Helices, spirals and phyllotaxis, Lecture notes of Statistics

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MAE 545: Lecture 21 (12/8)

Helices, spirals and phyllotaxis

2

Shaping of gel membrane

properties by halftone lithography

t curvatures (with radii of 2 resence of slight through- ns in swelling (see SOM for ontribute to the observed de- rogrammed curvature. Inter- t observe a boundary layer ssian curvature around the al cap as has been reported leads to a saturation in the number of wrinkles, because the bending energy arising from Gaussian curvature increases with n (for the films with h ≈ 7 mm in Fig. 4, a metric with n = 8 yielded only six wrinkles). However, given the subtle differences between the metrics plotted in Fig. 2F, the ability to accurately reproduce the pro- grammed number of wrinkles for n = 3 to 6 is a 1 þ jcn xþiy R j 1 ffiffi 2 p (^) j 2 where sn, cn, and dn are Jacobi elliptic func- tions, and x and y are the components of r. This metric still has four cusp-like singularities where Ω(r) = 0; however, one of its useful properties as a map projection is that only a small portion d disks met- ets pro- ate (A) rface h an C) a and ficit re- s of heets iew ssian ick- me- mm, ugh ness d due the d val- d if- b (see Eq. 3) (black solid circles) and the programmed d line). ( F) Swelling factors for the target metrics as a ized radial po sition on the unswelled disks r/R, with points rresponding to lattice points to indicate the resolu- is patterned. (G to J) Patterned sheets programmed to generate Enneper’s minimal surfaces with n = (G) 3, (H) 4, (I) 5, and (J) 6 wrinkles upon swelling as dictated by Eq. 5. 3D reconstructed images (top) and top-view surface plots of squared mean curvature H 2 and Gaussian curvature K (bottom). Initial thicknesses and disk diameters are 7 and 390 mm, respectively. 2A). Both values are in rea- t with the target values, al- cy of disks with uniform dot t curvatures (with radii of 2 resence of slight through- ns in swelling (see SOM for ontribute to the observed de- rogrammed curvature. Inter- t observe a boundary layer ssian curvature around the al cap as has been reported curvature and negative Gaussian curvature that matches closely with the target profile. For a given film thickness, increasing n eventually leads to a saturation in the number of wrinkles, because the bending energy arising from Gaussian curvature increases with n (for the films with h ≈ 7 mm in Fig. 4, a metric with n = 8 yielded only six wrinkles). However, given the subtle differences between the metrics plotted in Fig. 2F, the ability to accurately reproduce the pro- grammed number of wrinkles for n = 3 to 6 is a Wðx; yÞ ¼ 2 jdn xþiy R j 1 ffiffi 2 p   sn xþiy R j 1 ffiffi 2 p   j 2 1 þ jcn xþiy R j 1 ffiffi 2 p   j 2 h i 2 ð 6 Þ where sn, cn, and dn are Jacobi elliptic func- tions, and x and y are the components of r. This metric still has four cusp-like singularities where Ω(r) = 0; however, one of its useful properties as a map projection is that only a small portion d disks met- ets pro- ate (A) rface h an C) a and ficit re- s of heets iew ssian ick- me- mm, ugh ness d due the d val- d if- b (see Eq. 3) (black solid circles) and the programmed d line). ( F) Swelling factors for the target metrics as a generate Enneper’s minimal surfaces with n = (G) 3, (H) 4, (I) 5, and (J) 6 wrinkles upon swelling as dictated by Eq. 5. 3D reconstructed images (top) J. Kim et al., Science 335, 1201 (2012) ircles) and the programmed tors for the target metrics as a e unswelled disks r/R, with points ints to indicate the resolu- generate Enneper’s minimal surfaces with n = (G) 3, (H) 4, (I) 5, and (J) 6 wrinkles upon swelling as dictated by Eq. 5. 3D reconstructed images (top) and top-view surface plots of squared mean curvature H 2 and Gaussian curvature K (bottom). Initial thicknesses and disk diameters are 7 and 390 mm, saddle (Sa) cone with deficit angle (Cd) cone with excess angle (Ce) spherical cap (Sp) K - Gauss curvature H - mean curvature Enneper’s minimal surfaces (H=0) swelling profiles

4

Differential growth or differential shrinking

produces spontaneous curvature

faster growth

of the top layer

L

L(1 + ✏)

R R

W

L(1 + ✏)

L

R + W

R

Differential growth (shrinking) of the two

layers produces spontaneous curvature

K =

R

W

straight again and released m

shape, complete with the sam

Experiments were also perf

gravitational effects and dam

reproduced in File S1, captu

several transient features incl

Structural Filaments that are longer than ,

form helices to avoid steric interactions.

L > 2 ⇡R

more shrinking

of the bottom layer

Helix

x

y

z

2 r 0

p

t

~n 1

~n 2

pitch

diameter

Mathematical description

~r(s) =

r 0

cos(s/), r 0

sin(s/),

p

s

~ t(s) =

d~r

ds

=

⇣

r 0

sin(s/),

r 0

cos(s/),

p

2 ⇡

⌘

g =

~ t ·

~ t =

r

2

0

2

p

2

4 ⇡

2

2

= 1

q

r

2

0

• (p/ 2 ⇡)

2

Set to fix the metric

Cucumber tendril climbing via helical coiling

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

Cucumber tendrils

want to pull

themselves up above

other plants in order

to get more sunlight.

Helical coiling of cucumber tendril

young

tendril

old

tendril

extracted

fiber ribbon

tendril cross-section

band of immature g-fiber cells is barely visible by

using darkfield microscopy (Fig. 1B), with no

ultraviolet (UV) illumination signature, indicating

the absence of lignification (Fig. 1C). In coiled

tendrils (Fig. 1D), g-fiber cells are clearly visible

(Fig. 1E) and lignified (Fig. 1F). The fiber ribbon

consists of two cell layers, with the ventral layer

observations of extracted fiber ribbons that pas-

sively shrink and coil even further when dried but

regain their original shape when rehydrated

(movie S2). Dehydrated tendrils also exhibit this

behavior because they are dominated by the stiff

fiber ribbon (movie S3). Together, these facts

suggest that the biophysical mechanism for

mechanical behavior between the physical mod-

els and tendril fiber ribbons. When clamped at

both ends and pulled axially, the physical model

simply unwinds to its original uncoiled state (Fig.

2A and movie S4). In contrast, in fiber ribbons

we observed a counterintuitive “overwinding”

behavior in which the ribbon coils even further

A B

E

G H

C

F

D

I

Fig. 1. Tendril coiling via asymmetric contraction. During coiling, a strip of

specialized structural gelatinous fiber cells (the fiber ribbon) becomes lignified

and contracts asymmetrically and longitudinally. (A to C) A straight tendril

that has never coiled (A) lacks lignified g-fiber cells. In the tendril cross

section, darkfield (B) and UV autofluorescence (C) show no lignin signal. (D to

H) In coiled tendrils (D), the fully developed fiber ribbon consists of ∼2 layers

of highly lignified cells extending along the length of the tendril. In the tendril

cross section, darkfield (E) and UV autofluorescence (F) show strong lig-

nification in the fiber ribbon. In (G) and (H), increased magnification reveals

that ventral cells (top left) are more lignified than dorsal cells. (I) The extracted

fiber ribbon retains the helical morphology of the coiled tendril. (Inset) Higher

magnification shows the orientation of g-fiber cells along the fiber ribbon.

Scale bars, (B) and (C) 0.5 mm, (E) and (F) 100 mm, (G) and (H) 10 mm, (I)

1 mm.

1088 31 AUGUST 2012 VOL 337 SCIENCE www.sciencemag.org

0 .5mm 0 .5mm

100 μm 100 μm

10 μm^10 μm

1mm

Coiling in older tendrils is due to a thin layer of stiff, lignified

gelatinous fiber cells, which are also found in wood.

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

lignified g-fiber cells

Coiling of tendrils in opposite directions

right-handed helix left-handed helix perversion

) illumination signature, indicating

ignification (Fig. 1C). In coiled D), g-fiber cells are clearly visible gnified (Fig. 1F). The fiber ribbon cell layers, with the ventral layer regain their original shape when rehyd (movie S2). Dehydrated tendrils also exhib behavior because they are dominated by th fiber ribbon (movie S3). Together, these suggest that the biophysical mechanism

perversion Ends of the tendril are fixed and cannot rotate. This constraints the linking number. Link = Twist + Writhe Coiling in the same direction increases Writhe, which needs to be compensated by the twist. Note: there is no bending energy when the curvature of two helices correspond to the spontaneous curvature due to the differential shrinking of fiber. In order to minimize the twisting energy tendrils combine two helical coils of opposite handedness (=opposite Writhe).

Twist, Writhe and Linking numbers

Ln=Tw+Wr linking number: total number of turns of a particular end

twist: number of turns due to twisting the beam

Wr (^) writhe: number of crossings when curve is projected on a plane

Tw

Overwinding of tendril coils

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

Preferred curved state Flattened state

In tendrils the red inner layer

is much stiffer then the

outside blue layer.

In rubber models both layers

have similar stiffness.

High bending energy cost

associated with stretching

of the stiff inner layer!

Small bending energy.

D I

Fig. 1.

Tendril coiling via asymmetric contraction. During coiling, a strip of

specialized structural gela

tinous fiber cells (the fiber

ribbon) becomes lignified

and contracts asymmetrically and longitudinally. (

A

to

C

) A straight tendril

that has never coiled (A) lacks lignified g-fiber cells. In the tendril cross section, darkfield (B) and UV autoflu

orescence (C) show no lignin signal. (

D

to

H

) In coiled tendrils (D), the fully developed fiber ribbon consists of

∼

2 layers

of highly lignified cells extending along the length of the tendril. In the tendril

cross section nification in that ventral c fiber ribbon r magnification Scale bars, (B 1 mm.

31 AUGUST 2012

VOL 337

SCIENCE

w

1088

state as expected (movie S5). fiber ribbon unwinds, returning to a flat, uncoiled tually though, under high enough tension the perversion (Fig. 2A, right, and movie S5). Even- when pulled, adding turns on both sides of the

Inspi

of a coi relatively the inne lignifica

Tendrils try to keep the preferred

curvature when stretched!

14

Overwinding of rubber models with an

additional stiff fabric on the inside layers

state as expected (movie S5). fiber ribbon unwinds, returning to a flat, uncoiled tually though, under high enough tension the perversion (Fig. 2A, right, and movie S5). Even- when pulled, adding turns on both sides of the

Inspired by our observations of asymmetric

of a coiled physical model. To mimic lignified relatively inextensible fabric ribbon to the inside the inner layer is less extensible, we added a lignification in fiber ribbons, which suggest that

contraction. Together, these modifications increase gation, whereas the external copper wire prevents helix. The internal fabric ribbon prevents elon- compressible copper wire to the exterior of the cells that resist compression, we added an in-

Fig. 3.

Mechanical consequences of overwinding. (

A (^) and

(^) B )

separated into a segment conta substantial overwinding (blue curves). Each tendril was overwind (red curves) and one old tendril that exhibits Force extension curves for one young tendril that does not

ining the helical perversion

perversion (solid curves indi (dotted curves indicate perverted) and a segment with no

cate clamped). The dimension-

less force

F ∼

is plotted against the scaled displacement

∆ l

helical perversion materials) in (A). The difference in scaled force due to the (detailed definitions are available in the supplementary

∆ f =

f (perverted)

−

f (clamped) is plotted

against

(^) ∆

l in (B). The shaded range in (B) indicates variations

in the fitted initial slope value. (

C ) Dimensionless force-

extension curves are plotted for numerical filaments with

B /C

of the same data. ( values 1/5 (red), 1 (green), 5 (blue). (Inset) Log-linear plot

D ) The difference in force

∆ (^) F ∼

=

F ∼ (perverted)

(^) −

(^) F ∼ (clamped) highlights the mechanical effect

of the helical perversion. For

B

<

C , the perversion always

B decreases the force needed to axially extend the filament; for (^) >

(^) C , the perversion initially decreases the force needed but

eventually increases this necessary force at higher exten-

m

Fig. 2.

Twistless springs unwinding and overwinding. (

A ) A silicone twistless

spring with lower bending stiffness

B

than twisting stiffness

C

unwinds when

pulled, returning to its original flat shape. (

B ) When a fiber ribbon is pulled, it

of turns are indicated in white). ( initially overwinds, adding one extra turn to each side of the perversion (number

C ) Overwinding is induced in the silicone model

the helix and an inextensible (under compression) copper wire to the exterior. by adding a relatively inextensible (under tension) fabric ribbon to the interior of

Together, these increase the ratio

B /C

. ( D ) When

(^) B /C (^) > 1, numerical simulations

consistent with physical an of elastic helical filaments recapitulate this overwinding behavior, which is

d biological experiments. (

E ) Change in the number

of turns in each helix

(^) ∆

N (^) is plotted versus scaled displacement

(^) ∆

l (^) for

(^) B /C

(^) values

increasing 1/5 (red), 1 (green), and 5 (blue). Overwinding becomes more pronounced with

(^) B /C

. ( F ) Overwinding is also observed in old tendrils, which have dried

and flattened into a ribbon-like shape with

B /C

1. Scale bars, 1 cm.

REPORTS

Downloaded fromwww.sciencemag.org on August 30, 2012

relaxed

stretched

5

5

4

6

stiff fabric

soft rubber

Overwinding of helix with infinite bending modulus

x

y

z

2 r 0

p

pitch

diameter

Z

length of the

helix backbone

L

number

of loops

Helix pitch and radius

N =

Z

p

r 0

=

1

K

✓

1

Z

2

L

2

◆

p =

2 ⇡Z

KL

r

1

Z

2

L

2

Number of loops

N =

Z

p

=

KL

2 ⇡

p

1 (Z/L)

2

r 0

K

pK/(2⇡)

Z/L

2 ⇡N/(KL)

Overwinding

Spirals in nature

shells beaks claws

horns teeth tusks

What simple mechanism could produce spirals?

Equiangular (logarithmic) spiral

r(✓) = a

✓

= exp

(✓ cot ↵)

Growth of spiral structures

old structure

newly added

material

W

L

in

L

out

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.

v out

t

v in

t

v out

: v in

: v W

= L

out

: L

in

: W

W + v w

t

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