Helices and Spirals Helices in Plants

MAE 545: Lecture 10,11 (3/28, 3/30) Helices and spirals Helices in plants

How are helices formed?

2 Differential growth or differential shrinking produces spontaneous curvature L(1 + ✏) faster growth of the top layer L more shrinking of the bottom layer W

R R Differential growth (shrinking) of the two layers produces spontaneous curvature 1 ✏ L(1 + ✏) R + W K = = = R W L R

Filaments that are longer than L> 2 ⇡ R , Structural Transition from Helices to Hemihelices form helices to avoid steric interactions. straight again and released many times and each time the same shape, complete with the same number of perversions, reforms. Experiments were also performed under water to minimize gravitational effects and dampen vibrations. Video recordings, reproduced in File S1, capture the evolution of the 3D shapes, 3 several transient features including how perversions move along the bi-strip to form a regular arrangement as well as how an initial twisting motion is reversed. The experimental observations indicate that the number of perversions n is the critical geometric parameter that distinguishes which shape forms upon release. Assuming that the perversions are uniformly distributed along the length of the bistrip, the number that form can be expected to depend on the prestrain ratio, the cross-sectional aspect ratio and the length of the bi-strip. Figure 1. Illustration of a helix (top), a hemihelix with one Dimensional arguments then suggest that the number is given by: perversion marked by an arrow (middle) and a hemihelix with multiple perversions (bottom). The scale bar is 5 cm, and is the wn=L~g(x,h=w). To establish how the number of perversions same for each image. These different shapes were all produced in the depends on these variables, a series of experiments were same way as shown in figure 2 with the same value of pre-strain x~1:5 performed with different values of pre-strain and cross-sectional but with decreasing values of the height-to-width ratio of the bi-strip’s aspect ratio. The results of these experiments are shown in the cross-section. L~50cm, w~3mm, h~12,8,2:5mm). structural phase diagram in Fig. 3 where the numbers associated doi:10.1371/journal.pone.0093183.g001 with the symbols indicate the number of perversions observed. The boundary between the formation of helices and hemihelices is rectangular cross-section. The two strips are then glued together shown shaded. The data in Fig. 3 indicates that increasing the h=w side-by-side along their length. At this stage, the bi-strips are flat ratio drives the strip from the hemihelical configurations to helices. and the red strip is under a uniaxial pre-strain, defined as On the other hand, the prestrain ratio x has only a weak influence x~(L{L’)=L’. Being elastomers, volume conservation requires on both the helix-to-hemihelix transition and the number of that the heights are related by h’~hp1zx. Then, in the final perversions. This phase diagram (Fig. 3) was established under operation, the force stretching the ends of the bi-strip is gradually ffiffiffiffiffiffiffiffiffiffi experimental conditions that allowed both ends to freely rotate as released, with the ends free to rotate. More details of the the stretching force was reduced. A similar phase diagram (Fig. S5 manufacturing and experimental procedures are given in Materials in File S1) but notable by the absence of any helices was obtained and Methods. upon unloading when the ends were constrained from rotating (see Upon release, the initially flat bistrips start to bend and twist out File S1 for details). of plane and evolve towards either a helical or hemihelical shape, depending on the cross-sectional aspect ratio. As indicated by the Finite element simulations images in Fig. 1, when the aspect ratio h=w is small, we observe the formation of periodic perversions, separating helical segments of Numerical simulations to explore the morphological changes alternating chiralities, whereas when the bi-strips have a large occurring during the release in the bi-strip system were conducted aspect ratio, they spontaneously twist along their length to form a using detailed dynamic finite element simulations. In our analysis, regular helix. Significantly, these three-dimensional shapes form spontaneously and do so irrespective of whether the release is abrupt or the ends are slowly brought together. Furthermore, it is also observed that after release, the bi-strip can be stretched

Figure 3. The number of perversions observed as a function of both the prestrain and the cross-section aspect ratio, h=w. The Figure 2. Sequence of operations leading to the spontaneous data indicates that there is a transition between the formation of helixes creation of hemihelices and helices. Starting with two long at larger aspect ratios and hemihelices at smaller aspect ratios. The elastomer strips of different lengths, the shorter one is stretched to precise phase boundary cannot be determined with any precision be the same length as the other. While the stretching force, P, is experimentally and so is shown shaded but there is evidently only a maintained, the two strips are joined side-by-side. Then, as the force is weak dependence on the value of the pre-strain. In some cases, bistrips slowly released, the bi-strip twists and bends to create either a helix or a made the same way produce either one or the other of the two hemihelix. perversion numbers indicated. doi:10.1371/journal.pone.0093183.g002 doi:10.1371/journal.pone.0093183.g003

PLOS ONE | www.plosone.org 2 April 2014 | Volume 9 | Issue 4 | e93183 z y Helix x Mathematical description ~n2 ~n p 1 ~r(s)= r cos(s/),r sin(s/), s 0 0 2⇡ ✓ ◆ ~t Set to ﬁx the metric p in natural parametrization:

pitch d~r r r p ~t(s)= = 0 sin(s/), 0 cos(s/), ds 2⇡ ⇣ r2 p2 ⌘ g = ~t ~t = 0 + =1 · 2 4⇡22

2 2 = r0 +(p/2⇡) 2r0 q diameter

4 z Helix y Mathematical description x p ~r(s)= r cos(s/),r sin(s/), s 0 0 2⇡ ~n2 ✓ ◆ = r2 +(p/2⇡)2 ~n1 0 q Tangent and normal vectors ~t d~r r r p ~t(s)= = 0 sin(s/), 0 cos(s/), ds 2⇡ ⇣ ⌘ ~n (s)= cos(s/), sin(s/), 0 p 1 pitch p p r ~n (s)= sin(s/), cos(s/), 0 2 2⇡ 2⇡ ✓ ◆ Helix curvatures

2 d ~r r0 r0 ~n1 2 = 2 = 2 2 = K · ds r0 +(p/2⇡) 2r0 d2~r ~n =0 diameter 2 · ds2

5 Cucumber tendril climbing via helical coiling

Cucumber tendrils want to pull themselves up above other plants in order to get more sunlight.

Already studied by Charles Darwin in 1865:

S. J. Gerbode et al., Science 337, 1087 (2012)

6 REPORTS

theoretical treatments have incorporated in- on the inside of the helix showing increased lig- tendril coiling is provided by the asymmetric con- trinsic curvature or differential growth without nification relative to the dorsal outer layer (Fig. 1, traction of the stiff fiber ribbon, whose resulting addressing its origin or mechanical consequences GandH),whichisconsistentwithearlierobserva- curvature is imposed on the surrounding soft (6, 15, 16). Recent studies of tendril anatomy tions of increased lignification on the stimulated tendril tissue. The perversions in a doubly sup- (17, 18) have provided a new twist by revealing side of the tendril (17, 18). When a fiber ribbon is ported tendril follow naturally from the topo- an interior layer of specialized cells similar to extracted from the coiled tendril by using fungal logical constraint imposed by the prevention of the stiff, lignified gelatinous fiber (g-fiber) cells carbohydrolases [Driselase (Sigma-Aldrich, St. twist at its ends. found in reaction wood (19). These cells provide Louis, MO)] to break down the nonlignified epi- To better understand the origin of curvature in structural support in reaction wood via tissue dermal tendril tissue (20), it retains the helical fiber ribbons, we reconstituted the underlying morphosis driven by cell-wall lignification, water morphology of a coiled tendril, and furthermore, mechanism using a physical model composed of flux, and oriented stiff cellulose microfibrils. The lengthwise cuts do not change its shape (Fig. 1I two bonded, differentially prestrained silicone rub- presence of a similar ribbon-like strip of g-fiber and fig. S2). ber sheets, similar to rubber models for shaping cells in tendrils suggests that the coiling of the These observations suggest that tendril coil- sheets (21–23). The first silicone sheet was uni- soft tendril tissue may be driven by the shaping of ing occurs via asymmetric contraction of the fiber axially stretched, and an equally thick layer of this stiff, internal “fiber ribbon” (18). ribbon; the ventral side shrinks longitudinally rel- silicone sealant was spread onto the stretched We investigated the role of the fiber ribbon ative to the dorsal side, giving the fiber ribbon its sheet. After the sealant was fully cured, thin strips during tendril coiling in both Cucumis sativus intrinsic curvature. The asymmetric contraction were cut along the prestrained direction, yielding (cucumber) and Echinocystis lobata (wild cu- may be generated by a variety of dorsiventral bilayer ribbons (Fig. 2A) with intrinsic curvature cumber) (20). The g-fiber cells, identified in wild asymmetries, including the observed differential set by the relative prestrain, thickness, and stiff- cucumber by using xylan antibodies in (18), are lignification (Fig. 1H), variations in cellulose mi- ness of the two layers (fig. S3) (20). Like fiber easily distinguished as a band of morphologically crofibril orientation as in reaction wood, or dif- ribbons, the initially straight physical models spon- differentiated cells consistently positioned along ferential water affinities. For example, because taneously form coiled configurations with two the inner side of the helical tendril that lignify lignin is hydrophobic the ventral cells may expel opposite-handed helices connected by a helical during coiling (17, 18). In straight tendrils that more water during lignification, driving increased perversion (Fig. 2A, left). have not yet attached to a support (Fig. 1A), a faint cell contraction. This would be consistent with However, there is an unexpected difference in band of immature g-fiber cells is barely visible by observations of extracted fiber ribbons that pas- mechanical behavior between the physical mod-

using darkfield microscopy (Fig. 1B), with no sively shrink and coil even further when dried but els and tendril fiber ribbons. When clamped at on August 30, 2012 ultraviolet (UV) illuminationsignature,indicating regain their original shape when rehydrated both ends and pulled axially, the physical model the absence of lignification (Fig. 1C). In coiled (movie S2). Dehydrated tendrils also exhibit this simply unwinds to its original uncoiled state (Fig. tendrilsHelical (Fig. 1D), g-fiber cells arecoiling clearly visible behavior of because cucumber they are dominated by the stiff tendril2A and movie S4). In contrast, in fiber ribbons (Fig. 1E) and lignified (Fig. 1F). The fiber ribbon fiber ribbon (movie S3). Together, these facts we observed a counterintuitive “overwinding” consists of two cell layers, with the ventral layer suggest that the biophysical mechanism for behavior in which the ribbon coils even further tendril cross-section

A B C www.sciencemag.org young tendril 0.5mm 0.5mm

D E F

old Downloaded from tendril

I 100µm 100µm

extracted GH ﬁber ribbon 1mm 10µm 10µm

Fig. 1. Tendril coiling via asymmetric contraction. During coiling, a strip of cross section, darkfield (E) and UV autofluorescence (F) show strong lig- specialized structural gelatinous fiber cells (the fiber ribbon) becomes lignified nification in the fiber ribbon.lignified In (G) and (H),g-fiber increased magnification cells reveals and contracts asymmetrically and longitudinally. (A to C)Astraighttendril that ventral cells (top left) are more lignified than dorsal cells. (I)Theextracted that has never coiled (A) lacks lignified g-fiber cells. In the tendril cross fiber ribbon retains the helical morphology of the coiled tendril. (Inset) Higher section, darkfield (B) and UV autofluorescence (C) show no lignin signal. (D to magnification shows the orientation of g-fiber cells along the fiber ribbon. CoilingH)Incoiledtendrils(D),thefullydevelopedfiberribbonconsistsof in older tendrils is due∼ 2layersto a Scalethin bars, (B)layer and (C) 0.5 of mm, (E)stiff, and (F) 100 lignifiedmm, (G) and (H) 10 mm, (I) ofgelatinous highly lignified cells extending fiber along the cells, length of the tendril. which In the tendril are1mm. also found in wood.

1088 31 AUGUST 2012 VOL 337 SCIENCE www.sciencemag.org 7 S. J. Gerbode et al., Science 337, 1087 (2012) REPORTS theoretical treatments have incorporated in- on the inside of the helix showing increased lig- tendril coiling is provided by the asymmetric con- trinsic curvature or differential growth without nification relative to the dorsal outer layer (Fig. 1, traction of the stiff fiber ribbon, whose resulting addressing its origin or mechanical consequences GandH),whichisconsistentwithearlierobserva- curvature is imposed on the surrounding soft (6, 15, 16). Recent studies of tendril anatomy tions of increased lignification on the stimulated tendril tissue. The perversions in a doubly sup- (17, 18) have provided a new twist by revealing side of the tendril (17, 18). When a fiber ribbon is ported tendril follow naturally from the topo- an interior layer of specialized cells similar to extracted from the coiled tendril by using fungal logical constraint imposed by the prevention of the stiff, lignified gelatinous fiber (g-fiber) cells carbohydrolases [Driselase (Sigma-Aldrich, St. twist at its ends. found in reaction wood (19). These cells provide Louis, MO)] to break down the nonlignified epi- To better understand the origin of curvature in structural support in reaction wood via tissue dermal tendril tissue (20), it retains the helical fiber ribbons, we reconstituted the underlying morphosis driven by cell-wall lignification, water morphology of a coiled tendril, and furthermore, mechanism using a physical model composed of flux, and oriented stiff cellulose microfibrils. The lengthwise cuts do not change its shape (Fig. 1I two bonded, differentially prestrained silicone rub- presence of a similar ribbon-like strip of g-fiber and fig. S2). ber sheets, similar to rubber models for shaping cells in tendrils suggests that the coiling of the These observations suggest that tendril coil- sheets (21–23). The first silicone sheet was uni- soft tendril tissue may be driven by the shaping of ing occurs via asymmetric contraction of the fiber axially stretched, and an equally thick layer of this stiff, internal “fiber ribbon” (18). ribbon; the ventral side shrinks longitudinally rel- silicone sealant was spread onto the stretched We investigated the role of the fiber ribbon ative to the dorsal side, giving the fiber ribbon its sheet. After the sealant was fully cured, thin strips during tendril coiling in both Cucumis sativus intrinsic curvature. The asymmetric contraction were cut along the prestrained direction, yielding (cucumber) and Echinocystis lobata (wild cu- may be generated by a variety of dorsiventral bilayer ribbons (Fig. 2A) with intrinsic curvature cumber) (20). The g-fiber cells, identified in wild asymmetries, including the observed differential set by the relative prestrain, thickness, and stiff- cucumber by using xylan antibodies in (18), are lignification (Fig. 1H), variations in cellulose mi- ness of the two layers (fig. S3) (20). Like fiber easily distinguished as a band of morphologically crofibril orientation as in reaction wood, or dif- ribbons, the initially straight physical models spon- differentiated cells consistently positioned along ferential water affinities. For example, because taneously form coiled configurations with two the inner side of the helical tendril that lignify lignin is hydrophobic the ventral cells may expel opposite-handed helices connected by a helical during coiling (17, 18). In straight tendrils that more water during lignification, driving increased perversion (Fig. 2A, left). have not yet attached to a support (Fig. 1A), a faint cell contraction. This would be consistent with However, there is an unexpected difference in band of immature g-fiber cells is barely visible by observations of extracted fiber ribbons that pas- mechanical behavior between the physical mod- using darkfield microscopy (Fig.Helical 1B), with no coilingsively shrink and coilof even cucumber further when dried but els tendril and tendril fiber ribbons. When clamped at on August 30, 2012 ultraviolet (UV) illuminationsignature,indicating regain their original shape when rehydrated both ends and pulled axially, the physical model the absence of lignification (Fig. 1C). In coiled (movie S2). Dehydrated tendrils also exhibit this simply unwinds to its original uncoiled state (Fig. tendrilsDrying (Fig. 1D), g-fiber of cellsﬁbber are clearly ribbon visible behavior becauseDrying they are of dominated tendril by the stiff 2ARehydrating and movie S4). In contrast, of tendril in fiber ribbons (Fig. 1E) andincreases lignified (Fig. 1F). coiling The fiber ribbon fiber ribbonincreases (movie S3). Together, coiling these facts we observedreduces a counterintuitive coiling“overwinding” consists of two cell layers, with the ventral layer suggest that the biophysical mechanism for behavior in which the ribbon coils even further

A B C www.sciencemag.org

D E F Downloaded from

I inside outside During the ligniﬁcation of g-ﬁber cells layer layer water is expelled, which causes shrinking. GH

The inside layer is more ligniﬁed and therefore shrinks more and is also stiffer than the outside layer. Fig. 1. Tendril coiling via asymmetric contraction. During coiling, a strip of cross section, darkfield (E) and UV autofluorescence (F) show strong lig- specialized structural gelatinous fiber cells (the fiber ribbon) becomes lignified 8nificationS. J. in Gerbode the fiber ribbon. et al., In (G) Science and (H), increased 337, 1087 magnification (2012) reveals and contracts asymmetrically and longitudinally. (A to C)Astraighttendril that ventral cells (top left) are more lignified than dorsal cells. (I)Theextracted that has never coiled (A) lacks lignified g-fiber cells. In the tendril cross fiber ribbon retains the helical morphology of the coiled tendril. (Inset) Higher section, darkfield (B) and UV autofluorescence (C) show no lignin signal. (D to magnification shows the orientation of g-fiber cells along the fiber ribbon. H)Incoiledtendrils(D),thefullydevelopedfiberribbonconsistsof∼2layers Scale bars, (B) and (C) 0.5 mm, (E) and (F) 100 mm, (G) and (H) 10 mm, (I) of highly lignified cells extending along the length of the tendril. In the tendril 1mm.

1088 31 AUGUST 2012 VOL 337 SCIENCE www.sciencemag.org REPORTS theoretical treatments have incorporated in- on the inside of the helix showing increased lig- tendril coiling is provided by the asymmetric con- trinsic curvature or differential growth without nification relative to the dorsal outer layer (Fig. 1, traction of the stiff fiber ribbon, whose resulting addressing its origin or mechanical consequences GandH),whichisconsistentwithearlierobserva- curvature is imposed on the surrounding soft (6, 15, 16). Recent studies of tendril anatomy tions of increased lignification on the stimulated tendril tissue. The perversions in a doubly sup- (17, 18) have provided a new twist by revealing side of the tendril (17, 18). When a fiber ribbon is ported tendril follow naturally from the topo- an interior layer of specialized cells similar to extracted from the coiled tendril by using fungal logical constraint imposed by the prevention of the stiff, lignified gelatinous fiber (g-fiber) cells carbohydrolases [Driselase (Sigma-Aldrich, St. twist at its ends. found in reaction wood (19). These cells provide Louis, MO)] to break down the nonlignified epi- To better understand the origin of curvature in structural support in reaction wood via tissue dermal tendril tissue (20), it retains the helical fiber ribbons, we reconstituted the underlying morphosis driven by cell-wall lignification, water morphology of a coiled tendril, and furthermore, mechanism using a physical model composed of flux, and oriented stiff cellulose microfibrils. The lengthwise cuts do not change its shape (Fig. 1I two bonded, differentially prestrained silicone rub- presence of a similar ribbon-like strip of g-fiber and fig. S2). ber sheets, similar to rubber models for shaping cells in tendrils suggests that the coiling of the These observations suggest that tendril coil- sheets (21–23). The first silicone sheet was uni- soft tendril tissue may be driven by the shaping of ing occurs via asymmetric contraction of the fiber axially stretched, and an equally thick layer of this stiff, internal “fiber ribbon” (18). ribbon; the ventral side shrinks longitudinally rel- silicone sealant was spread onto the stretched We investigated the role of the fiber ribbon ative to the dorsal side, giving the fiber ribbon its sheet. After the sealant was fully cured, thin strips during tendril coiling in both Cucumis sativus intrinsic curvature. The asymmetric contraction were cut along the prestrained direction, yielding (cucumber) and Echinocystis lobata (wild cu- may be generated by a variety of dorsiventral bilayer ribbons (Fig. 2A) with intrinsic curvature cumber) (20). The g-fiber cells, identified in wild asymmetries, including the observed differential set by the relative prestrain, thickness, and stiff- cucumber by using xylan antibodies in (18), are lignification (Fig. 1H), variations in cellulose mi- ness of the two layers (fig. S3) (20). Like fiber easily distinguished as a band of morphologically crofibril orientation as in reaction wood, or dif- ribbons, the initially straight physical models spon- differentiated cells consistently positioned along ferential water affinities. For example, because taneously form coiled configurations with two the inner side of the helical tendril that lignify lignin is hydrophobic the ventral cells may expel opposite-handed helices connected by a helical during coiling (17, 18). In straight tendrils that more water during lignification, driving increased perversion (Fig. 2A, left). have not yet attached to a support (Fig. 1A), a faint cell contraction. This would be consistent with However, there is an unexpected difference in band of immature g-fiber cells is barely visible by observations of extracted fiber ribbons that pas- mechanical behavior between the physical mod- using darkfield microscopy (Fig. 1B), with no sively shrink and coil even further when dried but els and tendril fiber ribbons. When clamped at on August 30, 2012 ultraviolet (UV) illuminationsignature,indicating regain their original shape when rehydrated both ends and pulled axially, the physical model the absence of lignification (Fig. 1C). In coiled (movie S2). Dehydrated tendrils also exhibit this simply unwinds to its original uncoiled state (Fig. tendrils (Fig. 1D), g-fiberCoilingcells are clearlyof tendrils visible inbehavior opposite because directions they are dominated by the stiff 2A and movie S4). In contrast, in fiber ribbons (Fig. 1E) and lignified (Fig. 1F). The fiber ribbon fiber ribbonEnds of (movie the tendril S3). are Together, ﬁxed and these facts we observed a counterintuitive “overwinding” consists of two cell layers, with the ventral layer suggestcannot that the rotate. biophysical This constraints mechanism for behavior in which the ribbon coils even further perversion the linking number.

Link = Twist + Writhe A B C www.sciencemag.org

left-handed right-handed helix helix

perversion D E F Downloaded from

9 I

Fig. 1. Tendril coiling via asymmetric contraction. During coiling, a strip of cross section, darkfield (E) and UV autofluorescence (F) show strong lig- specialized structural gelatinous fiber cells (the fiber ribbon) becomes lignified nification in the fiber ribbon. In (G) and (H), increased magnification reveals and contracts asymmetrically and longitudinally. (A to C)Astraighttendril that ventral cells (top left) are more lignified than dorsal cells. (I)Theextracted that has never coiled (A) lacks lignified g-fiber cells. In the tendril cross fiber ribbon retains the helical morphology of the coiled tendril. (Inset) Higher section, darkfield (B) and UV autofluorescence (C) show no lignin signal. (D to magnification shows the orientation of g-fiber cells along the fiber ribbon. H)Incoiledtendrils(D),thefullydevelopedfiberribbonconsistsof∼2layers Scale bars, (B) and (C) 0.5 mm, (E) and (F) 100 mm, (G) and (H) 10 mm, (I) of highly lignified cells extending along the length of the tendril. In the tendril 1mm.

1088 31 AUGUST 2012 VOL 337 SCIENCE www.sciencemag.org Twist, Writhe and Linking numbers

Ln=Tw+Wr linking number: total number of turns of a particular end Tw twist: number of turns due to twisting the beam Wr writhe: number of crossings when curve is projected on a plane

10 REPORTS theoretical treatments have incorporated in- on the inside of the helix showing increased lig- tendril coiling is provided by the asymmetric con- trinsic curvature or differential growth without nification relative to the dorsal outer layer (Fig. 1, traction of the stiff fiber ribbon, whose resulting addressing its origin or mechanical consequences GandH),whichisconsistentwithearlierobserva- curvature is imposed on the surrounding soft (6, 15, 16). Recent studies of tendril anatomy tions of increased lignification on the stimulated tendril tissue. The perversions in a doubly sup- (17, 18) have provided a new twist by revealing side of the tendril (17, 18). When a fiber ribbon is ported tendril follow naturally from the topo- an interior layer of specialized cells similar to extracted from the coiled tendril by using fungal logical constraint imposed by the prevention of the stiff, lignified gelatinous fiber (g-fiber) cells carbohydrolases [Driselase (Sigma-Aldrich, St. twist at its ends. found in reaction wood (19). These cells provide Louis, MO)] to break down the nonlignified epi- To better understand the origin of curvature in structural support in reaction wood via tissue dermal tendril tissue (20), it retains the helical fiber ribbons, we reconstituted the underlying morphosis driven by cell-wall lignification, water morphology of a coiled tendril, and furthermore, mechanism using a physical model composed of flux, and oriented stiff cellulose microfibrils. The lengthwise cuts do not change its shape (Fig. 1I two bonded, differentially prestrained silicone rub- presence of a similar ribbon-like strip of g-fiber and fig. S2). ber sheets, similar to rubber models for shaping cells in tendrils suggests that the coiling of the These observations suggest that tendril coil- sheets (21–23). The first silicone sheet was uni- soft tendril tissue may be driven by the shaping of ing occurs via asymmetric contraction of the fiber axially stretched, and an equally thick layer of this stiff, internal “fiber ribbon” (18). ribbon; the ventral side shrinks longitudinally rel- silicone sealant was spread onto the stretched We investigated the role of the fiber ribbon ative to the dorsal side, giving the fiber ribbon its sheet. After the sealant was fully cured, thin strips during tendril coiling in both Cucumis sativus intrinsic curvature. The asymmetric contraction were cut along the prestrained direction, yielding (cucumber) and Echinocystis lobata (wild cu- may be generated by a variety of dorsiventral bilayer ribbons (Fig. 2A) with intrinsic curvature cumber) (20). The g-fiber cells, identified in wild asymmetries, including the observed differential set by the relative prestrain, thickness, and stiff- cucumber by using xylan antibodies in (18), are lignification (Fig. 1H), variations in cellulose mi- ness of the two layers (fig. S3) (20). Like fiber easily distinguished as a band of morphologically crofibril orientation as in reaction wood, or dif- ribbons, the initially straight physical models spon- differentiated cells consistently positioned along ferential water affinities. For example, because taneously form coiled configurations with two the inner side of the helical tendril that lignify lignin is hydrophobic the ventral cells may expel opposite-handed helices connected by a helical during coiling (17, 18). In straight tendrils that more water during lignification, driving increased perversion (Fig. 2A, left). have not yet attached to a support (Fig. 1A), a faint cell contraction. This would be consistent with However, there is an unexpected difference in band of immature g-fiber cells is barely visible by observations of extracted fiber ribbons that pas- mechanical behavior between the physical mod- using darkfield microscopy (Fig. 1B), with no sively shrink and coil even further when dried but els and tendril fiber ribbons. When clamped at on August 30, 2012 ultraviolet (UV) illuminationsignature,indicating regain their original shape when rehydrated both ends and pulled axially, the physical model the absence of lignification (Fig. 1C). In coiled (movie S2). Dehydrated tendrils also exhibit this simply unwinds to its original uncoiled state (Fig. tendrils (Fig. 1D), g-fiberCoilingcells are clearlyof tendrils visible inbehavior opposite because directions they are dominated by the stiff 2A and movie S4). In contrast, in fiber ribbons (Fig. 1E) and lignified (Fig. 1F). The fiber ribbon fiber ribbonEnds of (movie the tendril S3). are Together, ﬁxed and these facts we observed a counterintuitive “overwinding” consists of two cell layers, with the ventral layer suggestcannot that the rotate. biophysical This constraints mechanism for behavior in which the ribbon coils even further perversion the linking number.

Link = Twist + Writhe A B C Coiling in the same direction www.sciencemag.org increases Writhe, which needs to left-handed right-handed be compensated by the twist. helix helix In order to minimize the twisting energy tendrils combine two helical coils of opposite handedness (=opposite Writhe). D perversion Note: there is no bending energy E F

when the curvature of two helices Downloaded from correspond to the spontaneous curvature due to the differential shrinking of ﬁber. 11 I

Fig. 1. Tendril coiling via asymmetric contraction. During coiling, a strip of cross section, darkfield (E) and UV autofluorescence (F) show strong lig- specialized structural gelatinous fiber cells (the fiber ribbon) becomes lignified nification in the fiber ribbon. In (G) and (H), increased magnification reveals and contracts asymmetrically and longitudinally. (A to C)Astraighttendril that ventral cells (top left) are more lignified than dorsal cells. (I)Theextracted that has never coiled (A) lacks lignified g-fiber cells. In the tendril cross fiber ribbon retains the helical morphology of the coiled tendril. (Inset) Higher section, darkfield (B) and UV autofluorescence (C) show no lignin signal. (D to magnification shows the orientation of g-fiber cells along the fiber ribbon. H)Incoiledtendrils(D),thefullydevelopedfiberribbonconsistsof∼2layers Scale bars, (B) and (C) 0.5 mm, (E) and (F) 100 mm, (G) and (H) 10 mm, (I) of highly lignified cells extending along the length of the tendril. In the tendril 1mm.

1088 31 AUGUST 2012 VOL 337 SCIENCE www.sciencemag.org Overwinding of tendril coils Old tendrils overwind when stretched. Rubber model unwinds when stretched. ihteeprmna data. exten- experimental the higher with at force (Inset) necessary sions. this increases eventually ( data. same plot Log-linear the (Inset) of (blue). with ( 5 (green), filaments 1 (red), numerical value. 1/5 for values slope plotted are initial curves extension fitted the in against erae h oc eddt xal xedteflmn;for filament; the extend B axially to needed force the decreases dte uvsidct evre)adasgetwt no with force segment less indi a curves and (solid was perverted) perversion exhibits indicate tendril curves that conta Each (dotted segment tendril a curves). old into (blue separated one not overwinding and does substantial that curves) tendril (red young one overwind for curves extension Force (number ( perversion white). the in of indicated side are each turns to of turn extra ( one shape. adding flat overwinds, initially original its to returning pulled, F ftehlclpreso.For perversion. helical the of the to due force scaled supplementary in the difference perversion The helical in (A). available in materials) are definitions (detailed i.3. Fig. exterior. the to wire copper compression) (under of inextensible interior an the to and ribbon helix fabric tension) the (under inextensible relatively a adding by stiffness bending lower with spring 2. Fig. S5). (movie the expected uncoiled as flat, tension state a to enough returning unwinds, high ribbon fiber under Even- the though, S5). of movie tually sides and right, both 2A, on (Fig. perversion turns adding pulled, when ∼ ihteeprmna data. exten- experimental the higher with at force (Inset) necessary sions. this increases eventually ( data. same plot Log-linear the (Inset) of (blue). with ( 5 (green), filaments 1 (red), numerical value. 1/5 for values slope plotted are initial curves extension fitted the in against erae h oc eddt xal xedteflmn;for filament; the extend B axially to needed force the decreases dte uvsidct evre)adasgetwt no with force segment less indi a curves and (solid was perverted) perversion exhibits indicate tendril curves that conta Each (dotted segment tendril a curves). old into (blue separated one not overwinding and does substantial that curves) tendril (red young one overwind for curves extension Force (number ( perversion white). the in of indicated side are each turns to of turn extra ( one shape. adding flat overwinds, initially original its to returning pulled, F ftehlclpreso.For perversion. helical the of the to due force scaled supplementary in the difference perversion The helical in (A). available in materials) are definitions (detailed i.3. Fig. exterior. the to wire copper compression) (under of inextensible interior an the to and ribbon helix fabric tension) the (under inextensible relatively a adding by stiffness bending lower with spring 2. Fig. S5). (movie the expected uncoiled as flat, tension state a to enough returning unwinds, high ribbon fiber under Even- the though, S5). of movie tually sides and right, both 2A, on (Fig. perversion turns adding pulled, when ∼ (perverted) (perverted) > > C C ,t h ep e r v e r s i o ni n i t i a l l yd e c r e a s e st h ef o r c en e e d e db u t ,t h ep e r v e r s i o ni n i t i a l l yd e c r e a s e st h ef o r c en e e d e db u t ∆ ehnclcneune foewnig ( overwinding. of consequences Mechanical wsls pig nidn n vridn.( overwinding. and unwinding springs Twistless l ∆ F n() h hddrnei B niae variations indicates (B) in range shaded The (B). in ehnclcneune foewnig ( overwinding. of consequences Mechanical ∼ wsls pig nidn n vridn.( overwinding. and unwinding springs Twistless l spotdaanttesae displacement scaled the against plotted is ∆ − F n() h hddrnei B niae variations indicates (B) in range shaded The (B). in ∼ f F ∼ spotdagainst plotted is capd ihihstemcaia effect mechanical the highlights (clamped) spotdaanttesae displacement scaled the against plotted is ∆ ∆ − f f F = ∼ D spotdagainst plotted is capd ihihstemcaia effect mechanical the highlights (clamped) )T h ed i f f e r e n c ei nf o r c e f (perverted) ∆ f aecapd.Tedimension- The clamped). cate = B D C nn h eia perversion helical the ining )Overwindingisinducedinthesiliconemodel < )T h ed i f f e r e n c ei nf o r c e f

(perverted) 5 C ∆ C B ,theperversionalways − l )Dimensionlessforce- hntitn stiffness twisting than o ietcomparison direct for f aecapd.Tedimension- The clamped). cate capd splotted is (clamped) www.sciencemag.org B B relaxed C )Whenafiberribbonispulled,it nn h eia perversion helical the ining )Overwindingisinducedinthesiliconemodel < C ∆ C B ,theperversionalways A relaxed − l )Dimensionlessforce- ∆ hntitn stiffness twisting than and o ietcomparison direct for facie hsclmdl ommclignified mimic To model. a physical added coiled inside we a the of extensible, to ribbon less fabric inextensible is relatively layer that inner suggest which the ribbons, fiber in lignification f F capd splotted is (clamped) A ∼ www.sciencemag.org B )Asiliconetwistless ∆ nprdb u bevtoso asymmetric of observations our by Inspired B / = C ) l B C )Whenafiberribbonispulled,it5 nid when unwinds SCIENCE A ∆ and facie hsclmdl ommclignified mimic To model. a physical added coiled inside we a the of extensible, to ribbon less fabric inextensible is relatively layer that inner suggest which the ribbons, fiber in lignification F A ∼ B )Asiliconetwistless ∆ nprdb u bevtoso asymmetric of observations our by Inspired B featchlclflmnsrcptlt hsoewnigbhvo,wihis which helix behavior, each in overwinding turns of this an recapitulate physical with filaments consistent helical elastic of ratio the increase these Together, n ltee noarbo-iesaewith shape ribbon-like a into with flattened pronounced and more becomes Overwinding (blue). 5 increasing and (green), 1 (red), 1/5 / = C ) O 3 1AGS 2012 AUGUST 31 337 VOL l C nid when unwinds B / C .( SCIENCE F )O v e r w i n d i n gi sa l s oo b s e r v e di no l dt e n d r i l s ,w h i c hh a v ed r i e d ∆ N spotdvru clddisplacement scaled versus plotted is db i o l o g i c a le x p e r i m e n t s .( 6 increase modifications these Together, prevents contraction. elon- wire copper prevents external the ribbon the whereas gation, fabric of internal in- exterior The an the helix. to added wire we copper compression, compressible resist that cells featchlclflmnsrcptlt hsoewnigbhvo,wihis which helix behavior, each in overwinding turns of this an recapitulate physical with filaments consistent helical elastic of ratio the increase these Together, n ltee noarbo-iesaewith shape ribbon-like a into with flattened pronounced and more becomes Overwinding (blue). 5 increasing and (green), 1 (red), 1/5 B / C O 3 1AGS 2012 AUGUST 31 337 VOL .( stretched D )When B / C >1.Scalebars,1cm. B B / / C C >1,numericalsimulations .( E )Changeinthenumber F stretched )O v e r w i n d i n gi sa l s oo b s e r v e di no l dt e n d r i l s ,w h i c hh a v ed6 r i e d ∆ ∆ l for N B spotdvru clddisplacement scaled versus plotted is REPORTS / C db i o l o g i c a le x p e r i m e n t s .( otato.Tgte,teemdfctosincrease modifications these Together, prevents contraction. elon- wire copper prevents external the ribbon the whereas gation, fabric of in- internal exterior The an the helix. to added wire we copper compression, compressible resist that cells values 1089 B / C .(

D Downloaded from www.sciencemag.org on August 30,12 2012 S. J. Gerbode et al., Science 337, 1087 (2012) )When B / C >1.Scalebars,1cm. B / C >1,numericalsimulations E )Changeinthenumber ∆ l for B REPORTS / C values 1089

Downloaded from www.sciencemag.org on August 30, 2012

on August 30, 2012 30, August on www.sciencemag.org from Downloaded ” m, (I) m ). Like fiber overwinding )Theextracted I “ 20 prestrained silicone rub- m, (G) and (H) 10 m C F ). The first silicone sheet was uni- g-fiber cells along the fiber ribbon. tofluorescence (F) show strong lig- gy of the coiled tendril. (Inset) Higher 23 – 21 However, there is an unexpected difference in To better understand the origin of curvature in ribbons, the initially straight physicaltaneously models spon- form coiledopposite-handed configurations helices with connected two byperversion (Fig. a 2A, helical left). mechanical behavior between the physicalels mod- and tendril fiberboth ends ribbons. and When pulled clamped axially,simply the unwinds at physical to its model original uncoiled2A state and (Fig. movie S4).we In observed contrast, in abehavior fiber counterintuitive in ribbons which the ribbon coils even further tendril coiling is provided by thetraction asymmetric of con- the stiff fibercurvature ribbon, is whose resulting imposedtendril on tissue. The the perversionsported surrounding in tendril a soft doubly followlogical sup- naturally constraint imposed from bytwist the the at topo- its prevention ends. of fiber ribbons, wemechanism using reconstituted a physical the modeltwo composed underlying bonded, of differentially ber sheets, similar tosheets rubber ( models for shaping axially stretched, andsilicone an equally sealant thick wassheet. layer After spread the of sealant was onto fullywere cured, the thin cut strips along stretched the prestrainedbilayer direction, ribbons yielding (Fig. 2A) withset intrinsic by curvature the relativeness prestrain, of thickness, the and two stiff- layers (fig. S3) ( B E GH www.sciencemag.org Overwinding of tendril coils cross section, darkfieldnification (E) in the and fiberthat UV ribbon. ventral In cells au (G) (top left) and arefiber (H), more ribbon lignified increased retains than magnification the dorsal helical reveals cells. morpholo magnification ( shows the orientationScale of bars, (B) and1mm. (C) 0.5 mm, (E) and (F) 100 ), it retains the helical riations in cellulose mi- ). When a fiber ribbon is SCIENCE 20 18 , Preferred curved stateto Flattened state 17 D 2layers ∼ )Astraighttendril These observations suggest that tendril coil- C on the inside of thenification relative helix to showing the increased dorsal outer lig- GandH),whichisconsistentwithearlierobserva- layer (Fig. 1, tions of increased lignificationside of on the the tendril stimulated ( extracted from the coiledcarbohydrolases tendril [Driselase by using (Sigma-Aldrich,Louis, fungal St. MO)] to break downdermal the nonlignified tendril epi- tissuemorphology ( of a coiled tendril,lengthwise and cuts furthermore, do notand change fig. its S2). shape (Fig. 1I ing occurs via asymmetric contraction of theribbon; fiber the ventral side shrinksative longitudinally to rel- the dorsal side, givingintrinsic the curvature. fiber ribbon The its asymmetricmay contraction be generatedasymmetries, by including a the observed varietylignification differential (Fig. of 1H), va dorsiventral crofibril orientation as inferential reaction water wood, affinities. orlignin dif- For is example, hydrophobic the because ventralmore cells water may during expel lignification, driving increased cell contraction. This wouldobservations of be extracted consistent fiber with sively ribbons shrink and that coil even pas- further whenregain dried but their original(movie S2). shape Dehydrated tendrils when alsobehavior exhibit because rehydrated this they are dominatedfiber by the ribbon stiff (moviesuggest S3). that Together, these the facts biophysical mechanism for to ribbon) becomes lignified A ), are 18

(wild cu- In tendrils the red inner layer 31High AUGUST 2012 VOL 337 bending energy cost ). is much stiffer then theorescence (C) show no lignin signal. ( associated with stretching 18 ( Cucumis sativus ” outside blue layer. of the stiff inner layer! ). These cells provide nsignature,indicating cells are clearly visible tinous fiber cells (the fiber 19

). In straight tendrils that Tendrils try to keep the preferred 18 ihteeprmna data. exten- experimental the higher with at force (Inset) necessary sions. this increases eventually ( data. same plot Log-linear the (Inset) of (blue). with ( 5 (green), filaments 1 (red), numerical value. 1/5 for values slope plotted are initial curves extension fitted the in against erae h oc eddt xal xedteflmn;for filament; the extend B axially to needed force the decreases dte uvsidct evre)adasgetwt no with force segment less indi a curves and (solid was perverted) perversion exhibits indicate tendril curves that conta Each (dotted segment tendril a curves). old into (blue separated one not overwinding and does substantial that curves) tendril (red young one overwind for curves extension Force (number ( perversion white). the in of indicated side are each turns to of turn extra ( one shape. adding flat overwinds, initially original its to returning pulled, F ftehlclpreso.For perversion. helical the of the to due force scaled supplementary in the difference perversion The helical in (A). available in materials) are definitions (detailed i.3. Fig. exterior. the to wire copper compression) (under of inextensible interior an the to and ribbon helix fabric tension) the (under inextensible relatively a adding by stiffness bending lower with spring 2. Fig. S5). (movie the expected uncoiled as flat, tension state a to enough returning unwinds, high ribbon fiber under Even- the though, S5). of movie tually sides and right, both 2A, on (Fig. perversion turns adding pulled, when ∼ (perverted) > fiber ribbon , C “ Echinocystis lobata curvature when stretched! ,t h ep e r v e r s i o ni n i t i a l l yd e c r e a s e st h ef o r c en e e d e db u t 17 ∆ ehnclcneune foewnig ( overwinding. of consequences Mechanical wsls pig nidn n vridn.( overwinding. and unwinding springs Twistless l F n() h hddrnei B niae variations indicates (B) in range shaded The (B). in ∼ spotdaanttesae displacement scaled the against plotted is ∆ − ). The g-fiber cells, identified in wild f F ∼ spotdagainst plotted is ). Recent studies of tendril anatomy capd ihihstemcaia effect mechanical the highlights (clamped) 20 ∆ f Tendril coiling via asymmetric contraction. During coiling, a strip of 16 = D ) have provided a new twist by revealing )T h ed i f f e r e n c ei nf o r c e A D I , f (perverted) In rubber models both layers 18 aecapd.Tedimension- The clamped). cate

15 Small bending energy. We investigated the role of the fiber ribbon , B C , nn h eia perversion helical the ining )Overwindingisinducedinthesiliconemodel )Incoiledtendrils(D),thefullydevelopedfiberribbonconsistsof < 17 6 C ∆ C have not yet attached to a supportband (Fig. of 1A), immature a faint g-fiber cellsusing is barely darkfield visible by microscopyultraviolet (UV) (Fig. illuminatio 1B),the with absence no oftendrils lignification (Fig. (Fig. 1D), 1C). g-fiber (Fig. In 1E) and coiled lignified (Fig. 1F).consists The of fiber two ribbon cell layers, with the ventral layer cucumber by using xylaneasily antibodies distinguished in as a ( band ofdifferentiated morphologically cells consistently positionedthe along inner sideduring of coiling the ( helical tendril that lignify cumber) ( during tendril coiling(cucumber) in and both structural support in reactionmorphosis driven wood by cell-wall via lignification, water tissue flux, and oriented stiff cellulose microfibrils.presence The of a similarcells ribbon-like in strip tendrils of g-fiber suggestssoft tendril that tissue may the be driven coiling bythis the stiff, of shaping internal of the ( an interior layer ofthe specialized stiff, lignified cells gelatinous similar fiber (g-fiber) to cells found in reaction wood ( H of highly lignified cells extending along the length of the tendril. In the tendril theoretical treatments havetrinsic curvature incorporated or in- addressing differential its origin growth or mechanical without consequences ( have similar stiffness.Fig. 1. specialized structural gela and contracts asymmetrically andthat longitudinally. has ( neversection, coiled darkfield (B) (A) and UV lacks autoflu lignified g-fiber cells. In the tendril cross B ,theperversionalways − l )Dimensionlessforce- hntitn stiffness twisting than o ietcomparison direct for f capd splotted is (clamped) www.sciencemag.org B )Whenafiberribbonispulled,it REPORTS 1088 A ∆ and facie hsclmdl ommclignified mimic To model. a physical added coiled inside we a the of extensible, to ribbon less fabric inextensible is relatively layer that inner suggest which the ribbons, fiber in lignification F A ∼ B )Asiliconetwistless ∆ nprdb u bevtoso asymmetric of observations our by Inspired B /

= S. J. Gerbode et al., Science 337, 1087 (2012) C 13 ) l C nid when unwinds SCIENCE featchlclflmnsrcptlt hsoewnigbhvo,wihis which helix behavior, each in overwinding turns of this an recapitulate physical with filaments consistent helical elastic of ratio the increase these Together, n ltee noarbo-iesaewith shape ribbon-like a into with flattened pronounced and more becomes Overwinding (blue). 5 increasing and (green), 1 (red), 1/5 O 3 1AGS 2012 AUGUST 31 337 VOL B / C .( F )O v e r w i n d i n gi sa l s oo b s e r v e di no l dt e n d r i l s ,w h i c hh a v ed r i e d ∆ N spotdvru clddisplacement scaled versus plotted is db i o l o g i c a le x p e r i m e n t s .( otato.Tgte,teemdfctosincrease modifications these Together, prevents contraction. elon- wire copper prevents external the ribbon the whereas gation, fabric of in- internal exterior The an the helix. to added wire we copper compression, compressible resist that cells B / C .( D )When B / C >1.Scalebars,1cm. B / C >1,numericalsimulations E )Changeinthenumber ∆ l for B REPORTS / C values 1089

Downloaded from www.sciencemag.org on August 30, 2012 Overwinding of rubber models with an ihteeprmna data. exten- experimental the higher with at force (Inset) necessary sions. this increases eventually ( data. same plot Log-linear the (Inset) of (blue). with ( 5 (green), filaments 1 (red), numerical value. 1/5 for values slope plotted are initial curves extension fitted the in against erae h oc eddt xal xedteflmn;for filament; the extend B axially to needed force the decreases dte uvsidct evre)adasgetwt no with force segment less indi a curves and (solid was perverted) perversion exhibits indicate tendril curves that conta Each (dotted segment tendril a curves). old into (blue separated one not overwinding and does substantial that curves) tendril (red young one overwind for curves extension Force (number ( perversion white). the in of indicated side are each turns to of turn extra ( one shape. adding flat overwinds, initially original its to returning pulled, F ftehlclpreso.For perversion. helical the of the to due force scaled supplementary in the difference perversion The helical in (A). available in materials) are definitions (detailed additional 3. Fig. exterior. the to wire copper compression) (under of inextensible interior an the to and ribbon helix fabric tension) the (under inextensible relatively a adding by stiffness stiff bending lower with spring 2. Fig. fabric on the inside S5). (movie the expected uncoiled as flat, tension state a to enough returning unwinds, high ribbon fiber under Even- the though, S5). of movie tually sides and right, both 2A, on (Fig. perversion turns adding pulled, when layers ∼ (perverted) > C ,t h ep e r v e r s i o ni n i t i a l l yd e c r e a s e st h ef o r c en e e d e db u t ∆ ehnclcneune foewnig ( overwinding. of consequences Mechanical wsls pig nidn n vridn.( overwinding. and unwinding springs Twistless l F n() h hddrnei B niae variations indicates (B) in range shaded The (B). in ∼ spotdaanttesae displacement scaled the against plotted is ∆ − f F ∼ spotdagainst plotted is capd ihihstemcaia effect mechanical the highlights (clamped) ∆ f = D 5 )T h ed i f f e r e n c ei nf o r c e f (perverted) aecapd.Tedimension- The clamped). cate B C nn h eia perversion helical the ining )Overwindingisinducedinthesiliconemodel < C ∆ C B ,theperversionalways − l )Dimensionlessforce- hntitn stiffness twisting than o ietcomparison direct for 4 f capd splotted is (clamped) www.sciencemag.org B relaxed )Whenafiberribbonispulled,it

soft rubberA ∆ and facie hsclmdl ommclignified mimic To model. a physical added coiled inside we a the of extensible, to ribbon less fabric inextensible is relatively layer that inner suggest which the ribbons, fiber in lignification F A ∼ B )Asiliconetwistless ∆ nprdb u bevtoso asymmetric of observations our by Inspired B / = C ) l C nid when unwinds SCIENCE featchlclflmnsrcptlt hsoewnigbhvo,wihis which helix behavior, each in overwinding turns of this an recapitulate physical with filaments consistent helical elastic of ratio the increase these Together, n ltee noarbo-iesaewith shape ribbon-like a into with flattened pronounced and more becomes Overwinding (blue). 5 increasing and (green), 1 (red), 1/5 O 3 1AGS 2012 AUGUST 31 337 VOL 6 B / C .( F )O v e r w i n d i n gi sa l s oo b s e r v e di no l dt e n d r i l s ,w h i c hh a v ed r i e d

stiff fabric ∆ N spotdvru clddisplacement scaled versus plotted is db i o l o g i c a le x p e r i m e n t s .( otato.Tgte,teemdfctosincrease modifications these Together, prevents contraction. elon- wire copper prevents external the ribbon the whereas gation, fabric of in- internal exterior The an the helix. to added wire we copper compression, compressible resist that cells B / C .( stretched D )When B / C >1.Scalebars,1cm. B

/ 5 C >1,numericalsimulations E )Changeinthenumber ∆ l for B REPORTS / C values 1089

Downloaded from www.sciencemag.org on August 30,14 2012 Overwinding of helix with infinite bending modulus z y Mathematical description p x ~r(s)= r cos(s/),r sin(s/), s 0 0 2⇡ ✓ ◆ 2 2 = r0 +(p/2⇡) Z = pN = p(L/2⇡) p q pitch Z Infinite bending modulus fixes the helix curvature during stretching

r0 K = 2 2 r0 +(p/2⇡) 2r0 diameter Helix pitch and radius

length of the 1 Z2 L r0 = 1 Number of loops helix backbone K L2 ✓ ◆ Z number 2⇡Z Z2 Z KL N = p = 1 N = = of loops 2 p 2⇡ 1 (Z/L)2 p KL r L 15 p Overwinding of helix with inﬁnite bending modulus z y Helix pitch and radius

x 1 Z2 r0 = 1 Number of loops K L2 ✓ ◆ 2⇡Z Z2 Z KL p = 1 N = = p 2 p 2⇡ 1 (Z/L)2 KL r L pitch Z p 4 3.5 3 Overwinding 2r0 2.5 diameter 2 2⇡N/(KL) 1.5 length of the pK/(2⇡) L 1 helix backbone 0.5 r0K Z number 0 N = 0 0.2 0.4 0.6 0.8 1 p of loops Z/L 16 Spirals in nature shells beaks claws

horns teeth tusks

What simple mechanism could produce spirals?

17 Equiangular (logarithmic) spiral

in polar coordinates radius ↵ = 82 grows exponentially

↵ r(✓)=a✓ =exp(✓ cot ↵) ↵ cot ↵ =lna ↵ name logarithmic spiral: ln r ↵ ✓ = ln a Ratio between growth velocities in the radial and azimuthal directions ↵ velocities is constant! ↵ dr dr/dt v cot ↵ = = = r rd✓ rd✓/dt v✓

18 Equiangular (logarithmic) spiral

↵ = 85 ↵ = 82 ↵ = 80

↵ ↵ ↵

↵ = 75 ↵ = 60 ↵ = 45

↵ ↵ ↵ r(✓)=a✓ =exp(✓ cot ↵) 19 Growth of spiral structures newly added voutt material

Lout W + vwt W old structure

vint Lin

New material is added at a constant ratio of growth velocities, which produces spiral structure with side lengths and the width in the same proportions.

vout : vin : vW = Lout : Lin : W

Note: growth with constant width (vW=0) produces helices

20 Growth of spiral structures

✓ cot ↵ Assume the following spiral proﬁles rout(✓)=e of the outer and inner layers: ✓ cot ↵ rin(✓)=e < 1

e2⇡ cot ↵ > 1 e2⇡ cot ↵ < 1 = 75 = 86 ↵ ↵ , , 5 5 . . =0 =0

In some shells the inner layer does not grow at all 21 3D spirals

3D spiral of ram’s horns is due to the triangular vb Shells of mollusks are cross-section of the va often conical horn, where each side vc grows with a different velocity.