (biometrics, tree architecture, wood quality).

Adaptations and physiological reactions of plants to their mechanical state are commonly observed; Stem taper seems to be optimized to withstand wind or gravitational forces (see, e.g., Greenhill, 1881; Schwendener, 1884; Metzger, 1893; Esser, 1946; Larson, 1963; McMahon, 1975; King and Loucks, 1978; Wilson and Archer, 1979; Dean and Long, 1986; West et al., 1989; Bertram, 1989; Mattheck, 1991); moreover, stems react to mechanical stress and grow faster in height and slower in diameter when they are prevented from swaying (Larson, 1965; Quirk et al., 1975; Quirk and Freeze, 1976a, b; Kellog and Steucek, 1977, 1980; Telewski and Jaffe, 1986a-c; Grace, 1977; Timell, 1987; Osawa, 1993); finally, reaction wood is a response to an abnormal orientation and allows secondary movements of the stem, often gravitropic (Wilson and Archer, 1979; Fisher and Wassmer-Stevenson, 1981; Timell, 1987; Archer, in press). A third step is thus to model these biological reactions or adaptations. Such models are based on a previously built mechanical model and postulate a
biological control of some outputs of this model. They are called "biomechanical models of regulation."

To apply these general principles to the biomechanical study of the tree growth, a mechanical model for stresses and strains in the woody cross section of a growing tree is developed. We emphasize growth and maturation pressures. In addition, we develop examples of the variation of maturation strain related to the morphology and architecture of the tree and to anatomical and biochemical features of the wood. Other work concerns the bending mechanics of trees, stems, and branches submitted to external loads (wind and snow) and involves biomechanical characterization of parameters as moduli of elasticity or rupture (Cannell and Morgan, 1989; Langbour et al., 1988; Nakatani et al., 1984; Koizumi and Ueda 1986; Mamada et al., 1984).

Finally, there are biomechanical models such as the regulation of cambial growth based on the constant stress hypothesis (Mattheck's model) and the regulation of lean by asymmetric maturation prestresses induced by the differentiation of reaction wood.

## THE MECHANICAL CONSEQUENCES OF RADIAL GROWTH

A mechanical model of the cross section of the woody stem loaded by external or internal forces can use classical tools for the mechanical analysis of cylinders, made of anisotropic elastic materials (Lekhnitskii, 1963; Maiti and Adams, 1968). The model can consider

(i) some heterogeneities, since further biomechanical characterization shows radial, longitudinal, and circumferential variation of wood properties (Zobel and van Buijtenen, 1989);

(ii) the possible discrepancy between the cylinder axes and the directions of anisotropy (the grain is sometimes wavy or helicoïdal and the tree rings not concentric);

(iii) noncircular cross sections; and

(iv) nonelastic materials.

Even though the material is anisotropic, beam theory gives a very good approximation of longitudinal stresses and strains (Fournier, 1989).

There is another specificity of tree mechanics that is rarely mentioned: The cross section grows while it is loaded. Usual mechanical analyses are not suitable for radially growing structures, which are loaded by continuously changing forces, while new tissues are differentiated progressively. Therefore, based on Eulerian formalism, an original mechanical analysis has been developed, taking into account relative kinetics of both loads and growth (Fournier, 1989; Fournier et al., 1992). It generalizes the original idea of Martley (1928) and is achieved by two processes:
(1) an incremental problem between two times $t$ and $t + dt$, is stated as a classical problem since growth remains infinitesimal between these two instants;

(2) the mechanical state (i.e., the position of each material point and the stress field at a final time $t_f$ is calculated by superimposing the successive incremental solutions calculated from process 1).

For each material point, the superposition of stress and strain begins from the time it was differentiated. That allows for the assumption that a tissue cannot be loaded before it exists. In addition, the mechanical analysis requires data about the whole history of the structure, since the relative kinetics of growth and loads are necessary to solve the incremental problem.

Many numerical simulations both of stresses in a growing cross-section (Fournier, 1989; Fournier et al., 1990, 1991a, b) and of successive forms of stems (Schaeffer, 1990; Castera and Morlier, 1991; Fournier and Moulia, 1994) have been performed. Great discrepancies can be shown between classical beam theory and the previous analysis, when both are applied to the modeling of the self-weight effects in a branch or a leaning tree cross section: At the level of the cross section, the distribution of bending and compressive stresses given by classic strength of materials varies linearly along the radius, from tensile value on the upper side to a compressive one on the lower side. The improved model predicts very low support stresses everywhere at the stem surface (Fournier et al., 1991a) (Fig. 1). Actually, the very young peripheral wood supports only the last increment of weight added since it was differentiated (i.e., a very small load in most cases). Furthermore, the radial position of maximum stress depends not only on the present weight $P$ but also on the relative kinetics of both $P$ and radial growth.

At the level of the whole stem, the predicted deflections of a leaning stem are also different and cannot be qualified by a simple tip deflection in the most general case. The initial lean of the apical growth is defined by the stem architecture. At a given part of the stem, the lean increases with time as the stem becomes older and loaded. When considering the stem at a given age, this increase took longer in the older stem closer to the base. However, at each lapse of time, it was less important near the base which is closer to the fixed restraint. Therefore, the shape of the growing stem is an "S" form, due to the competitive effect of age and spatial position (toward the restraint) along the stem (Schaeffer, 1990). This shape can be computed from (i) the original orientation and kinetics of primary growth, (ii) the kinetics of loads, and (iii) the kinetics of secondary growth (Fig. 2). Small differences in radial growth can lead to great changes in form.
<figure><img src="image_243.png" /><figcaption>Fig. 1. Distribution of support longitudinal stresses on a leaning tree (from Fournier et al., 1991a). The crown weight *P* varies as $R^{2.5}$ (R stem radius at the trunk base). The lever arm of the crown weight varies as $R^x$. The higher the value of *ex*, the greater the variation of the lever arm since the beginning of growth. Stresses are compared to the "nongrowing" distribution of usual strength of materials, computed from the final weight *P*<sub>f</sub>, the final lever arm *e*<sub>f</sub>, and the final second moment of area $\pi R_f^2/4$.</figcaption></figure>

<figure><img src="image_244.png" /><figcaption>Fig. 2. Gravitationally strained form of a radially and longitudinally growing branch. Growth is divided into seven units of equal length (30 cm) and initial orientation (45°). A load of 200 g is supported at the beginning of the last two growth units. The Young's modulus and the specific gravity of the homogeneous material are 5000 MPa and 1 g/cm³. The only difference between 1 and 2 is related to radial growth: for simulation 1, the increment of radius for each growth unit is 2 mm (everywhere and at any time); for simulation 2, this value is 1.5 mm for the three first increments and 2 mm thereafter.</figcaption></figure>
## PRESTRESSES DUE TO WOOD DIFFERENTIATION—
ASSOCIATED DEFORMATIONS IN THE GROWING STEM

Wood is internally prestressed in the living tree (Archer, 1986). Moreover, since material starts to be loaded after its formation, the high levels of stress observed in peripheral young wood must be explained by a phenomenon that occurs continuously just at the end of wood differentiation. Therefore, the origin of growth stress can be described by the following assumption: During radial growth, cells of the outermost layers, just after differentiation, have a tendency to strain. As the new cells are glued on the internal older wood, this tendency is impeded in longitudinal and tangential directions and the wood is prestressed. This assumption of macroscopic maturation strains can be related to underlying physical and chemical mechanisms at the scale of the cell wall; these mechanisms are still a matter of discussion and would be a swelling of the lignin matrix (Boyd, 1972) and a shrinkage of cellulose microfibrils (Bamber, 1987).

Stress and strain due to the maturation of a new peripheral layer can be calculated by an incremental problem, and stress at any final time is calculated by superimposition (Archer and Byrnes, 1974; Archer, 1986; Fournier et al., 1991b) (Fig. 3).

Finally, the consequence of an angular (i.e., circumferential) asymmetry of maturation strains on the cross section is reorientation of the stem, since the side with the higher tensile stress pulls the other one (Fig. 4 and the Appendix). A mechanical model has been derived to compute, at each step of radial growth, the bending movements of a stem, from (i) the variation of the gravitational bending moment and (ii) the differentiation of heterogeneous wood (Fournier, 1989; Fournier et al., 1991a, b, 1992). The model uses a superimposition of (i) a time-differentiated expression of classical equations of beams in pure bending (see the last paragraph and the simulations in Fig. 2) and (ii) the effect of asymmetric maturation strains (see the Appendix).

So far, we have considered the mechanical analysis of tree growth. Now we focus on specifically biomechanical questions.

## VARIATIONS OF MATURATION STRAINS RELATED TO
TREE ARCHITECTURE

The hypothesis is the functional interpretation of circumferential peaks of longitudinal maturation strains related to stem reorientations. These strains have been estimated by the measurement of released strains at the stem surface (i.e., strains of a small piece of wood isolated from the stem by cutting grooves, drilling holes, etc.). Different metrologies are available (Archer, 1986). We usually used the single-hole method or Wap's method (Fournier et al., 1994; Baillères, 1994).
<figure><img src="image_245.png" /><figcaption>Transverse stresses (MPa)</figcaption></figure>

<figure><img src="image_246.png" /></figure>

Fig. 3. Longitudinal and transverse maturation stresses in a standard stem (calculated from Fournier *et al.*, 1991b). Just after its differentiation, wood is under longitudinal tensile stress and compressive tangential stress. As it becomes older, the tangential stress becomes a tension, and the longitudinal stress a compression; the radial stress is a tension.
<figure><img src="image_247.png" /><figcaption>Fig. 4. Bending reorientation of tree stems due to the maturation of heterogeneous wood from one side to the other, usually related to the formation of reaction wood. (a) The case of angiosperms: The longitudinal maturation strain of tension wood is a higher shrinkage. (b) The case of gymnosperms: The longitudinal maturation strain of compression wood is dilatation.</figcaption></figure>

Maturation strains usually vary along the circumference (formation of "peaks" of strains) with two typical patterns (Fig. 5) (Archer, 1986): (i) a peak of longitudinal shrinkage (i.e., the initial stress is a longitudinal tension) on angiosperm trees and (ii) a peak of longitudinal dilatation (i.e., the initial stress is a compression) on gymnosperm trees. These peaks can be large: The higher value can be eight times the lower. They are mostly associated with sections of reaction wood: compression wood in gymnosperms (Timell, 1987) and tension wood in angiosperms (Scurfield, 1973). Notice, however, that the dichotomy between angiosperms and gymnosperms may be not so contrasted since very low values of shrinkage have been measured in some angiosperms (Baillères et al., 1994), and quite high values in gymnosperms. These peaks are not an anomaly but most stems have to differentiate reaction wood during specific periods of their life. For instance, in very homogeneous plantations of *Eucalyptus*, we find great longitudinal shrinkage on the upper side, although the trees were very slightly tilted (Fig. 6) (Baillères, 1994). This observation does not contradict the functional interpretation of peaks of strain: Verticality may be the consequence of efficient reorientation.

Peak strain usually corresponds to the lean of the stem (peak tensions are on the upper side and peak compressions are on the lower face) but not always, since other situations lead to reorientations: For example, most conifers are monopodial: The trunk is normally formed by a single meristem during the whole life of the tree (Edelin, 1989; Bell, 1991). However, the apical bud can
<figure><img src="image_248.png" /><figcaption>Fig. 5. Typical circumferential variations of longitudinal maturation strains at the stem periphery of two *Fagus sylvatica* (Chanson *et al.*, 1992) and one *Pinus pinaster* (Loup *et al.*, 1991). The measurements were made by the single-hole method (CIRAD's sensor): 100 µm corresponds approximately to a strain of 0.12%. (Baillères, 1994). In angiosperms, peaks of high tensile maturation strains (i.e., high longitudinal shrinkages) are related to the formation of tension wood. In gymnosperms, peaks of compressive strains (i.e., high longitudinal dilatation) are related to the formation of compression wood.</figcaption></figure>

<figure><img src="image_249.png" /><figcaption>Fig. 6. Lean and peaks of maturation strains in *Eucalyptus* [clone 1-45; industrial plantations of Pointe-Noire, Congo, Africa (from Baillères, 1994)]. (a) Lean of the trunk (at breast height): values (%) and angular direction of the lower side (polar graphic). (b) Maximal maturation strain (from eight measurements) measured on the trunk circumference (at breast height): values (microstrains) and angular direction (polar graphic).</figcaption></figure>
die accidentally. A branch then rights itself and becomes the new leader. This is linked to a reorientation that can require the formation of compression wood. We observed on a *Pinus pinaster* Ait. that the induced "flows" of compression wood can spread widely in space and time (Fig. 7a) (Loup *et al.*, 1991a, b). Other examples concern peak strains associated with the maintenance of verticality in trees that support an asymmetric crown. Radi (1992) found a great amount of compression wood in a *Pinus pinaster* Ait., almost vertical at the base but strongly tilted in the crown. We measured peak strains in an almost-vertical *Fagus sylvatica* L. that were not opposite to the direction of lean but to a great branch. These peaks disappeared above the branch (Fig. 7b) (Chanson *et al.*, 1992; Chanson, 1993).

A last observation is much more puzzling: Two opposite peaks (on the same circumference) have been measured on *Eperua falcata* Aubr. (French Guyana), *Sacoglottis gabonensis*, *Dichostemma* sp., *Alstonia* sp., (Cameroon), and *Castanea sativa* Mill (France, Italy) (Chanson, 1993; Fournier *et al.*, 1994). The functional explanation is not obvious, as the consequence for the tree is a tendency to split. Each side pulls the stem toward itself (Fig. 7c). Actually, these patterns of maturation strains have been observed on sympodial trees, where the stem is made of successive growth units from axillary buds (Bell, 1991), and thus, these double peaks can be interpreted as conflicting signals coming from the former and the new leader.

## RELATIONS BETWEEN MACROSCOPIC PROPERTIES OF WOOD AND HISTOLOGICAL FEATURES (MICROBIOMECHANICS)

The understanding of macroscopic mechanical properties of wood from the microstructure of the material at the scale of cells, cell walls, etc., (Barber and Meylan, 1964; Mark, 1967; Cave, 1968; Barrett, 1973; Archer, 1987a, b; Gibson and Ashby, 1988; Vieville, 1992) provides the mechanical basis for further biomechanical studies. The cell wall can be modeled as a composite made of a matrix of lignins, hemicelluloses, and amorphous cellulose, reinforced by long bundles of crystalline cellulosic fibrils. In the S2 layer—which represents the greatest part of the cell wall in fibers or tracheids—the fibrils are oriented helicoïdally and the angle between the fibrils and the cell axis (the "microfibril angle;" MFA) can vary from 0 to 50° among woods. Models derived from mechanics of composite materials emphasize the importance of the MFA, as well of other parameters such as the crystalline cellulose content or the mechanical properties of the matrix, to explain the variations of macroscopic properties of wood. Longitudinal shrinkage (longitudinal tensile stress) decreases when the MFA or the lignin content increases, and increases with the crystalline cellulose content; it becomes a dilatation (a compressive stress) for a high MFA or high lignin content (Yamamoto and Okuyama, 1988; Okuyama *et al.*, 1993).
<figure><img src="image_250.png" /></figure>

Fig. 7. Unusual peaks of maturation strains related to tree morphology. (a) Accidental death of the apex and traumatic reorientation in a *Pinus pinaster* Ait. (b) asymmetry of crown in a *Fagus sylvatica* L.; (c) double peak of tension in a *Saccoglottis gabonensis*. (From Loup *et al.*, 1991; Chanson, 1993; Fournier *et al.*, 1994.)
<figure><img src="image_251.png" /><figcaption>Fig. 7. Continued.</figcaption></figure>

The biomechanical problem is then to establish the laws of variations of microstructural parameters within a sample of woods that characterize a biologically significant range of situations. The variations of structural variables usually show intricate interactions: For example, in conifers, the MFA increases with the lignin content since a higher MFA and lignin content is typical of compression wood or juvenile wood (Timell, 1987; Zobel and van Buijtenen, 1989).

A similar law has been observed in a *Eucalyptus* (clones planted in Congo, Africa) by Baillères (1994; Baillères *et al.*, 1994 (Fig. 8), correlated with variations in the longitudinal maturation shrinkage (Fig. 8). However, we cannot really validate the mechanical model since the effects of the variations of MFA and lignin content cannot be tested independently. Moreover, a very good cor-
<figure><img src="image_252.png" /><figcaption>Fig. 8. Relations between longitudinal maturation and some histological features—the microfibril angle (MFA), lignin content, and monomeric composition—in clones of *Eucalyptus* (Congo, Africa). (From Baillères, 1994; Baillères et al., 1994.)</figcaption></figure>

relation between the longitudinal maturation strain and the monomeric composition of lignins (the ratio of syringil-to-gaiacyl units; S/G) has been found (Fig. 8). This correlation is difficult to explain from a mechanical model. It could be explained by variations of the mechanical properties of the lignin matrix due to monomeric composition (direct physical cause) but also by a more complicated correlation between the S/G ratio and the microfibril angle (or the lignin content) due to biological mechanisms related to the cambial activity and the secondary wall formation and lignification.

We now focus on biomechanical models of regulation. Two examples are studied: Mattheck’s “constant stress” growth and the regulation of lean by the differentiation of heterogeneous wood.

## BIOMECHANICAL MODELS OF REGULATION

These models assume, in addition to the mechanical model, biological control of output (i.e., control of strained forms, supported stresses, safety factors, etc.).

Since trees seem to grow faster in diameter when they are loaded, cambial growth can be postulated to be proportional to the stress level at the stem periphery. From a mechanical model giving the distribution of stresses from the applied loads and the stem geometry, the more stressed zones are assumed to grow faster, and thus the distribution of growth along and around the stem is modeled. From this idea, Mattheck (1990a–d; Mattheck and Korseka, 1989;
Mattheck and Prinz, 1991; Mattheck, in press) developed a finite element method for computing shapes of trees or of parts of trees. It has been applied to study forks, branch joints, welds, stems near stakes and buttresses loaded by bending forces, wounds submitted to tensile surface stresses, and mechanical contacts at the tree surface (stones, other trees). The level of stress is characterized by the Von Mises stress, which is independent of the material direction, usually not used by engineers in highly anisotropic materials such as wood. However, in each case, the computed form is checked to be in good visual agreement with the natural form observed.

Another example comes from compression (resp. tension) wood on the lower (resp. upper) side of leaning trunks or branches of gymnosperms (resp. angiosperms), associated with high asymmetry of maturation strains, as shown in our previous biomechanical characterization. From our preliminary mechanical analysis (see the Appendix), we know that this asymmetry (represented by a parameter $\alpha_1$) allows the stem to bend upward and we can compute the successive forms of the growing leaning stem, loaded by gravitational forces and maturation prestresses, if the value of $\alpha_1$ is given at each step of growth. We can estimate $\alpha_1$, independently of the mechanical computation, by measurements of released strains or observations of histological features. But $\alpha_1$ can also be modeled assuming that the formation of reaction wood is locally related to the stem lean. When this lean overcomes a given value, reaction wood is differentiated to restore the stem verticality, and thus $\alpha_1$ becomes different from zero. The sign of $\alpha_1$ is given by the sign of the lean. Its intensity has been assumed from different laws: (i) a given value (based on the assumption of a unique "quality" of reaction wood), (ii) a value proportional to the lean, (iii) a value proportional to the velocity of the lean variation, and (iv) a combination of the last two factors.

We applied this model (under the simplest assumption, i) to the description of the reactions of a young maritime pine artificially tilted (Fournier and Chanson, 1994). The results (Fig. 9) show a good qualitative agreement with the experiment: First, the whole stem bends and compression wood is formed on the lower side. The base goes on bending up, while the top straightens itself by bending down forming compression wood on the upper side. However, in the model the overcorrection is activated when the stem passes the vertical, while in the experiment the top of the plant anticipates the curvature of the base and begins to bend down when the whole stem is still severely tilted. Moreover, the model underestimates the curvatures of the base and overestimates those of the top. Therefore, the necessity for a more complicated regulation law (iv) is demonstrated. The parameters that govern such a law remain to be determined.

Finally, the regulation has been based on the cambial perception of lean (graviperception). However, a perception of stresses or strains, or an apical perception diffused along the stem, could also be tested. We prefer the first one
<figure><img src="image_253.png" /><figcaption>Fig. 9. Biomechanical regulation of lean on a *Pinus pinaster* Ait. artificially tilted. (a) Observed successive forms, from digitized photographs; (b) observed distributions of compression wood (C.W.) (drawn from microscopic observations) (c) simulation by the biomechanical model of both righting movements and formation of C.W. The asymmetry of maturation strains $\alpha_t$ is 0.25%; the mean radial growth and diameters were measured on the previous drawings, at five heights. Computations use 10 spatial elements and 10 temporal units of growth, and thus the See Loup *et al.* (1991) for more details about a and b.</figcaption></figure>

since it fits most of the experiments that have been done on conifers [artificial leans, loops (Timell, 1987)]. For example, we observed that an artificially tilted but not stressed (sustained in a pipe) stem of maritime pine differentiates compression wood associated with a high asymmetry of maturation strains (Chanson and Fournier, unpublished).

## FUNCTIONAL INTERPRETATION OF BIOMECHANICAL MODELS OF REGULATION

A further problem is to ask whether the regulation leads to or restores a characteristic mechanical state. For example, Mattheck's model leads to a morphology where peaks of stress at the stem periphery are avoided since the faster growth in more loaded areas homogenizes the distribution of loads. As a consequence, the safety factor (i.e., the ratio between the critical stress at failure and the supported stress) is kept constant along and around the stem periphery. Such a mechanical state is called "constant stress design." Our model refers to an equilibrium lean, which we assumed to be vertical in our example, but it could be oblique or horizontal for branches (Sinott, 1952). Hence, the mechanism of regulation generally induces peculiar mechanical states.
<figure><img src="image_254.png" /><figcaption>length: 90 mm</figcaption></figure>

<figure><img src="image_255.png" /><figcaption>length: 390 mm</figcaption></figure>

<figure><img src="image_256.png" /></figure>

1 mm

gravity

<figure><img src="image_257.png" /><figcaption>length: 240 mm</figcaption></figure>

<figure><img src="image_258.png" /></figure>

<figure><img src="image_259.png" /></figure>

traumatic tissue

<figure><img src="image_260.png" /></figure>

resin ducts

<figure><img src="image_261.png" /></figure>

compression wood

<figure><img src="image_262.png" /><figcaption>length: 30 mm</figcaption></figure>

Fig. 9. Continued.

These mechanical states can be interpreted as "targets," which can be related to principles of economy or efficiency (optimal design). Constant stress design usually ensures the minimal quantity of material required, for a given level of this safety factor and a given load (Fleury, 1978). Actually, Mattheck
<figure><img src="image_263.png" /><figcaption>Fig. 9. Continued.</figcaption></figure>

put the emphasis on the optimality of the computed morphologies toward this principle of both economy (minimal quantity of material) and efficiency (given level of safety factor). Concerning our second example, the vertical position of our model is also a mechanical advantage for a trunk since it minimizes the bending moments due to weights and the risks of torsional strains in the case of lateral winds.

However, optimality is a deep problem in biology. Its interpretation as a consequence of a neo-Darwinian process of adaptation is usually the matter of deep discussion (Maynard Smith, 1978; Givnish, 1986; Gould, 1986; Niklas, 1988). Therefore, the existence of a mechanical target cannot be proved by a simple principle of efficiency and economy and must be validated carefully.

## CONCLUSION

We have tried to organize and summarize the biomechanics of tree growth, mainly at the level of the organ (the stem). As these studies require the use of both mechanical and biological concepts and methods, three methodological steps have been carefully distinguished, from classical mechanical analysis (which includes, however, the surprising effects of radial growth) to the more original biomechanics of regulation. This last step is much more ambitious than the first approach, as it aims at predicting the geometry of the structure or the mechanical properties of materials (which are data and not results in the preliminary mechanical problems). However, the hypothesis of a biological control
must be assumed and validated. Up to now, biomechanical models of regulations are based on very simple hypotheses, while the biological control of growth and differentiation is obviously much more complex.

The biomechanics of tree growth and wood formation provides methods of analysis as well as practical results for forestry, wood technology, and arboriculture. The biomechanical analysis of the righting movements of an artificial stem, in relation to the formation of reaction wood, could be the basis of a test for breeding selection applied to the ligniculture of pines, poplars, eucalyptus, etc., where reaction wood due to a gravitropic response and associated great curvatures of trunks are supposed to be responsible for a lower quality of trees.

Finally, trees are interesting plants to perform biomechanical investigations, since wood has a tremendous memory, which allows one to read a posteriori the plant's reactions.

## APPENDIX: LONGITUDINAL PRESTRESSES AND ASSOCIATED BENDING MOVEMENTS

This Appendix develops the mechanical model of longitudinal strains and prestresses due to the cellular maturation of a new peripheral thin layer. This model allows calculation of the reorientation (curvature) of the stem due to asymmetry of maturation strains, taking into account the geometry (radius and thickness of the peripheral layer) of the stem.

Assuming that between $t$ and $t + dt$, a new layer of wood (thickness $dR$) has been added on the surface of the tree (radius $R$), and that stresses other than longitudinal are of negligible magnitude, unidirectional, very simple behavior laws can be written as

$$ \partial \sigma_L^* = E(\epsilon_L - \alpha_L \Delta M) = E(\epsilon_L - \alpha_L) \quad \text{in the new layer} \quad (1) $$

$$ \partial \sigma_L = E(\epsilon_L) \quad \text{in the older wood} \quad (2) $$

where $E$ is the longitudinal Young's modulus, $\alpha_L$ is the tendency of the new layer to shrink longitudinally during cellular maturation, and $\partial \sigma_L$ and $\epsilon_L$ are, respectively, the incremental stresses and strains in the stems between $t$ and $t + dt$.

Then assuming that the experimentally observed datum $\alpha_L$ varies with the angular position $\theta$ as

$$ \alpha_L(\theta) = \alpha_0 + \alpha_1 \sin \theta \quad (3) $$

and that the cross section remains plane so that $\epsilon_L$ varies linearly along the y axis ($y = r \sin \theta$),

$$ \epsilon_L(r, \theta) = \epsilon_0 + \epsilon_1 \frac{r}{R} \sin \theta \quad (4) $$
behavior laws can be rewritten as

$$ \partial\sigma_L^* = -E[(\alpha_0 - \epsilon_0) + (\alpha_1 - \epsilon_1) \sin \theta] \quad \text{in the new layer} \quad (r = R) \qquad (5) $$

$$ \partial\sigma_L = E \left( \epsilon_0 + \epsilon_1 \frac{r}{R} \sin\theta \right) \quad \text{in the older wood} \quad (r < R) \qquad (6) $$

Thus, $\epsilon_1$ follows from the equilibrium of bending moments within the cross section:

$$ \int_{0}^{2\pi} \int_{0}^{R} \partial\sigma_L r \sin\theta r \, dr \, d\theta + \int_{0}^{2\pi} \partial\sigma_L^* R \sin\theta (R \, dR \, d\theta) = 0 \qquad (7) $$

so (by replacing $\partial\sigma_L$ and $\partial\sigma_L^*$ by their above-written expressions and after integrating)

$$ \epsilon_1 = 4 \frac{dR}{R} \alpha_1 \qquad (8) $$

Note that $\epsilon_1 \ll \alpha_1$ (it could be shown as well that $\epsilon_0 \ll \alpha_0$) since $dR \ll R$. Therefore,

$$ \partial\sigma_L^* = -E[\alpha_0 + \alpha_1 \sin\theta] = -E\alpha_L \qquad (9) $$

Maturation strains are entirely impeded and converted into stresses in the new thin layer. The variation of curvature is finally,

$$ \partial C = \frac{\epsilon_1}{R} = 4 \frac{dR}{R^2} \alpha_1 \qquad (10) $$

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# Stem and Root Tree Architecture: Questions for Plant Biomechanics<fnref n="1" />

C. Edelin<fnref n="2" /> and C. Atger<fnref n="2" />,<fnref n="3" />

*The analysis of the architecture and ontogenesis of trees reveals growth processes which bring into play the mechanical reaction of the organism. We present some of these processes concerning both shoot and root systems. The problems they entail might serve as study models for biomechanics.*

**KEY WORDS:** tree architecture; development; stems; roots.

## INTRODUCTION

Trees are branched organisms. Growing from a fixed point, the collar, they develop below and above the ground a system of axes which is grouped in three parts: the crown, the trunk, and the root system. The stability of the whole system, which can be huge, is devoted to support tissues which are variously located on axes, wood being functionally the most important of these tissues. The knowledge of wood properties is essential for understanding the mechanics of a tree and is the subject of numerous studies (Kollmann et al., 1968). However, this knowledge is not enough to illustrate the way by which a plant organism supports itself. The development of wood, its location, and its structure are the tree's adaptative response to mechanical problems inherent in its architecture and vary with its immediate needs. This tissue has been selected during evolutionary processes because of its efficiency to solve problems, but it is an executor and not a cause. We will understand its role better when these mechanical problems are clearly exposed.

<footnote n="1">¹This is the published version of a paper presented at the Plant Biomechanics Congress, Montpellier, France, September 5-9, 1994.</footnote>

<footnote n="2">²Laboratoire de Botanique, CNRS URA 327, 163 rue Broussonet, 34000 Montpellier, France.</footnote>

<footnote n="3">³To whom correspondence should be addressed.</footnote>
In studies on whole-tree mechanics, the plant is often considered as a simple geometrical shape quite far from the botanical reality. This reduction is justifiable for the needs of modeling, but we think that, far from being a difficulty, to take into account the real tree shape is, on the contrary, a sign of progress. By posing new problems, this approach can stimulate reflection, open new fields of research, or give some interesting information that leads to a better view of problems already in course of study. That is why, in this paper, we present real phenomena of tree mechanics which, to our knowledge, have not been already studied. First, we analyze some problems inherent in shoot and root systems; second, we try to replace them on a larger scale of reflection.

# VARIOUS MECHANICAL PROBLEMS ENCOUNTERED IN SHOOT SYSTEMS

## Growth and Straightening Up

Trunk or branches straightening up is a frequent phenomenon that has been little studied. It occurs frequently, without any trauma, during the ontogeny of numerous species and allows the tree to express its specific architecture. Two examples taken from tropical trees allow us to illustrate different ways of becoming erect.

### *The Erection of Tree Trunks in Troll's Model*

Architectural models (Hallé *et al.*, 1978) are fundamental shapes which can be encountered in the plant kingdom. There are about 20 which account for all the known species. Architectural models are based on the analysis of some fundamental characters which establish the tree shape: pattern of branching, pattern of growth, growth direction of axes, and location of flowers. One of the most frequent is the Troll model. According to its definition, plants conforming to this model are built with plagiotropic axes only, growing in a horizontal direction. Such a direction, which concerns branches as well as the trunk, seems at first sight to be incompatible with arborescent status. However, some of the biggest trees in the plant kingdom conform to this model, such as *Koompasia malaccensis*, which, with its height of 70 m, dominates the forest in Malaya. To gain in height, trees conforming to the Troll model have developed various growth strategies, some of which involve mechanical problems. We study these processes on *Muntingia calabura*.

*Muntingia calabura* (Elaeocarpaceae) is a small tree from Central America, widespread in the tropics. It is usually named the cherry tree. The young individual (Fig. 1a) is composed of a monopodial first-order axis (A1), with distinct phyllotaxis: Leaves are placed on a horizontal plane, alternately at the
<figure><img src="image_264.png" /><figcaption>Fig. 1. Three stages in the development of *Muntingia calabura*.</figcaption></figure>

right and the left side of the bearing axis. The base of this axis is vertical (orthotropic); its distal part is horizontal (plagiotropic). This growth direction is the result of the apical meristem activity. After germination, the apical meristem grows orthotropically a few centimeters. This way of growing allows the axis to emerge from the soil, then its direction changes and it becomes plagiotropic: It grows perpendicularly to the gravity gradient.

When the tree reaches 1 or 2 m in height (Fig. 1b), its trunk bears long branches alternately disposed to the left and the right side of the bearing axis. These branches are disposed in a horizontal plane. They are composed of three orders of axes, all disposed in the same horizontal plane. The trunk is vertical except for its distal part, which is horizontal and illustrates the plagiotropic behavior of its apical meristem.

The adult does not exceed 10 m in height (Fig. 1c). Its trunk bears long plagiotropic branches composed of five orders of axes, the most peripheral order bearing lateral flowers. The trunk is vertical to oblique for the first two-thirds of its length and then its apex is horizontal. During the later stage, the latter grows definitively plagiotropically. In this species, the development of an orthotropic trunk is a consequence of the main axis straightening up. This phenomenon occurs 10 cm behind the apex. It begins exactly in a zone where the leaves begin to fall as a consequence of cambial growth: It is a process linked to wood formation.

Many trees conforming to the Troll model show such a development. Secondary from the viewpoint of morphogenesis, this process is of great importance because it allows the tree to gain in height. It depends on the surrounding
conditions: When the tree grows in open light, growth stops quickly and the tree remains small. On the other hand, in a dense population, growth goes on for a long time: The trunk is vertical for the greater part of its length, and only its tip when located in the forest canopy is horizontal.

### The Turning-Up of Branches

Some trees conform to the same architecture all their life. Others change in growth patterns during their ontogenesis. In those trees that show metamorphosis (Edelin, 1990), one very important process that accompanies such a transformation is the progressive setting-up of branches when they become larger and larger and more and more branched, as can be seen, for example, in *Shorea leprosula*.

Young *S. leprosula* has an architecture conforming to Roux's model. A monopodial orthotropic trunk bears plagiotropic branches subtended by each leaf axil (Figs. 2a and b). These branches are made of two types of axis (Fig. 2c); the second-order axis (A2) is horizontal to oblique, with distichous phylLOTAXIS. It is branched in one plane. Its third-order laterals (A3) are also pla-

<figure><img src="image_265.png" /><figcaption>Fig. 2. (a, b, d) Three stages in the development of *Shorea leprosula*; (2c) view of a plagiotropic branch.</figcaption></figure>
giotropic, with distichous phyllotaxis. They are short-lived and remain un-branched.

At the top of the trunk, the branching angle is more acute and branches are oblique. Nevertheless, they become nearly horizontal away from their apices. This event is not the consequence of simple drooping. The branching angle varies during the first stages of cambial growth of the trunk.

When the tree reaches 10 to 20 m in height (Fig. 2d), its architecture is different. Its crown is composed of three types of branches. The first is plagiotropic branches, with three orders of axis. They are quite short and similar to those already described. The second is orthotropic branches, which develop from the upper part of the trunk. They are vertical and their branching angle is acute and remains constant during the cambial growth of the trunk. Nevertheless, during their growth they subside later, so that, finally, their distal part becomes oblique. These branches have a spiral phyllotaxis and their branching is continuous and radial. They bear plagiotropic order 3 and 4 axes. The third type is architecturally intermediate between the first two. Their orientation and their organization change during their ontogenesis. At its basal part, the order 2 axis, A2, is plagiotropic; its branching angle is wide and its phyllotaxis is distichous. It bears twigs disposed on a horizontal plane. At its distal part, the order 2 axis is orthotropic and its growth direction vertical. Its phyllotaxis is spiral and its branching is continuous and radial. The connection zone between these two parts is short. The turning-up takes place on only a few internodes.

When these branches are disposed at the base of the crown, the connection zone between the orthotropic and plagiotropic parts is located near the end of the branches. When these branches are located in the middle of the crown, the change of orientation takes place earlier. In older trees these intermediate branches disappear: Mostly orthotropic branches occur, separated here and there by a few plagiotropic branches, which soon fall. The turning-up observed in *S. leprosula* is a process which accompanies axial growth. It differs from the situation described in Troll's model by the fact that it takes place in the growing end, and not behind it. It originates in apical meristem activity and is reinforced later by wood development.

## The Strengthening of Branches

The mechanical stability, or even the integrity, of the organism may be endangered by the development of certain formations inside the crown of trees. For example, an abnormally heavy load of fruits can cause a branch to break. As soon as there is a change in the mechanical balance of the crown, the tree generally reacts by reinforcing its existing structure. Provided that the reaction is quick and intense enough, it manages to compensate for the mechanical loss of balance which has begun to appear. This adaptive process occurs continually
in the life of the organism and even plays an essential part for certain species. We illustrate such a phenomenon with the example of the crown development of *Virola surinamensis* (Barthélémy *et al.*, 1991).

*Virola surinamensis* is a tree growing in the swamp forest of Amazonia. It can reach a height of 35 m and form a crown 25 m in diameter. The architecture of the young tree conforms to Massart’s model: The trunk is an orthotropic monopodium which bears whorls of regularly spaced plagiotropic branches (Figs. 3a and b). Its structure changes completely on becoming an adult. At this stage we can observe small trees developing from resting buds located on the large plagiotropic branches growing on the upper part of the crown (Fig. 3c). They are easy to recognize in the crown, as the vertical orientation of their small trunk contrasts with the horizontal direction of its bearing branches. They develop quickly, and after a few years each one makes a heavy load for its bearing branch. The latter reacts by increasing its diameter in its basal part to maintain it in a horizontal position. On the other hand, the end part dies and degenerates gradually, so that eventually the branch seems to end in a small upright tree. When it has spread over the whole crown (Fig. 3d), this process gives the adult tree its very characteristic “candelabra” habit. The development of these small trees in the crown can cause the trunk to bend, though generally only slightly,

<figure><img src="image_266.png" /><figcaption>Fig. 3. (a, b) *Virola surinamensis*—young stages conforming to Massart's model; (c) development of small trees on some branches; (d) crown architecture of the adult tree.</figcaption></figure>
as their pattern of appearance on each branch whorl allows for a symmetrical
distribution of weight.

The shoot system of the tree must react against the tendency of its different
parts to droop. Nevertheless, the examples we have analyzed show that wood
production can also respond to other imperatives. It may be linked with the
expression of ontogenetic processes—differentiation, metamorphosis, and re-
iterative patterns—which have nothing to do with support problems.

VARIOUS MECHANICAL PROBLEMS ENCOUNTERED
IN ROOT SYSTEMS

In subterranean systems, roots are also sensitive to gravity but they are
included in the soil and they have no problem about support. They are under
the influence of the shoot systems that they have to stabilize. This role needs
several mechanical properties. Stabilization of the tree is based on the general
shape of its root system. In outline it is a disk-like structure (Fig. 4) composed
of big perennial roots which are superficially buried in the soil and radiate far
from the collar (Atger et al., 1994). This disk is sometimes strengthened with
oblique roots (Raimbault, 1991). This group of roots prevents the tree from
falling over. Furthermore, one or several tap roots anchor it. We have to notice
that, contrary to what can be seen in the literature, the horizontal extent of roots
can be much greater than the height of the trunks and the tap roots are generally
much smaller. We study several variations of this organization.

Tree Anchorage

There are several species in which stability and anchorage are provided by
the tap root, for instance, *Anaxagorea dolichocarpa* (Fig. 5). This is a small
tree from the swamp forest of the Guyanas. It does not exceed 15 m in height.

<figure><img src="image_267.png" /><figcaption>Fig. 4. General architecture of shoot and root system in trees.</figcaption></figure>
<figure><img src="image_268.png" /><figcaption>Fig. 5. *Anaxagorea dolichocarpa*: a view of the root system.</figcaption></figure>

Its root system is composed of a straight tap root bearing, near the collar, a whorl of three or four sinuous perennial horizontal roots. Laterally, these horizontal roots bear two rows of thin roots which are short-lived and disappear quickly. The tap root is 1.5 to 2 m in length and 10 cm in diameter near the collar, whereas its perennial laterals are only 3 m in length and 3 cm in diameter near its base. During the disinterring of the root system, the cutting of these horizontal laterals does not result in the tree falling but only reduces its stability. In this species, the tap root represents the greater part of the mechanical support of the plant during its whole life.

In other species, such a situation can be observed only at the beginning of ontogenesis. With germination, the tap root develops more quickly than its laterals or the shoot; in the same way, its diameter increases more quickly than that of the lateral roots. The tap root is then the major part of the vegetative system of the seedling. For several years, it provides the stability of the young plant. Later this function is transferred to the horizontal roots which develop more than the tap root.

## Horizontal Stability

The greater part of tree stability is devoted to the horizontal system. Its organization, like a disk, is adapted to this need provided that it is well secured and rigid enough. Trees can respond to such a need by different properties such as root tension, cambial growth, and forking of roots.

### Root Tension

In several species of *Clusia* (Fig. 6), the seedling germinates in the crown of a host tree; more frequently, in a fork of its main branches. To reach the soil, it quickly develops a tap root, which goes down freely along the trunk. As
<figure><img src="image_269.png" /><figcaption>Fig. 6. Development of the root system of *Clusia* sp. and experimental demonstration of tension in tap roots.</figcaption></figure>

its distal part touches the soil, it branches and several horizontal roots begin to form. At the same time, the tap root begins to be in tension. If one cuts it at the height of a man, one can indeed observe that its two ending parts separate and cannot be joined again without forcing them together. This retraction of tap root tissues is the consequence of the anchorage of its end part. When such a phenomenon occurs on horizontal laterals, as can be observed in several other species such as *Cecropia obtusa* and *Ficus* spp, it is the whole root disk which goes under tension and braces the tree.

## Formation of the Forks

The pattern of branching in horizontal perennial roots leads to the formation of forks, which play a direct part in tree stability. *C. obtusa* shows such a pattern of branching (Fig. 7). Its root system is composed of a tap root 60 cm in length bearing, near the collar, a whorl of horizontal perennial roots. The basal part of these roots forms several successive forks composed of three members: two horizontal ones and a third vertical one, which branches from the lower face of the fork and penetrates deeply into the soil. At the end of these perennial horizontal roots, some forks can be found, but they are quite different. They are
<figure><img src="image_270.png" /><figcaption>**Fig. 7.** Horizontal stability of the *Cecropia obtusa* root system: development of forks and supplementary tap roots on the perennial horizontal roots.</figcaption></figure>

composed of three horizontal members: the perennial root and two of its shedding laterals, which are subopposed. These two patterns of branching allow the root system to radiate widely on soil surface and are the basis of the spreading of the root system. Furthermore, these three branched forks act as anchorage points which prevent any slipping of roots during an eventual tilting of the trunk.

### Cambial Growth

The large diameter of the perennial roots—about several dozens of centimeters—is one of the most important properties which contributes to the rigidity of the root disk and to the stability of the tree. But many structures increase this rigidity, such as root grafts, butresses, and adventitious roots.

**Root Grafts.** The root system of the plane tree (*Platanus hybrida*, Plataneceae) is composed of a whorl of perennial horizontal roots branching in one or two forks near the collar (Fig. 8a). Some additional roots appear on that structure later (Fig. 8b). During their growth these additional roots intertwine and come into contact one with another; their cambium goes into fusion (Fig. 8c). Gradually, they develop as a net of grafted roots which, with cambial growth, turns into a solid plinth around the base of the tree (Fig. 8d). The development of these additional roots goes on for a long time so that the plinth is continually strengthened.

In several species, the capacity of roots to enter into fusion can play a major role in the mechanical support of the plant. Strangling figs, for example, begin their life by germinating in the crown of a host tree (Fig. 9a). They quickly develop two or three tap roots which go down along the trunk, entwining it and branching (Fig. 9b). All these roots go into fusion and build something
<figure><img src="image_271.png" /><figcaption>Fig. 8. Four stages in the development of the plane tree root system, showing the progressive grafting of roots.</figcaption></figure>

like a framework around the host trunk, which gradually prevents its cambial growth and may eventually cause its death (Fig. 9c). In the same way, the entwining and the strengthening of these grafted roots lead to the formation of a root trunk whose rigidity allows the tree to be self-supported (Ng, 1975).

*Buttresses and Stilt Roots.* At the base of the trunk, roots can form some
<figure><img src="image_272.png" /><figcaption>Fig. 9. Progressive grafting of roots allowing the strangling fig tree to self-support.</figcaption></figure>

structures which prevent the tree from bending over. Buttresses are one example of such a structure (Fisher, 1982). Buttresses are a strip-like basal part of the perennial horizontal roots showing various shapes, which strengthen the meeting point between root and shoot. They are the result of an enhanced cambial growth of the upper part of the root, which leads to a vertical strip of wood joining the base of the trunk to that of these laterals. They have two origins: first, cambial growth concerns horizontal perennial roots already developed (Figs. 10a and b); and second, buttresses are formed from adventitious roots developing later at the base of the trunk (Fig. 10c). In these two cases the effect on tree support is the same. Some adventitious roots can act differently. *C. obtusa*, for example, strengthens its root system by developing stilt roots around the base of its trunk later. These roots are like arches and firmly hold up the trunk (Fig. 11).

## DISCUSSION AND CONCLUSION

This rapid survey has enabled us to present a few situations which may serve as models for biomechanics research. Many other situations occur, particularly those that are the result of trauma. We know that when a tree bends
<figure><img src="image_273.png" /><figcaption>Fig. 10. (a, b) Various shapes of buttresses in tropical trees; (c) buttresses developed from adventitious roots.</figcaption></figure>

<figure><img src="image_274.png" /><figcaption>Fig. 11. Adventitious stilt roots in *Cecropia obtusa*.</figcaption></figure>

as the result of an accident—say, when it has been partially uprooted—this
deformation may cause the trunk and the branches to straighten up, and reaction
wood is formed. Such an incident also has repercussions on the root system.
On the other hand, when a part of the crown disappears after a drastic natural
or artificial pruning, the tree reacts to reestablish harmony in the mass weighing
on the crown. These phenomena are often mentioned, so we preferred to discuss
less common situations. If we consider all these examples, we notice that the
study of tree mechanics in the field must take into account two complementary
aspects. First, we must analyze the static structure of the tree and understand
the forces which assure the stability of a shoot or root branched system spreading around one central axis. But this is not enough. The tree is an organism which undergoes constant changes. Its ontogeny is constantly creating new situations that the organism must control from a mechanical point of view by a specifically adapted production of wood according to problems that arise: development of the existing structures, changes in the growth direction of axes, torsion, etc. To understand tree mechanics, one must necessarily take these factors into account.

The study of trees shows how careful one must be in defining general models. Architecture and ontogenesis, cambial activity, and wood structure vary according to species, thus creating a great range of different mechanical problems and specifically adapted answers. The latter vary according to the organs involved. The ability to produce wood is not the same in the trunk, branches, or twigs, and the age and physiological state of the tree influence this faculty to a great extent. Faced with so much diversity and detail, engineers and botanists need to collaborate closely so that the study of tree mechanics in the field may give good results.

## ACKNOWLEDGMENT

This is a publication of the Institut des Sciences de l'Évolution.

## REFERENCES

* Atger, C.; and Edelin, C. (1994). Premières données sur l'architecture comparée des systèmes racinaire et caulinaire des arbres. *Can. J. Bot.* (in press).

* Barthélémy, D., Edelin, C., and Hallé, F. (1991). Canopy architecture. In Raghavendra, A. S. (ed.), *Physiology of Trees*, John Wiley & Sons, New York, Chichester, pp. 1-20.

* Edelin, C. (1991). The monopodial architecture: The case of some tree species from tropical Asia. *Research Pamphlet 105*, Forest Research Institute Malaysia,, Kuala Lumpur, Malaysia.

* Fischer, J. B. (1982). A survey of buttresses and aerial roots of tropical trees for presence of reaction wood. *Biotropica* 14(1):56-61.

* Hallé, F., Oldeman, R. A. A., and Tomlinson, P. B. (1978). *Tropical Trees and Forest. An Architectural Analysis*, Springer Verlag, Berlin.

* Kollmann, F. F. P., and Coté, W. A. (1968). *Principles of Wood Science and Technology*, Springer Verlag, Berlin.

* Ng, F. S. P. (1975). A note on natural root grafts in Malaysian trees. *Malays. Forest.* 38(2):153-159.

* Raimbault, P. (1991). Quelques observations sur les systèmes racinaires des arbres de parcs et alignements: diversité architecturale et convergences dans le développement. In Edelin, C. (ed.), *L'Arbre. Biologie et Développement*, *Naturalia Monspeliensia*, H.S., pp. 85-96.
# The Biomechanics of Leaf Rolling<fnref n="1" />

Bruno Moulia<fnref n="1" />

The transverse rolling of the graminaceous leaves submitted to water depletion is studied from a biomechanical point of view. It is shown that the "poor" rolling of some species such as maize is not related to a local "low ability" of transverse curvature, but to a structural effect due to the coupling of curvatures within the shell structure made by the whole leaf. This is demonstrated through both an experimental study of free curvatures and a mechanical modeling of leaf rolling at the whole-leaf level. It is thus argued that the differences in the rolling response between species or cultivars may be linked to structural mechanical effects which have been overlooked in past studies, and which can be studied only by a biomechanical approach.

**KEY WORDS:** leaf rolling; structure mechanics; water stress; Graminaceae.

## INTRODUCTION

Leaf movements are a common response to water depletion (Begg, 1980). In many graminaceous plants, the major change in leaf shape associated with water stress is a transverse rolling of the blade (Shield, 1951; Sobrado, 1987; O'Toole and Cruz, 1979, 1980; Hsiao et al., 1984). Several authors have argued that leaf rolling is of adaptive value. Studies have demonstrated that it decreases the transpiration through changes of the "effective" leaf area (i.e., submitted to incident radiation) and of the boundary layer of the leaf (Lemée, 1950; O'Toole and Cruz, 1979; Begg, 1980). The efficiency of this water-saving mechanism may, however, vary with species (see for a minireview see Moulia, 1993). Thus, throughout the century, a research effort has been directed to the study of the

<footnote n="1">¹This is the published version of a paper presented at the Plant Biomechanics Congress, Montpellier, France, September 5-9, 1994.</footnote>

<footnote n="2">²U.R. Bioclimatologie, INRA, F-78 850 Thiverval-Grignon, France.</footnote>
relationship between rolling and water status of the leaf, both by botanists [to establish the anatomical basis of such an adaptation (e.g., Burström, 1942; Shield, 1951)] and by agronomists [to improve our knowledge of the behavior of crop canopies under water deficit (e.g., O'Toole and Cruz, 1979, 1980; Hsiao et al., 1984; Matthews et al., 1990)].

At the whole-leaf level, there seem to be two kinds of relations between the mean hydric status (usually measured as the leaf water potential $Ψ$; see Appendix) and leaf rolling [measured mostly by visual scoring; see Moulia (1993) for more details on the various measurements of leaf rolling]. These typical relations are sketched in Fig. 1. Some species, such as rice and tall fescue, display a rather complete rolling (type 1), whereas others, such as maize and some sorghum, roll much less (type 2). Moreover, both the range of water potential in which the leaf rolls and the shape of the rolling curve are clearly distinct; Type 1 leaves display an almost-linear increase in rolling over a large range of water potentials (approx 2 MPa, after an initial threshold), whereas type 2 leaves show a sigmoid curve, in which the rolling is limited to a much smaller range (approx. 0.6 MPa).

At the anatomical level, the focus has been mainly on the putative role of special motor cells, also called bulliform cells (Burström, 1942; Shield, 1951; Ellis, 1976). The discussion is based mainly on considerations of the correlation between the degree of leaf rolling and the amount and deformation of bulliform cells, that is, on a possible cellular basis for differences in the rolling response.

However, from a mechanical point of view, the leaf blade of the Graminaceae is a shell-like structure (not developable within a plane). Thus, there is a mechanical coupling of the changes in curvature within the leaf structure. If

<figure><img src="image_275.png" /><figcaption>Fig. 1. The two types of relations between the mean water potential and the transverse rolling of graminaceous leaves. The degree of transverse rolling of the whole leaf is usually estimated by visual comparison to a scoring scale of typical transverse shapes (which are figured along the ordinate axis). The type 1 curve is redrawn from data for rice (O'Toole and Cruz, 1980; Hsiao et al., 1984) and tall fescue (Viratelle, 1992), and the type 2 curve from data for maize (Sobrado, 1987) and sorghum (Begg, 1980). See Moulia (1993) for a more complete review of the literature on leaf rolling.</figcaption></figure>
such coupling is important, it may prevent the display of the transverse rolling. As a matter of fact, it is noteworthy that type 1 rolling corresponds to species with very flexible blades, whereas type 2 plants have larger leaves with a stiff midrib.

The purpose of this paper is to report investigations of the importance of such structural effects in the determinism of leaf rolling. The general idea is to concentrate on the study of a type 2 plant, maize (Zea mays L.), and to try to determine the reasons for such a "poor" rolling behavior. The hypothesis to be tested is that the mechanical coupling of curvatures within the leaf structure is the major cause that explains the phenomenology of type 2 leaf rolling. Before going on to our results, however, it is necessary to recall the main features of the maize leaf.

GENERAL MORPHOLOGY AND MECHANICS
OF THE MAIZE LEAF

The mature (i.e., no longer growing) maize leaf is a slender structure, with
bilateral symmetry. Its size varies with leaf rank (see Appendix) and also grow-
ing conditions. It is composed of a central thickened, curved midrib and of a
much thinner part, on both sides of the midrib, which is henceforth referred to
as the lamina (Fig. 2). Previous studies in our laboratory demonstrated that the
midrib is the major stiffening element that influences the longitudinal bending
rigidity of the leaf (Moulia et al., 1994). It can be modeled as a tapered com-
posite beam (Moulia and Fournier, 1994). Moreover, it has been demonstrated

<figure><img src="image_276.png" /><figcaption>Fig. 2. General morphology of the mature maize leaf (redrawn from Moulia *et al*., 1994).</figcaption></figure>
(Moulia *et al.*, 1994) that, in mature leaves (see Appendix), the midrib is curved downward even when the instantaneous bending under its own weight is removed. More details on the morphology and the internal material structure (anatomy) of maize leaves are given by Sharman (1942) and Moulia (1993).

When submitted to water depletion, the maize leaf displays, as stated, type 2 transverse rolling. However, the amount of rolling at a given water status varies among the leaves, with the leaf rank (upper younger leaves usually roll more than lower ones), and within the leaf (the tip is usually more rolled than the middle of the leaf). Moreover, leaf rolling in maize is compounded with a trend to longitudinal straightening (seen mostly in the upper leaves), as seen in photographs of plants profiles and leaf habits (Downey and Miller, 1971; Viratelle, 1992). Notice that this empirical correlation is qualitatively in accordance with our hypothesis of mechanical coupling of the curvatures but could also be explained by a decrease in the self-weight loading due to water loss and, hence, concomitant to the rolling.

## EXPERIMENTAL STUDY OF LOCAL "FREE" CURVATURES

If our hypothesis is correct, then a corollary is that a piece of the lamina isolated from the leaf blade should display increased transverse rolling when submitted to water depletion. This corollary has been tested experimentally. A detailed description of this experiment is given by Moulia (1993) and is being published elsewhere. The general principle has been to submit a transverse specimen from the lamina to various controlled water statuses. The water conditioning has been obtained by bathing the specimens in solutions of polyethylene glycol (PEG 6000) of various known water potentials. The water potentials range from 0 to -3.5 MPa, as rolling typically occurs between these values. PEG was used, as it does not enter the cells, avoids respiration losses, and has been shown to have no toxic effects on maize tissues (Hohl and Schopfer, 1991). The transverse shape of the specimen at hydric equilibrium was recorded by photographs and two-dimensional (2D) digitizing (Fig. 3), and the transverse curvatures estimated using the sliding vicinity program Courb2D (Moulia *et al.*, 1994).

Figure 4 shows three typical transverse shapes at $Ψ = 0, -2.8$, and $-3.5$ MPa. It is clear that transverse rolling increases with drying. Moreover, at a given water potential, transverse rolling of a specimen isolated from the structure can be much more important than that observed in the whole leaf. This is particularly clear at -3.5 MPa, where rolling is almost complete and very similar to that observed in rice or tall fescue (compare with the sketches in Fig. 1).

When the curvatures along the specimen are calculated, there is no significant trend of change with the distance from the midrib (i.e., from the midrib
<figure><img src="image_277.png" /><figcaption>Fig. 3. Experimental design for the study of the free transverse curvatures related to changes in water status.</figcaption></figure>

side to the edge of the transverse specimen). Thus the mean curvature can be estimated for each specimen. The graph of mean transverse curvature versus water potential is given in Fig. 5, showing an exponential relation. No effect of the initial position within the leaf or of the leaf of origin was detectable. Finally, the transverse curvatures are completely reversible (data not shown), so that the response curve of curvature vs potential at static water equilibrium is biunivocal (at least for the studied population of leaves, obtained under homogeneous growing conditions).

As a conclusion for this experimental study, our results are completely in accordance with our initial hypothesis. Free lamina transverse rolling is much more important than in the whole leaf structure and is homogeneous along the leaf, as between leaves. The next step is to study the mechanical (structural) effects within the leaf, to test if they can effectively explain the hindering of the rolling behavior in the whole leaf. This requires mechanical modeling.

## MECHANICAL MODELING OF LEAF ROLLING

Dealing with such a complicated structure as plant organs, there is a *priori* a very large number of material and structural characteristics that may play a part in the determination of their mechanical behavior. Much of the challenge of mechanical modeling is thus to incorporate enough complexity to be sufficiently realistic, but not too much, in order for the model to remain tractable
<figure><img src="image_278.png" /><figcaption>Fig. 4. Typical transverse shapes of the lamina specimens at three water potentials $Ψ$: (a) $Ψ = 0$ MPa (pure water); (b) $Ψ = -2.8$ MPa; (c) $Ψ = -3.5$ MPa. The shapes are obtained from the digitizing of the photographs. The black triangle represents the position of the midrib, with the top of the triangle directed to the upper (adaxial) part of the leaf.</figcaption></figure>
<figure><img src="image_279.png" /><figcaption>Fig. 4. Continued.</figcaption></figure>

<figure><img src="image_280.png" /><figcaption>Fig. 5. Relation between mean transverse free curvature and mean water potential (at hydric equilibrium).</figcaption></figure>
and understandable. It is worth discussing these underlying modeling choices, in order for the model to be understandable.

## Structural Geometry

The maize leaf has been modeled as a thin shell with a varying width, stiffened by a tapering and initially curved beam, representing the midrib. The lamina structure is thus seen as an assembly of shell elements, and the purpose of the mechanical model is to determine the behavior of the whole structure, given the constitutive laws of the shell elements of the "lamina" and of the beam elements of the "midrib." The whole structure is taken as symmetrical about the vertical plane. The width of the lamina is given by a biometrical model (Prevot *et al.*, 1991). No attempt has been made to model the wavy shape of its edges.

## Rheology of the Shell and Beam Elements

The constitutive law of the lamina has been considered as analogous to the linear thermoelastic rheological model. This is the simplest way to generalize the linear elasticity [described experimentally for grass lamina by Vincent (1982) and Moulia (1993) among others] to take into account the occurrence of the (previously described) free strains due to changes in the water status. It states only that any mechanical configuration in which the shape of a piece of the lamina differs from its free shape at a given local water status generates elastic internal stresses (which is called hydric autostresses). Such a thermoelastic analogy was also proposed by Vincent and Jeronimidis (1991), but at a tissue level. It is based on the underlying assumption that the present local water status is not influenced, at a first order, by the changes in the mechanical state.

The material within the lamina has been taken as axially anisotropic, in terms both of free strains and of elastic rigidities. The only water-induced free strain that has been retained in the model concerns the transverse curvature of the lamina, as described in the previous section (notice that the relation between water potential and free curvature is nonlinear).

No attempt was made to incorporate the effects of changes in lamina thickness, which were also reported to occur with changes in water status (Rashke, 1970). In thin shells, these former strains are thought to be geometrically compatible and, thus, of no autostressing effect. (Their only influence is thus on the linear transverse rigidities of the lamina, an effect that can be tested through the sensitivity analysis of the model.) As no longitudinal and transverse shrinkage has (to our knowledge) ever been reported within the ecophysiological ranges of water status, they have not been taken into consideration either.

The linear membrane and bending rigidities of the lamina are also taken as axially anisotropic, with a principal axis of anisotropy in the longitudinal
direction [the maize lamina, as other grass leaves, has longitudinal and parallel ribs, which makes the lamina similar to an oriented fibrous composite (Vincent, 1982; Moulia, 1993)]. The local elastic bending behavior of the midrib is given by a composite beam model (Moulia and Fournier, 1994). Here again, and for the same reasons as for the lamina, no transverse shrinking with water status has been retained.

## Loading

The only loads that have been taken into account is the occurrence of the autostress field related to changes in free curvatures due to changes in water status (i.e., a rheological autostressing). It is assumed also that the water status can vary only longitudinally (i.e., the water status is homogeneous along any transverse line going from the midrib to the lamina edge). Thus, the loading is also symmetrical about the vertical plane. Finally, for the sake of simplicity, self-weight has been neglected, as it can take action only as resisting bending moments and is, thus, equivalent to increased bending rigidity.

## Kinematics

When the maize leaf rolls, it is obvious that large rotations occur. Thus, for the model to provide morphologically realistic outputs, it is necessary to include the structural nonlinearity due to the large displacements. In other respects, it has been assumed in this first version of the model that the transverse shape of the lamina is always an arc of a circle (as free curvature was previously shown to be constant across the lamina). Moreover, we have assumed pure 2D bending [as the mechanical problem has a bilateral symmetry and as the whole leaf remains a very slender structure with a longitudinal aspect ratio (length/width) larger than 10]. As a consequence, the transverse bending plane is supposed to remain orthogonal to the central line of the midrib (no intralamina shearing).

## Mechanical Solving of the Problem

This mechanical problem cannot be handled analytically. The complexity of the structure forces to a discretization, to reduce the domain of possible strain fields. We have used semianalytical modeling. The basic element is presented in Fig. 6. It is truss, with a midrib subelement composed of articulated bars (discretizing the longitudinal beam in pure bending) and a lamina subelement with transverse curved beams (discretizing the transverse bending behavior) related by longitudinal ropes (representing the tensile behavior of the lamina). The whole structure is assembled as described in Fig. 7.

The total mechanical energy of the whole discretized structure at a given
<figure><img src="image_281.png" /><figcaption>Fig. 6. Schematic drawing of the basic element of the leaf model (see text for explanations).</figcaption></figure>

<figure><img src="image_282.png" /><figcaption>Fig. 7. The assembled whole structure (i.e., the modeled "leaf"). Two successive elements have one longitudinal bar and two transverse beams in common (C1 continuity).</figcaption></figure>

"water status" can thus be calculated, given the initial "natural" (i.e., un-loaded) geometry, the water-driven change in free curvature, and the rigidities of the midrib and of the lamina (transverse bending lineic rigidity and longitu-dinal tensile lineic rigidity). The mathematical expression of the model is not detailed here, and is given by Moulia (1993).

The equilibrium configuration is then determined through a numerical min-imization of this total energy, the command variables being the longitudinal and transverse curvatures. Such an iterative method is required because of the non-linearity of the problem related to large displacements (Curnier, 1993). Although
the numerical details are not given here (for a more complete discussion see Moulia, 1993), it should be noted that one of the main problems with a structurally nonlinear problem is that the equilibrium solution is not necessarily unique. The model has been implemented as a C program, called Leafroll.

## SIMULATION RESULTS

The ranges of realistic values for the lamina rigidities required for the model were estimated experimentally using a miniaturized testing machine [see Moulia (1993) for a description of the experiments and of the way to estimate the model parameters from them]. These data being obtained, a series of simulations has been done to test the sensitivity of the model outputs to changes in the various geometrical and material parameters. The only results presented here concern the study of the effect of the initial longitudinal shape (habit) of the leaf. Two initial (really observed) longitudinal habits have been compared (Fig. 8a). The first one is "weeping," corresponding to a typical leaf from the basal part of the plant; the second one is less curved, representative of leaves higher up the plant. All the other parameters of the model are the same in the two cases (including the values for numerical initiation). The loading corresponds to a change in water potential $Ψ$ from -1.4 to -2.5 MPa, which is the range in which rolling has been reported to occur in maize (see Fig. 1b). The related change in free curvature corresponds to a free rolling of one complete whorl in the part of the leaf with the largest width. Notice also that the potential (and hence the free curvature) has been taken as spatially homogeneous (no gradient of $Ψ$ along the leaf). The initial transverse geometry (i.e., at $Ψ$ = -1.4 MPa) was supposed to be flat (i.e., corresponding to the free transverse curvature at this potential). The equilibrium configurations are presented in Fig. 8. Figure 8a shows the longitudinal habits, whereas Fig. 8b represents the width of the orthogonal projection of the lamina on the initial (flat) surface of the leaf. First it is clear that a large part of the transverse rolling cannot be expressed due to the structural shell effects. As stated, this is due to the structural coupling of transverse rolling and longitudinal straightening. Notice also that both the changes in longitudinal curvatures and the degree of transverse rolling are heterogeneous along the leaf. The tip (where the midrib is much less stiff) rolls and straightens much more than the rest of the leaf. Finally, there is a strong effect of the initial habit on both the transverse rolling and the longitudinal straightening. The initially curved leaf rolls and straightens less than the more erected one.

Thus, the shell structural effects existing on the maize leaf can explain (at least qualitatively) the observed morphology of leaf rolling in maize and its empirical correlation with a more or less important longitudinal straightening. Both the longitudinal changes in the degree of rolling along a given leaf and the changes in the degree of rolling between the droopy basal leaves and the upper,
<figure><img src="image_283.png" /><figcaption>Fig. 8. Effect of the initial longitudinal habit of the leaf on leaf transverse rolling and longitudinal straightening: (a) leaf longitudinal profiles (i.e., habit of the midrib); (b) changes in the projected width of the lamina along the leaf (the position being measured by the curvilinear abscissa along the midrib, with the origin at the base of the leaf). (1) Leaf with a "weeping" initial longitudinal habit; (2) leaf with a more straight longitudinal habit. (+ - - +) Initial (unloaded) state; (+ - - - +) equilibrium state. See text for explanations about the values of the parameters.</figcaption></figure>
more erect (but still adult) leaves can be simulated without any spatial gradient in the local water status of the lamina. Finally, the study of the sensitivity of the model to changes in its parameter values shows that the more important parameters are the initial longitudinal habit (as detailed above) and the bending stiffness of the midrib (data not shown; see Moulia, 1993). The more longitudinally curved and the stiffer the midrib, the less the observed rolling in the entire leaf.

## CONCLUSION

The initial purpose of the work reviewed here was to investigate the control of leaf rolling in graminaceous plants and to explain the "poor" rolling of maize- and sorghum-type leaves. It is demonstrated that this phenomenon is not related to some intrinsic inability of the maize lamina to roll (for example, an anatomical peculiarity). Moreover, it is shown that the structural mechanical effects within the leaf can explain the morphology of leaf rolling in maize. It is thought that this illustrates the usefulness of mechanical models in botany, to allow for a change of scale and to account for specific structural effects. Further work will concentrate on generalizing the discretized model (to test the effect of some assumptions concerning the transverse curvature or the shearing of the lamina), on microbiomechanical study of the determinism of free rolling, and also on the biomechanics of the curving of the midrib during the growth of the leaf.

## APPENDIX: DEFINITIONS OF THE MAIN TERMS USED

**Water potential:** This is one of the most popular variables used in plant sciences to characterize the hydric status. It is defined as the amount of work that must be given to 1 unit of volume of water in the considered system to bring it back to a reference state, called "free water" (i.e., pure water at the same temperature and normal atmospheric pressure), through an isothermal and quasi-static transformation. It is thus a local intensive variable, having the unit of a pressure (usual unit, MPa). Referring to "the water potential of a leaf" thus means a spatial homogenization, leading to an "equivalent water potential."

**Leaf rank:** The number of the leaf within the sequence of leaf emergence (the first leaf to emerge has rank 1).

**Mature leaf:** This is a leaf in which both growth and differentiation (especially lignification) are achieved.