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Physiology and Biochemistry |
2Department of Land, Air, and Water Resources, University of California, Davis, California 95616-8627 USA; 3Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138 USA
Received for publication May 9, 2005. Accepted for publication August 1, 2005.
ABSTRACT
The stability of twining vines under gravitational loads suggests an important role for friction. The coefficient of friction, µ, between vine stems and wood is high, often five times greater than between leather and wood, as determined by slip tests on an inclined plane. Stem trichomes function like ratchets to facilitate climbing upward (or to facilitate slipping if the stem is inverted). A mathematical model predicts large masses (kg) must be applied to the base of a twining vine to cause slipping. Vines slip as predicted when µ is low and arc length on the pole is short, and they break before slipping when µ is large or arc length is long. In contrast, twining vines are unstable in compression, collapsing when small masses (<10 g) are hung from the top of the vine. However, if the loads are applied below the uppermost gyre, the stabilizing tensional effect dominates. Therefore, in nature vines twining on a cylindrical support are stable under gravitational loads, unless these loads occur near the apex. A corollary is that a short apical coil can hold up large masses of maturing shoot.
Key Words: biomechanics • curvature • friction • helix • Ipomoea purpurea • structural stability • twining • vines
Curved and twisted structures are common in nature, yet there are few biomechanical studies involving large curvature or torsion. Twining vines are curved and twisted and have physical features similar to both mechanical springs and ropes. Like springs, twining vines have helical shapes. Like ropes, vines fall over when removed from rigid supports. The mechanical properties of vines are reminiscent of the novelties known as "finger trap" puzzles (Fig. 1). These toys, cylindrical shells made from helically woven straps, tighten to pinch inserted fingers when pulled apart, but loosen when compressed, widening the diameter of the trap. This paper explores the tightening and consequent stability of a helical coil in tension.
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), can be specified by the values of s, the arc length along the axis; 
, the associated curvature; and 
, the torsion. Thus an understanding of plant habit requires an understanding of the manner in which curvature and torsion are produced and maintained. Furthermore, and enter into force and moment balances of curved and twisted structures (Love, 1944
; Costello 1978
).
To understand axial curvature and torsion, we need some definitions based on the Frenet vectors, the unit tangent, normal and binormal to a curve in space (Fig. 2). The curvature vector, , lies along the direction of the unit normal (n) to the curve and is the rate of change of the unit tangent (t) as we move along the curve, = n = (
t/s). A more intuitive measure is the radius R of the circle that just fits the curve locally. An arc has large curvature if it is fit by a small circle, and the converse is true. It can be shown that is the reciprocal of R. For a plant stem the spatial distribution of curvature can be characterized by plots of or R vs. s, where s is the distance along the edge or a computed midline of a photographic projection of a stem (Silk and Erickson, 1978
; Whippo and Hangartner, 2003
). Since stems are often straight, having zero but infinite R, it is convenient to plot curvature rather than the radius of curvature as a function of position. A planar curve given by a set of values (s, ) retains its shape even if it is rotated and translated in space. If the curve lies in more than one plane, as is the case for twining vines, it is specified by s, , . The torsion is a measure of the rate at which the curve is twisting out of the local plane and is given by = n = – (b/s), where b is the binormal, i.e., the unit vector perpendicular to the plane of the curve.
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). Ascending a smooth vertical pole, a twining vine forms a helical tube of tissue (Fig. 2). The helix 
through the center of the vine is 
; the parameter a is the radius of the helix; c is the pitch, i.e., the gyre wavelength divided by 2
; and i, j, and k are the unit vectors of the cartesian reference frame. The curvature and torsion at all locations on the helix are
 | (2) |
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; Rowe and Speck, 1996
). The twining mode of growth has evolved in many plant families and enables vines to ascend vertical supports with a minimum expenditure of biomass. The evolutionary success of the twining habit suggests that good mechanical stability should be associated with the helical structure. Silk and Hubbard (1991)
showed that the mechanics of the twining habit involves an innate tendency for the vine to grow into a coil of smaller radius than the cylindrical supporting structure. This causes the stem to hug the support. Putz and Holbrook (1991)
explored the model of loosely coiled springs to show that friction must also play an important role. In this paper we extend the earlier analyses to quantify the role of frictional interactions in maintaining the stability of the twining habit. MATERIALS AND METHODS
Cultivation of vines
Morning glory vines were grown from seeds of Ipomoea purpurea (L.) Roth (Convolvulaceae) cv. Heavenly Blue. Plants were maintained in a growth chamber as described previously (Silk and Hubbard, 1991
). After eight leaves had emerged, stems were allowed to twine around a wooden dowel or the TWIFOR force measuring apparatus (Matista and Silk, 1997
).
Coefficients of friction
To determine the coefficient of friction between vine stems and differently textured surfaces, stem segments were placed on a flat piece of wood ("plane surface") either left bare, covered in contact paper (smoothest texture), or covered with wool fabric (roughest texture). The plane surface was then elevated at one side using a micromanipulator (Olympus America, San Diego, California, USA). Measured from the horizontal, the angle of inclination at which the stem segment slid down the plane was recorded. Two types of placement were used in the measurements. First the segments were firmly placed, so that the entire surface of the stem was in good contact with the plane. This placement was intended to mimic the stem experiencing a twining force so that it is in close contact with its support. In a second set of determinations, termed "gentle placement," the same stem segments were placed on the plane surface with as little pressure as possible. The slip test was also performed for two orientations: an "upright" configuration in which the apical end of the stem segment was at the higher end of the inclined plane and an "inverted" configuration in which the basal end of the segment was placed on the higher end of the inclined plane. An aluminum plate was used to determine frictional coefficients of the nylon string. Uncoiled vine stems were used for the slip tests on an aluminum pole. The coefficient of static friction, µ, was calculated as the tangent of the angle at which the stem or string first slips on the inclined plane. The coefficient of kinetic friction could be estimated as 0.75 µ (e.g., Beer and Johnston, 1977
).
Measurements of the loads produced by the twining stems
A load-measuring apparatus, the TWIFOR, was used to record the normal load exerted by the helical vine stem (Matista and Silk, 1997
; Scher et al., 2001
). The apparatus has a vertical support pole that is cut in half longitudinally to make two poles with semicircular cross sections. The half-poles are mounted in close proximity so that the twining vine acts to pull the halves together. One half-pole is mounted rigidly in a fixed platform, and the other is anchored in a swinging bar suspended at each end from the fixed platform. Suspending elements are thin plastic strips and a thin beam load cell, supplied with balancing, compensating, and conductive elements and a rated capacity for 113 g (Omega Engineering Inc., Stamford, Connecticut, USA). The normal load exerted by the plant on the pole is sensed as a slight horizontal movement of the bar and recorded with a datalogger (Campbell Scientific, Logan, Utah, USA) connected to the load cell. Calibration of the TWIFOR involves an analog to the twining stem: a helically wrapped nylon string of 1 mm diameter. The TWIFOR gives stable output for the calibrating string, with mV output constant in time (Silk and Matista, 1997). The force produced by the calibrating string rapidly drops to zero when the masses are removed from the string. For the experiments shown in Fig. 4, the vine stem was tied to pieces of string at each end and wrapped in a helical form around the TWIFOR. The ends of the string were draped over pulleys, and a mass of 12 g was attached to one end. Progressively greater masses were added at the other end of the system, and millivolt output was recorded as a function of the applied mass.
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Instability of the vine in compression
Small masses (2.5 g) were hung from a leaf axil near the top of the uppermost gyre of a twining vine of Ipomoea purpurea. Masses were added to the leaf axil until the helix collapsed. During the collapse of the helix, the coiled stem fell to the bottom of the support pole.
Statistics
Differences in sliding angle due to substrate, placement method, stem orientation, and presence of epidermal trichomes were analyzed with a general linear model from Minitab Statistical Software (Release 13 for Windows; Minitab Inc., State College, Pennsylvania, USA). Factors were assumed to be fixed and crossed. Analysis of variance was performed with an F test (P < 0.005). Tabular data are shown as means ± SD.
RESULTS
The stem of the twining vine has a high coefficient of friction
Frictional interactions between stems and surfaces were characterized by determining the angle at which stem segments began to slide off an inclined plane. The coefficient of static friction was calculated as the tangent of the angle of slip. Once the vine begins to move on its support, a continuing slide is related to the coefficient of kinetic friction, generally about 25% lower than the coefficient of static friction (e.g., Beer and Johnston, 1977
).
Two I. pupurea phenotypes, having hairless stems and hairy stems, were found in plants grown from a single seed packet. The frictional properties of these two stem types were compared. The effects of smooth and rough surfaces and the effect of placing segments upside down were also determined. In both stem phenotypes, the angle of slip was highest on the rough (wool covered) surface (Silk et al., 2000
; Table 1). In fact, for the hairy phenotype most segments did not slide at all on the woolen surface; rather, when the plane surface was inclined substantially past the vertical, the segments would fall off the plane ("undefined" coefficient of friction in Table 1). This suggests that the stem trichomes grip the wool fibers in some way. In the upright configuration, the angle of slip of the hairy phenotype was least on smooth paper, highest on wool, and intermediate for the bare wood surface. For the hairless phenotype, there was no significant difference between frictional interactions on the wood and paper surfaces.
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In general, Table 1 shows that the gently placed vine stem has a coefficient of friction in the range reported for other materials on dry surfaces: metal on wood has µ between 0.15 and 0.60, while wood on leather has µ between 0.25 and 0.50 (Beer and Johnston, 1977
). In contrast, the firmly placed upright vine stem, with µ in the range 0.61–3.12, has an unusually high coefficient of friction. This shows that in nature a twining vine segment will have large frictional interactions with supporting structures.
Relevance of the problem of a rope wrapped around a pole and generalization to helical forms
Analysis of the mechanical interaction between the vine and its supporting pole indicates that the vine puts itself into tension as it "grabs" the pole (Silk and Hubbard, 1991
). That is, the tendency of the stem to tighten its coil (make a helix of smaller radius and larger torsion than the helix on the supporting pole) produces a normal load (a force per unit length), acting toward the center of the pole, balanced by an axial tension in the vine (Fig. 2). Compressive loads tend to open the helix so that frictional interactions with the pole are eliminated (see Fig. 1 for another example). In contrast, added axial tension causes the vine to push harder on the pole, so that frictional interactions are facilitated, Thus twining stems should be stable in tension but weak in compression. Putz and Holbrook (1991)
showed that the problem of a rope wrapped around a pole was relevant to tendril twining. Here we extend their approach to assess the stability of the twining stem. We ask, what is the axial force that must be added to the vine to cause it to slip on its supporting pole?
To analyze the situation of a helically wrapped vine experiencing an additional axial load, a frictional term can be added to the basic equations of the force balance of a thin, elastic rod (Love, 1944
). We assume a frictional load acts along the tangent. As is conventionally assumed, the frictional load is proportional to the normal load with the proportionality constant set by the coefficient of friction. The complete balance of forces is given by
; Silk and Hubbard, 1991
).
As is conventional in the study of wire ropes (Costello, 1978
), we will ignore the resultant normal and binormal forces. The frictional load acts along the tangent and thus helps prevent the vine from slipping.
 | (4) |
 | (5) |
 | (6) |
. Equation 6 is in great contrast to the conventional force balances for either beams or helical springs. In cantilevers, for instance, curvature and torsion are assumed negligible; the normal load is integrated along the beam to give the shear force Fn; the shear force is integrated along the beam to give the bending moment Mb; and the axial force Ft sums to zero through the cross-section because the upper surface of the beam is in tension, while the lower surface is in compression (e.g., Beer and Johnston, 1977
). For helical springs Ft and Mb are important, and pn is usually neglected (Lin and Pisano, 1988
). In contrast, eq. 6 states that the twining stem has a normal distributed load, acting along the line of contact between the stem and the pole, balanced by an internal, axial tensile force.
 | (7) |
 | (9) |
Note that when the vine is wrapped in a circle, the curvature is 1/r, where r is the radius of the pole; and the arc length 
s can be written r
. Equation 9 then is seen as a generalization of Euler's familiar formula for the force Ft that can be held by a smaller initial force Fi at one end of a rope wrapped around a pole
 | (10) |
is the total angle of wrap of the rope around the pole (Hibbeler, 1983
; Putz and Holbrook, 1991
). The contrast between eq. 9 and eq. 10 illuminates the difference between the botanical system and a rope. In a rope wrapped helically around a pole, A of eq. 9 would be the same as Fi of eq. 10. Fi is a force applied to one end of the helically wrapped rope, and Ft would be the force that would need to be added to the other end of the rope to make it slip. In a twining stem, initial contact with a vertical pole induces the growth gradients that cause the plant to cling to the pole. By developing growth gradients within the stem (and with form change impeded by the pole), the vine generates its own twining force., thus A is the twining force, an axial tension measurable with the calibrated TWIFOR in contact with a gyre near the top of the vine and without added masses (see Silk and Matista, 1997). In contrast, the rope can only remain in place with an externally applied force Fi. In both helical systems, the force at the apical end can support a larger mass (applied load or maturing vine stem with leaves) below the coiled region.
Equation 9 can be used as a model to find the importance of various factors in maintaining the stability of the twining habit. Let us use eq. 9 and consider the case of a vine wrapped around a wooden pole of 1.91 cm diameter (Fig. 2). Silk and Hubbard (1991)
have tabulated values for , , and gyre arc lengths in I. purpurea. Typical values of µ = 1.5 (this paper), and = 0.25 cm–1 can be used with the arc length of the vine around the pole s = 12 cm. If we assume a twining force produced by the vine in the typical range, equivalent to 30 g of tension (Scher et al., 2001
), then eq. 9 predicts that more than 2700 g would need to be applied before the vine would begin to slip down the pole (Table 2). Note that the value of Ft from eq. 9 is very sensitive to both the coefficient of friction and the length of the vine in contact with the pole. If hairy stems with µ = 3 are used, or if the vine has two full gyres wrapped around the pole (s = 24 cm), then the mass needed to make the vine slip would be greater than 243 kg! At the other extreme, a vine lightly resting on a smooth pole (corresponding to our gentle placement, µ = 0.41), and before the development of a large twining force (A = 10 g, s = 12 cm) might slip downward if only 34 g were applied to the bottom of the vine.
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Empirical tests of equation for stem slipping
To test eq. 9 we used a smooth aluminum pole to reduce the coefficient of friction. We measured slipping first in nylon string and then in uncoiled lengths of I. purpurea vine so that A = Fi would be only the applied load of 12 g. Slip tests gave µ = 0.91 for the vine and µ = 0.48 for the string sliding on an aluminum plate. Observed values of the mass required to cause slipping were similar to the predicted values for masses in the range between 30–700 g (Table 2; Fig. 5). Larger masses tore the vine before causing any slippage. This supports the hypothesis that in situ the vine twining on a rough support would usually break before it slips in tension. The twining force that develops as the top of the vine hugs the pole is amplified exponentially with the coefficient of friction and the length of the helical contact.
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; Scher et al., 2001
) must involve repeated, active opening and closing of the helix over time.
The twining vine is weak in compression
To demonstrate the weakness of the twining habit in compression, we hung progressively heavier masses, in increments of 2.5 g, from the leaf axil located at the apical end of the uppermost gyre (Fig. 5, right). The helical form of the vine remained stable until a sudden collapse occurred as the helical gyres were compressed. Of the sample of 10 vines, one collapsed when 2.5 g was applied and two remained stable until 15 g had been applied. The average mass producing compressive failure was 8.25 ± 4.25 g. However, when the masses were applied below the uppermost gyre, the helix did not collapse. This is because the upper gyres were put into tension by the mass hanging below them, and the tendency to slip was resisted by the mechanism described above for the vine in tension.
Once the vine grows above a supporting structure, the unsupported part of the stem exerts a force and bending moment and tends to destabilize the older helical structure on the pole. Collapse of an existing helix is not observed, however. Instead, after the shoot apex reaches the end of a supporting pole the vine uncoils: the wavelengths of the upper gyres increase; the vine no longer touches the pole; and a reversion to an erect searching habit is observed.
During twining the apical stem segment remains uncoiled and is displaced during growth in an undulating, circular path. Unlike the searching habit that occurs in the absence of the support pole, the radius of the path is small, so the length of the unsupported portion of the stem remains short. In our I. purpurea stems twining on 6.2 mm poles, the unsupported stem segment is 2–4 cm in length and weighs 15–30 mg, well below the 8000 mg load required to collapse the helix; thus the twining habit appears to be engineered to avoid compressive loads.
DISCUSSION
Helical growth has been the subject of recent interest, as molecular and cellular determinants have been discovered for handedness of twisting growth forms in roots of Arabidopsis (Thitamadee et al., 2002
; Yuen et al., 2003
). In classical literature, the growth of twining stems has fascinated scientists since the time of Darwin (1876)
. Twining plants grow in a searching mode characterized by sweeping circular or elliptical movement of the apex. After the stem encounters a suitable support the growth pattern changes (see Baillaud, 1962
, for a review of the historical literature). A growth tensor in a natural coordinate system can be used to find the intrastem pattern of expansion and twisting that produces the well-known corkscrewlike shape of the twining vine (Silk, 1989
). The shift from selfsupporting to a supported growth habit in woody vines has been shown to involve an increase in stem flexibility resulting from both a smaller stem diameter and a more flexible wood type (Gartner, 1991; Gallenmueller et al., 2004
). Furthermore, an anatomical basis for the variation in material mechanical properties has been determined (Rowe and Speck, 1996
; Gallenmueller et al., 2004
). Recently the twining habit has been characterized in terms of wood chemistry and microfibril angle (Hoffmann et al., 2003
), thus there is historical and recent interest in the mechanical properties of vine stems. Nevertheless, the literature contains little analysis of the mechanical function of the twining habit.
In nature, some species of twining stems, including I. purpurea, appear to have mostly continuous contact between stem and support. Other species, such as Phaseolus spp., touch the support infrequently and appear to be more selfsupporting. A complete analysis of the mechanics of the twining habit would involve modeling the range of frictional interactions observed in nature, from species that twine smoothly with continuous contact to species with intermittent contact and a more self-supporting habit. Theoretical treatments of the dynamics of helical strips have identified instabilities that might be relevant to the problem of optimum mechanical design for those vines with intermittent contact (Goriely and Shipman, 2000
; Goriely et al., 2001
).
In this study, we put the vine into tension and thereby ensured continuous contact. The results of this manipulation demonstrate that in nature, gravitational loads (unless they are applied very near the vine apex) will stabilize the twining habit if the vine is wrapped around the pole. It is also instructive to think of the frictional interaction as a mechanism by which a small apical stem portion can hold up a large mass of older stem. This explains the observation that a loosely coiled, heavy, older vine segment can simultaneously stabilize and be held up by one or two tightly wound younger coils (Putz and Holbrook, 1991
).
The twining vine, unstable in compression and stable in tension, is similar to structures made of rope or cloth, such as clotheslines, tents, and "finger traps." Indeed, many plant structures are stabilized by tensile forces (Vogel, 1988
; Niklas, 1992
; Moulia, 2001). For the twining habit, an underlying mystery is how a vertical pole of a particular radius induces the tendency to produce stem coils of a particular "potential" radius (observable when the stem is removed from its support). Because the potential coil radius is smaller than the pole radius, contact with the supporting pole is assured. As shown in this paper, the frictional interactions arising from the contact produce stability under tensional forces and allow a few coils near the top of the vine to support a large mass of maturing stem against the pull of gravity.
FOOTNOTES
This work was supported by a grant from the National Science Foundation (IBN 9604230) to W.K.S. The visit of W.K.S. to the laboratory of N.M.H. was supported by a Bullard Fellowship from the Harvard Forest. We thank Christine Jessup, who conducted experiments, not described here, to show that twining forces are maintained after turgor pressure declines in severed stems. 
4 Author for correspondence (e-mail: wksilk{at}ucdavis.edu
) fax: 530-752-1552
LITERATURE CITED
Baillaud L. 1962 Les mouvements d'exploration et d'enroulement des plantes volubiles. Handbuch der Pflanzenphysiologie 12: 635-708
Beer F. P. E. R. Johnston Jr 1977 Vector mechanics for engineers: statics and dynamics, 3rd ed. McGraw-Hill, Boston, Massachusetts, USA
Costello G. A. 1978 Analytical investigation of wire rope. Applied Mechanics Reviews 31: 897-899
Darwin C. 1876 The movements and habits of climbing plants. Appleton, New York, New York, USA
Docarmo M. P. 1979 Differential geometry of curves and surfaces. Prentice Hall, Englewood Cliffs, New Jersey, USA
Gallenmuller F. N. Rowe T. Speck 2004 Development and growth form of the neotropical liana Croton nuntians: the effect of light and mode of attachment on the biomechanics of the stem. Journal of Plant Growth Regulation 23: 83-97[CrossRef]
Goriely A. M. Nizette M. Tabor 2001 On the dynamics of elastic strips. Journal of Nonlinear Science 11: 345
Goriely A. P. Shipman 2000 Dynamics of helical strips. Physical Review E 61: 4508-4517[CrossRef][ISI]
Hibbeler R. C. 1983 Engineering mechanics: statics. Macmillan, New York, New York, USA
Hoffmann B. B. Chabbert B. Monties T. Speck 2003 Mechanical, chemical and X-ray analysis of wood in the two tropical lianas Bauhinia guianensis and Condylocarpon guianense: variations during ontogeny. Planta 217: 32-40[ISI][Medline]
Lin Y. A. P. Pisano 1988 The differential geometry of the general helix as applied to mechanical springs. Journal of Applied Mechanics 55: 831-836[ISI]
Love A. E. H. 1944 A treatise on the mathematical theory of elasticity. Dover, New York, New York, USA
Matista A. A. W. K. Silk 1997 An electronic device for continuous, in vivo measurement of forces exerted by twining vines. American Journal of Botany 84: 1164-1168[Abstract]
Moulia B. 2000 Leaves as shell structures: double curvature, auto-stresses, and minimal mechanical energy constraints on leaf rolling in grasses. Journal of Plant Growth Regulation 19: 19-30[Medline]
Niklas K. J. 1992 Plant biomechanics: an engineering approach to plant form and function. The University of Chicago Press, Chicago, Illinois, USA
Putz F. E. N. M. Holbrook 1991 Biomechanical studies of vines. In F. E. Putz and H. A. Mooney [eds.], The biology of climbing plants, 53–78. Cambridge University Press, Cambridge, UK
Rowe N. P. T. Speck 1996 Biomechanical characteristics of the ontogeny and growth habit of the tropical liana Condylocarpon guianense (Apocynaceae). International Journal of Plant Sciences 157: 406-417[CrossRef][ISI]
Scher J. L. N. M. Holbrook W. K. Silk 2001 Temporal and spatial patterns of twining force and lignification in stems of Ipomoea purpurea. Planta 213: 192-198[CrossRef][ISI][Medline]
Silk W. K. 1989 Growth-rate patterns which maintain a helical tissue tube. Journal of Theoretical Biology 138: 311-327[CrossRef]
Silk W. K. S. Abou haidar 1986 Growth of the stem of Pharbitis nil—analysis of longitudinal and radial components. Physiologie Vegetale 24: 109-116
Silk W. K. R. O. Erickson 1978 Kinematics of hypocotyl curvature. American Journal of Botany 65: 310-319[CrossRef][ISI]
Silk W. K. N. M. Holbrook C. Jessup 2000 The forceful nature of twining vines. In H. C. Spatz and T. Speck [eds.], Plant biomechanics 2000. Proceedings of the Third Plant Biomechanics Conference Freiburg-Badenweiler, 638–646. Georg Thieme Verlag, Stuttgart, Germany
Silk W. K. M. Hubbard 1991 Axial forces and normal distributed loads in twining stems of morning glory. Journal of Biomechanics 24: 599-606[CrossRef][ISI][Medline]
Thitamadee S. K. Tuchihara T. Hashimoto 2002 Microtubule basis for left-handed helical growth in Arabidopsis. Nature 417: 193-196[CrossRef][Medline]
Vogel S. 1988 Life's devices. Princeton University Press, Princeton, New Jersey, USA
Whippo C. W. R. P. Hangarter 2003 Second positive phototropism results from coordinated co-action of the phototropins and cryptochromes. Plant Physiology 132: 1499-1507
Yuen C. Y. L. R. S. Pearlman L. Silo-suh P. Hilson K. L. Carroll P. H. Masson 2003 WVD2 and WDL1 modulate helical organ growth and anisotropic cell expansion in Arabidopsis. Plant Physiology 131: 493-506
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(angle, measured from the normal) are defined in terms of the Frenet vectors. The unit vectors e can be drawn tangent to the natural coordinate curves; and surfaces can be visualized corresponding to constant values of the natural coordinates (diagram C, lower right). The forces generated by the twining habit are dominated by axial tensions (tensile forces in the t direction) balanced by normal loads (forces per unit length in the n direction)
