Shape con ol o ac i e su aces inspi ed by he
mo emen o euglenids
Ma ino A oyo
LaC`aN, Uni e si a Poli `ecnica de Ca alunya-Ba celonaTech
Ba celona 08034, Spain
An onio DeSimone
SISSA, 34136 T ies e, I aly
June 6, 2013
Abs ac
We examine a no el mechanism o ac i e su ace mo phing inspi ed by he cell
body de o ma ions o euglenids. Ac ua ion is accomplished h ough in-plane simple
shea along p esc ibed slip lines deco a ing he su ace. Unde gene al non-uni o m
ac ua ion, such local de o ma ion p oduces Gaussian cu a u e, and he e o e leads o
shape changes. Geome ically, a de o ma ion ha ealizes he p esc ibed local shea
is an isome ic embedding. We explo e he possibili ies and limi a ions o his bio-
inspi ed shape mo phing mechanism, by i s cha ac e izing isome ic embeddings un-
de axisymme y, unde s anding he limi s o embeddabili y, and s udying in de ail
he accessibili y o su aces o ze o and cons an cu a u e. Modeling mechanically
he ac i e su ace as a non-Euclidean pla e (NEP), we u he examine he mechanism
beyond he geome ic singula i ies a ising om embeddabili y, whe e mechanics and
buckling play a decisi e ole. We also p opose a non-axisymme ic ac ua ion s a egy
o accomplish la ge ampli ude bending and wis ing mo ions o elonga ed cylind ical
su aces. Besides helping unde s and how euglenids delica ely con ol hei shape, ou
esul s may p o ide he backg ound o enginee so machines.
1 In oduc ion
So machines a e s uc u es made ou o lexible ac i e ma e ials ha unde go con ollable
shape ans o ma ions unde he ac ion o a a ie y o s imuli. Examples include swellable
hyd ogels sensi i e o ei he empe a u e (Hu e al.,1995)o elec ic- ield(Kwon e al.,2008),
elec oac i e polyme s (Jage e al.,2000), liquid-c ys al elas ome s (Wa ne and Te en je ,
2007;DeSimone and Te esi,2009) ac ua ed by empe a u e (deHaan e al.,2012), ligh
1
(Whi e e al.,2008;Camacho-Lopez e al.,2004), o elec ic ield (Fukunaga e al.,2008),
and biohyb id cons uc s consis ing o muscle cells a ached o elas ome shee s (Feinbe g
e al.,2007;Naw o h e al.,2012). Bioinspi a ion plays an impo an ole in his ield.
O en, ac ua ion is accomplished h ough bending, e.g. by non-uni o m swelling h ough
he hickness, ypically in quasi one-dimensional s uc u es (Hu e al.,1995;A mon e al.,
2011;Sawa e al.,2010;U ayama,2012). In quasi wo-dimensional s uc u es, imposed
bending alone, e en i wo-dimensional, p o ides access o a limi ed epe oi e o shapes as
i can only p oduce cu a u e in one di ec ion due o he dominan s e ching penal y o
doubly cu ed shapes (Wa ne e al.,2010). An al e na i e ac ua ion pa adigm o hin
ilms consis s o imposing an in-plane non-uni o m de o ma ion ield. I such an imposed
de o ma ion canno be ealized wi hin he plane, i leads o shape ans o ma ions (Sha on
and E a i,2010;Ben Belgacem e al.,2000;Bha acha ya e al.,1999).
A he oo o his concep , coined as non-Euclidean pla es (NEPs), is Gauss’ Eg egium
heo em, by which a gene ic in-plane de o ma ion (me ic enso ) p oduces Gaussian cu -
a u e. Geome ically, a a ge nonuni o m me ic is p esc ibed on he su ace ep esen ing
he hin ilm. Such me ic can some imes be ealized by an isome ic embedding, i.e. a
pa ame ized su ace in space wi h he speci ied me ic. In his si ua ion, he hin body will
change shape o mee he a ge me ic. Al hough in-plane elas ici y o e whelmingly domi-
na es bending elas ici y o hin ilms, and he e o e he geome y o isome ic embeddings
is he main shape selec ion p inciple in non-Euclidean pla es, elas ic s e ching and bending
play a ole in some si ua ions. Fo ins ance, geome y alone canno selec a shape when
mul iple isome ic embeddings exis , and hen mechanics p o ides he selec ion p inciple,
h ough bending ene gy minimiza ion (Lewicka and Reza Pakzad,2011). Fu he mo e, he
a ge me ic may no be ealizable, leading o us a ed shapes ha may elax in-plane
s e ching by cu ing in o possibly e y complex shapes (Sha on e al.,2002;Ma de ,2003).
This iewpoin has been mo i a ed by he shapes o lowe s and lea es as a esul o
non-uni o m g ow h (Nechae and Voi u iez,2001;Ma de and Papanicolaou,2006;Sha on
e al.,2004;Liang and Mahade an,2009;Ama e al.,2012), and has been implemen ed in
disk and ubula opologies by non-uni o m swelling/sh inkage o gels (Klein e al.,2007;
Kim e al.,2012) o non-uni o m swi ching o liquid c ys al elas ome s (deHaan e al.,2012;
Sawa e al.,2010;U ayama,2012). In hese and mos e e ences, he non-uni o mi y in he
de o ma ion is accomplished by pa e ning non-uni o m swelling o nema ic di ec o ields,
see also e.g. Modes e al. (2011), and applying a uni o m s imulus, al hough i has also
been accomplished by non-uni o m illumina ion (Camacho-Lopez e al.,2004). Such p o-
g ammed NEPs can he e o e execu e p ede ined shape changes, al hough he pa h be ween
con igu a ions can p esen signi ican a iabili y (Kim e al.,2012).
He e, inspi ed by he mo ili y o euglenids, we examine ano he mechanism o ac ua ing
he a ge me ic, which we ha e ecen ly iden i ied (A oyo e al.,2012). Euglenids a e
a amily o unicellula p o is s p esen wo ldwide in a wide ange o aqua ic en i onmen s,
wi h ypical sizes om ens o hund eds o mic ons (Leande ,2008;T ieme ,1999). They
a e e y di e se in e ms o mo phology, lexibili y, eeding s a egy ( eeding on o he cells,
pho osyn hesis, abso bing dissol ed nu ien s di ec ly om he en i onmen ), o mo ili y, and
2
ab
i
ii
iii
i
i
Figu e 1: Mo ili y o plas ic euglenids. (a) Eu ep iella species execu ing highly epe i i e
and nea ly axisymme ic s oke. O iginal ame images cou esy o Richa d E. T ieme . (b)
Euglena species pe o ming a mo e acilla ing mo ion, including bending and wis ing o he
cell. O iginal ame images cou esy o F ancisco Pujan e.
3
can be soli a y o li e in colonies. Al hough euglenids mo e p ima ily bea ing hei lagella,
some species exhibi la ge ampli ude and highly conce ed body dis o ions, called me aboly
o euglenoid mo emen , which ha e ascina ed mic obiologis s and physical scien is om
he ea lies days o mic oscopy o now (Dobell,1932;Fle che and The io ,2004). See
Figu e 1 o an illus a ion. The unc ion o me aboly emains unclea , al hough i seems
o ha e eme ged om he need o a malleable cell wall o engul la ge p ey, and may ha e
ound u ili y in locomo ion la e in he e olu iona y his o y. Fo his eason, me aboly
would ha e pe sis ed in pho osyn he ic and osmo ophic species, which do no engul o he
cells. We ha e p e iously shown ha me aboly is a slow bu e icien mo ili y mode in
a New onian luid, and sugges ed ha i seems pa icula ly well-sui ed o locomo ion in
complex o g anula media (A oyo e al.,2012). Me aboly is con olled by he pellicle, a
s ia ed en elope enclosing mos euglenids, see Figu e 2.
The cell shape excu sions o me aboly a e g ace ully execu ed in ime-scales o a ew
seconds, and include pe is al ic mo ions, bending, wis ing, ounding, and elonga ing (see
Figu e 1 o an illus a ion). The euglenoid mo emen is e y smoo h in space and ime,
sugges ing ha i does no in ol e geome ical ins abili ies. The s iking ac ha a uni-
cellula o ganism, wi h minimal ana omical and senso y machine y, is able o pe o m in
a seemingly con olled manne such di e se mo ions wi h ema kable agili y sugges s ha
me aboly is g ounded on a obus and e sa ile physical mechanism. In A oyo e al. (2012),
we de eloped a con inuum heo y o model he cell kinema ics, and used i o quan i a i ely
analyze mo ie eco dings o me abolic euglenids displaying axisymme ic mo ions. He e, we
e isi and expand his heo y o sys ema ically explo e he possibili ies and limi a ions o
his shape ac ua ion mechanism, ocusing on an idealized model cylind ical pellicle.
2 The pellicle and i s de o ma ion
The pellicle is an ac i e s ia ed su ace ha execu es me aboly, see Figu e 2a,b. The e is
s ong ana omical and unc ional e idence sugges ing ha his ac i e en elope can econ ig-
u e by mobilizing molecula mo o s along mic o ubulues a anged below he pellicle s ips,
see Figu e 2c. Mo e speci ically, he pellicle s ips ha e been shown o slide ela i e o each
o he du ing me aboly wi hou changing hei wid h o leng h (Suzaki and Williamson,1985,
1986). Since he lexible pellicles o de o mable euglenids exhibi a la ge numbe o s ips
(a ew ens), we adop a con inuum app oxima ion and idealize he kinema ics as consis ing
o simple shea along he pellicle s ips. Fo highly localized de o ma ion ea u es, in he
leng h scale o he wid h o he pellicle s ips, such a model may miss e↵ec s a ising om
he disc e e na u e o he pellicle. We de elop nex a ma hema ical model o he pellicle
kinema ics.
Conside a ma e ial su ace 0⇢R3, pa ame ized as x0(u, ) o (u, )2¯
, modeling
he pellicle a a gi en e e ence s a e. The na u al basis o he angen bundle o he e e ence
su ace is ob ained by pa ial di↵e en ia ion (deno ed wi h a comma) wi h espec o uand ,
{x0,I ,I =u, }. By ei,i=1,2,3 we deno e he Ca esian basis ec o s. We conside now a
de o med con igu a ion x(u, )=P3
i=1 xi(u, )ei, pa ame izing he su ace . The in-plane
4
Scanning elec on mic og aphs o Eu ep ia pe yi (A) and Euglena spi ogy a (B) (Scale ba = 10 mic ons).
T ansmission elec on mic og aphs o he subs uc u al ea u es o he euglenid pellicle (he e Pe anema
ichopho um, scale ba = 1 mic on)
s0
m0
ab
c
Figu e 2: Scanning elec on mic og aphs o Eu ep ia pe yi (a) and Euglena spi ogy a
(b) (Scaleba = 10 µm), showing he pellicle, a s ia ed en elope co e ing mos euglenids.
Amongs euglenids, lexible species exhibi a la ge numbe o s ips ( ens), whose helici y
co ela es wi h body de o ma ion. (c) T ansmission elec on mic og aph o he subs uc-
u al ea u es o he euglenid pellicle (he e Pe anema ichopho um, scaleba = 1 µm). The
pellicle is a co ical complex including he plasma memb ane, a se o in e locking p o eina-
ceous s ips, mic o ubules, and ubula cis e nae o endoplasmic e iculum a anged along
he s ips. Mic og aphs om Leande e al. (2001). We also illus a e he angen ec o
ields s0and m0along and pe pendicula o he pellicle s ips.
5

de o ma ion g adien is a 2 ⇥2 enso ield mapping he he angen bundle o 0 o ha o
,andcanbeexp essedasF(P)=Dx(x1
0(P))[Dx0(x1
0(P))]1 o P20(Ma sden and
Hughes,1983). By exp essing Dx0and Dxin he canonical basis o ¯
and he na u al bases
o 0and , hei componen s a e he iden i y ma ix, and he e o e he componen s o he
igh Cauchy-G een de o ma ion enso in he basis {x0,I ,I =u, }a e equal o hose o he
su ace me ic enso , and gi en by he scala p oduc s o angen ec o s o he de o med
su ace
CIJ =gIJ =x,I ·x,J .(1)
The su ace 0is deco a ed by a angen ial uni ec o ield s0, whose in eg al lines can
be hough o as he con inuum pellicle s ips. Wi h he con inuum app oxima ion o he
pellicle, we suppose ha he ac i e su ace is capable o p oducing simple shea (u, ) along
s0. I we deno e by m0a uni ec o ield pe pendicula o s0, he in-plane de o ma ion
g adien due o he pellicle shea can be w i en as
F=R(Id+s0⌦m0),(2)
whe e Ris an unde e mined o a ion enso ield, which can be a oided by conside ing he
igh Cauchy-G een de o ma ion enso
C=FTF=Id+(s0⌦m0+m0⌦s0)+2m0⌦m0.(3)
This a ge me ic has uni de e minan (is a ea-p ese ing). Compa ing Eqs. (1)and(3)
p o ides a link be ween ac ua ion ()andshape(x). In ecen wo k (A oyo e al.,2012), we
ha e used hese ela ions o “measu e” he spa io- empo al pa e ns o he ac ua ion shea ,
(u, , ), om ideo eco dings o mo ile cells. Figu e 3illus a es he co ela ion be ween
shape changes and pellicle shea ac ua ion o he pe is al ic mo ion o a mo ile Euglenid
exhibi ing an axisymme ic mo ion cap u ed in ideo and p ocessed wi h he me hod p e-
sen ed in his e e ence. He e, we examine he shapes esul ing om a p esc ibed pellicle
shea ield, which is an amoun o inding a global isome ic embedding in R3(x)o a
Riemannian 2-mani old wi h he p esc ibed me ic enso gi en by Eq. (3).
I is wo h compa ing his shape ac ua ion mechanism wi h ano he a ea-p ese ing mech-
anism gi en by he igh s e ch enso U=1/q0⌦q0+p0⌦p0, whe e p0and q0a e
mu ually o hogonal ec o ields. This mechanism can be ele an o hin ilms made o
nema ic glasses o elas ome s (Modes e al.,2011). By choosing p0and q0so ha hey o m
a45
angle wi h s0and m0, i is easy o see ha U2is only equi alen o Eq. (3) a ound in-
ini esimal de o ma ions, while o ini e de o ma ions he wo mechanisms a e undamen ally
di↵e en .
Fo conc e eness, we ix h oughou he pape a cylind ical e e ence con igu a ion o
leng h L0, adius R0, and whose pellicle s ips a e s aigh and aligned wi h i s axis, see
Figu e 4-i o an illus a ion. This minimal model sys em is ep esen a i e o euglenids, and
po en ially in e es ing in applica ions. We can con enien ly choose u(aligned wi h he axis
o he cylinde ) and (along he azimu hal di ec ion) such ha x0,u =s0and x0, =m0.
Then, gi en a unc ion (u, ), a de o med con igu a ions ha ealizes he p esc ibed simple
6
0
1
2
3
(, )
Figu e 3: Ma hema ical model o a mo ile euglenid du ing a ull s oke, cap u ed on ideo
and p ocessed as desc ibed in A oyo e al. (2012). The colo map ep esen he magni ude
o he pellicle shea , which eaches o e 300%, he black lines a e pellicle s ips, and he
colo ed lines and poin s a e Lag angian ma ke s, which help isualize he ne o a ion o he
cell a ound i s symme y axis. Du ing he powe s oke o me aboly, a bulge slides down he
elonga ed cell, accomplishing signi ican o wa d mo ion when placed on a New onian luid
a anishing Reynolds numbe . Du ing he eco e y s oke, he bulge a he ail disappea s
a he expense o he bulge a he head, in ol ing s ong cy oplasmic s eaming, and he
cell mo es back a li le. The igu es highligh s he s ong co ela ion be ween shape, pellicle
helici y, and azimu hal mo ions.
7
shea along he pellicle should sa is y he h ee independen nonlinea pa ial di↵e en ial
equa ions in x,u ·x,u x,u ·x,
x,u ·x, x, ·x, =1
1+2.(4)
The a ge me ic in he igh -hand side o his equa ion is no la in gene al. I s Gaussian
cu a u e ( he p oduc o he p incipal cu a u es) can be compu ed by di↵e en ia ing he
componen s o he me ic enso as (do Ca mo,1976)
K=(, ,u),u.(5)
The sys em in Eq. (4) is highly non i ial and many open ques ions emain (Han and
Hong,2006). The equa ions change cha ac e depending on he sign o he Gaussian cu a-
u e associa ed wi h he a ge me ic. Exis ence o global solu ions canno be expec ed in
gene al, and when hey exis hey can be non unique. A i ial example is = 0, ealizable
by in ini ely many cylind ical su aces wi h non-ci cula c oss-sec ion. To analyze local and
global exis ence o isome ic embeddings, Eqs. (4)ha ebeen ecas inequi alen o ms,asa
Monge-Amp`e e equa ion called he Da boux equa ion, o as he Maina di-Codazzi sys em.
He e, we a e in e es ed in he global p oblem o non-compac su aces, and o Gaussian
cu a u es changing sign, o which he e is no gene al ma hema ical esul a ailable. In-
s ead, we p oceed by i s analyzing elemen a y axisymme ic examples in Sec ion 3, whe e
he di↵e en ial equa ions go e ning he de o med shape can be sol ed explici ly, and hen by
examining non-axisymme ic examples h ough nume ical simula ions based on he heo y
o non-Euclidean pla es (E a i e al.,2009), in Sec ion 4.
3 Axisymme ic shapes wi hou s e ch
We examine he e simple shapes esul ing om axisymme ic ac ua ion, i.e. , =0,inwhich
he a ge me ic can be me exac ly. Thus, mechanically, such de o ma ions in ol e no
s e ch, which is he dominan ene ge ic con ibu ion o hin bodies. The mos i ial
example, uni o m pellicle shea , helps unde s and how simple shea along he pellicle s ips
esul s in shape changes. Figu e 4p o ides a pic o ial depic ion o his si ua ion, which
exploi s he ac ha K= 0, and he e o e he pellicle su ace can be de eloped on o a plane.
F om elemen a y geome ical conside a ions, he inal cylinde has leng h L0/p1+2and
adius R0p1+2. This example also illus a es he s ong co ela ion be ween adius and
local pellicle o ien a ion obse ed in mic og aphs o euglenids, see Figu es 2and 3, as well
as he coupling be ween shape changes and azimu hal mo ions.
3.1 Kinema ics
We pa ame ize he e e ence con igu a ion as
x0(u, )={R0cos ( /R0),R
0sin ( /R0),u},u2[0,L
0], 2[0,2⇡R0],(6)
8
un olded
e e ence
s a e
simple shea
along s ips o a ion
oll-up in o a hicke
and sho e cylinde
shape change
by pellicle shea
w

i
=/w
ii iii i
Figu e 4: Main idea o he shape ac ua ion p inciple by simple shea along he pellicle s ips
deco a ing he ac i e su ace. In he simples si ua ion, he e e ence shape is a cylinde wi h
he pellicle lines o ien ed pa allel o i s axis, and he p esc ibed shea is uni o m. Uni o m
pellicle shea esul s in a sho e and hicke cylinde wi h helical pellicle s ips. This can
be unde s ood by cu ing and un olling he e e ence pellicle (i) in o a plana ec angle (ii),
shea ing i uni o mly (iii), and hen olling he esul ing pa allelog am along a di ec ion
pe pendicula o he ee ends (i ) o p oduce he de o med shape ( ).
9
and and µa e Lame’s cons an s. In his model, he cu a u e ene gy is ela i e o a la
s a e, al hough i could measu e cu a u e de ia ions om 0in a heo y o non-Euclidean
shells.
Le us assume i s ha ¯
Cis embeddable. I is small, he bending ene gy imposes a small
bias o he s e ching ene gy, and he e o e minimiza ion o he elas ic ene gy is expec ed
o lead o a su ace ha closely ealizes he a ge me ic. By educing , we expec o
con e ge o an exac embedding o he a ge me ic, and he e o e he scaled memb ane
ene gy Em/ =(1/ )R0wmdS0should end o ze o. See Lewicka and Reza Pakzad (2011)
o ma hema ical esul s along hese lines. I he a ge me ic is no embeddable, he
mechanics o he NEP will selec mo phologies ha app oxima e he a ge me ic, bu
wi h ini e scaled memb ane ene gy e en as ends o ze o. In his si ua ion, he us a ed
embeddings depend undamen ally on he choice o , and he heo y o NEP can be ega ded
as a plausible mechanical model o he ac ual sys em, a he han a de ice o explo e he
geome y o isome ic embeddings.
He e, we implemen his heo y nume ically wi h subdi ision ini e elemen s, and mini-
mize he elas ic ene gy in Eq. (23) wi h a limi ed memo y BFGS quasi-New on algo i hm
(A oyo and Bely schko,2004), which esul s in a leas locally s able equilib ium con igu-
a ions. I mus be no ed ha beyond embeddabili y, we may expec mul iple equilib ium
b anches. In he simula ions we ollow pa hs in which ¯
Cis p og essi ely modi ied, s a ing
wi h a ield close o he iden i y (close o ze o), and also ollow pa hs in which changes.
While his nume ical s a egy is e y obus , as , and lexible, a wo d o cau ion should be
men ioned: he nume ically explo a ion o embeddabili y o la ge shape de o ma ions mus
be done ca e ully. Embeddabili y may be moni o ed by checking nume ically i Em/ =
(1/ )R0wmdS0 ends o ze o as !0, which equi es e y accu a e ene gy minimiza ion.
The e o s associa ed wi h he nume ical disc e iza ion ( he mesh size) and wi h he ene gy
minimiza ion algo i hm mus be ca e ully con olled o meaning ul esul s. Howe e , his
is challenging due o he la ge mesh dis o ions associa ed wi h la ge pellicle shea s and he
ill-condi ioning ypical o he mechanics o e y hin shells.
4.2 Isome ic embeddings beyond he singula i y
We i s in es iga e he pseudo-sphe e beyond embeddabili y, as epo ed in Figu e 8, o
NEPs o di↵e en hickness. We ind ha be o e he geome ic singula i y is eached (¯<
1), he heo y o NEPs p o ides a good app oxima ion o he isome ic embedding ound
di ec ly in he p e ious sec ion, wi h e y small scaled memb ane ene gy, i.e. e y small non-
compliance wi h isome y. We obse e ha as ¯inc eases, bu emains smalle han 1, he
scaled memb ane ene gy sligh ly inc eases. We ha e checked by mesh e inemen ha his is
due o he ini e elemen disc e iza ion e o s, as mo e closely analyzed la e . As expec ed,
he geome ic singula i y mani es s i sel in he con ex o NEPs as us a ed con igu a ions,
which buckle in o di↵e en symme y-b eaking shapes despi e he ac ha he a ge me ic
is axisymme ic. A he ini ial s ages beyond ¯= 1, he sys em emains axisymme ic and
s o es inc easingly la ge amoun s o scaled memb ane ene gy. A a ce ain poin , he sys em
16

= 0
= 0/2
= 0/4
= 0/8
0 0.5 1 1.5
0
1
2
x 10−3
¯
Em/
=0.05
=0.05/2
=0.05/4
=0.05/8
3
5
4
5
9
4

0
1
2
3
4
Figu e 8: The pseudo-sphe e beyond embeddabili y. We conside a pellicle shea dis ibu ion
gi en by =¯(L0/R0)⇠ o se e al alues o ¯0, i.e. we ollow he ho izon al axis in he
phase diag am in Figu e 6 om 0 owa ds nega i e alues. A ¯= 1, we hi he geome ic
singula i y, and hen u he inc ease his pa ame e beyond embeddabili y. We examine
NEPs o di↵e en hicknesses ( 0/R0=0.05), and epo on he scaled memb ane ene gy
Em/ and he esul ing mo phologies. The numbe o downwa d pe als is ma ked on he
buckled con igu a ions.
17
i
ii
iii
i


a b
c
d e
Figu e 9: Examples o cylind ical pellicles o hickness h/R0=0.01 b ough beyond he
geome ic singula i y (a-c). In (a), (i) is subjec ed o a pellicle shea ollowing Eq. (13),
which esul s in cones o la annuli i exac ly embedded. In (b) and (c), a Gaussian pellicle
dis ibu ion as in Eq. (22)o di↵e en wid h and in ensi y is p esc ibed, esul ing in buck-
led con igu a ions wi h sel -in e sec ions. An anomalous euglenid, displaying ab up shape
changes and collapsed con o ma ions is shown in (d) and (e). O iginal ame images cou esy
o F ancisco Pujan e.
18
pa ially elaxes by buckling in o pe aled con igu a ions, o which de ia ion om isome y
g ows a a slowe a e. While be o e he geome ic singula i y he beha io o he sys em is
only sligh ly dependen on , beyond he singula i y he dependence is s ong, in e ms o he
buckling poin and he mo phology (e.g. numbe o pe als). We obse e ha hinne NEPs
exhibi ine ea u es and a e able o be e app oxima e isome y. Fo a ixed hickness, we
also obse e mode swi ching, by ansi ioning om 4 o 3, o om 5 o 4 pe als. No e he
e y la ge a ge shea s ains imposed o he sys em, o up o 400%.
Figu e 9(a-c) shows a galle y o NEPs subjec o axisymme ic pellicle shea s and b ough
beyond he geome ic singula i y. Theo e ically, he a ge me ic in (a-ii) is isome ically
embeddable as a la annula disk. Ins ead, he ini e hickness NEP s e ches i s ou e im,
while comp essing i s inne im and emaining axisymme ic. This is consis en wi h he
embeddabili y condi ion in Eq. (15). Upon u he shea ing, as in he p e ious example,
he NEP b eaks symme y and de elops a wa y buckled s a e. A simila beha io is ob-
se ed when a localized pellicle shea dis ibu ion is b ough beyond he geome ic limi s o
embeddabili y. A small ampli ude localized w inkling ini ially de elops, and hen swi ches
o la ge-ampli ude de o ma ions ha a↵ec he whole sys em, and lead o sel -in e sec ing
con o ma ions. S ikingly, hese de o ma ion modes bea simila i y wi h hose o an anoma-
lous euglenid shown in Figu e 9(d,e). Al hough mos euglenids exhibi smoo h mo ions
seemingly wi hou geome ic ins abili ies, his pa icula specimen unde goes ab up shape
changes, which a e indica i e o excessi e ac ua ion s ains, buildup o s e ching ene gy,
and sudden elease by buckling. The esul ing con o ma ions a e collapsed, and exhibi
e-en an olds. This seems o indica e ha no only geome y, bu also mechanics, a e
impo an in unde s anding he mo ili y o euglenids.
4.3 Bending and wis ing
To mimic he bending and o sional mo ions o euglenids and a e some expe imen a ion,
we examine he e a s a egy based on helical pa e ns o pellicle shea , while s ill conside ing
as e e ence pellicle a cylind ical su ace wi h he pellicle s ips aligned wi h he symme y
axis o he cylinde . Fo his pu pose, we de ine a helix by i s azimu hal angle as a unc ion
o longi udinal coo dina e, ✓+
0(u)=2⇡u/B, whe e Bis he pi ch o he helix. I B=L0,
he helix pe o ms a ull u n a ound he cylind ical e e ence pellicle. We de ine a second
helix ✓
0(u)=2⇡u/B +⇡, symme ic o he p e ious helix wi h espec o he axis o he
cylinde . A each poin o he e e ence pellicle labelled by (u, ), i s azimu hal angle can
be w i en as ✓( )= /R0. By deno ing wi h ✓±(u, )2[0,⇡] he magni ude o he angle
disc epancy be ween ✓( )and✓±
0(u), we de ine
(u, )=A+exp h✓+(u, )/C2i+Aexp h✓(u, )/C2i,(26)
whe e A±cha ac e izes he s eng h o shea in he ±helices, and C he la e al sp ead o he
shea dis ibu ion a ound he helices. Figu e 10 (a) p o ides an illus a ion o A+=0.25,
A=0.5, B=L0and C=0.3.
19
10−310−2
10−6
10−5
10−4
Em
Coa se mesh
Fine mesh
a
b
c
d
e

Em/
Figu e 10: Nume ical es o embeddabili y o a non-axisymme ic a ge me ic. A e e ence
cylind ical pellicle o uni adius is subjec ed o a shea dis ibu ion shown in (a), wi h
a spi al egion wi h la e al Gaussian modula ion o posi i e pellicle shea ( ed), and an
opposing spi al egion wi h a la ge nega i e shea (blue). We es wo meshes (coa se and
ine) shown in (b), and minimize he NEP ene gy o dec easing shell hickness . The
esul ing de o med shapes o a gi en hickness and he coa se and ine meshes a e shown in
(c) and (d). The scaled memb ane ene gy Em/ is ep esen ed agains he shell hickness in
a log-log scale in (e).
20

0
1
-1
a
b c
on iew
side iew
Figu e 11: De o ma ions esul ing om he a ge pellicle shea in Eq. (26), wi h pa ame e s
A+=A=1.5, C=0.3, and B=L0(a,b) o B=2L0(c). The NEP hickness is
= 0=0.05 R0in (a,c) and = 0/4 in (b).
In gene al, such a pellicle shea dis ibu ion p oduces a combina ion o bending and
o sion o he cylind ical su ace, as shown by he nume ical esul s o NEP illus a ed in
Figu e 10 (d). The immedia e ques ion ha a ises is whe he such a non-axisymme ic
a ge me ic is embeddable, wi hou s e ch. We analyze his by nume ically acking
he scaled memb ane ene gy Em/ as !0, which should con e ge o ze o i he a ge
me ic can be exac ly ealized, see Figu e 10 (e). We ind ha o a coa se mesh, he
scaled memb ane ene gy does no educe as he shell becomes hinne . Howe e , by e ining
he mesh signi ican ly, we ind ha his is due o he disc e iza ion e o s in oduced by
he ini e elemen app oxima ion, which can be a oided sys ema ically by mesh e inemen .
The con e gence o he scaled memb ane ene gy o ze o o he ine mesh p o ides s ong
nume ical e idence ha indeed he a ge me ic is embeddable wi hou memb ane ene gy.
We now examine u he he ac ua ion mechanism embodied in Eq. (26), and conside
s onge pellicle shea s. Figu e 11 shows esul s in which A+=A. We ind ha i he
opposing helical pa e ns o shea in he cylinde a e e y dissimila in magni ude, he NEP
is e y p one o buckling. We obse e ha wi h a sho pi ch, (a), he cylinde s ongly
bends and wis s, olding on o i sel a la e s ages. Fo a longe pi ch, (c), one can achie e
21

signi ican bending wi h li le wis . I he NEP is e y hin, (b), ins ead o bending and
wis ing as a whole, while keeping he c oss-sec ion close o ci cula , he sys em elaxes by
se e ely dis o ing he c oss sec ion, a he expense o li le global bending ac ua ion. These
esul s sugges ha his mechanism can accomplish la ge planned bending and o sional
mo ions i c oss-sec ional ins abili ies a e con olled. Fu he mo e, o he nume ical es s
indica e ha o a gi en s eng h o he pellicle shea , |A±|, i is possible o adjus he pi ch
Bso ha he esul ing de o ma ion is pu e bending, wi hou wis . While his obse a ion
is po en ially in e es ing in applica ions, we no e ha in mo ies o mo ile euglenids bending
and wis ing mo ions o he cell body appea o be sys ema ically coupled, sugges ing he
cell does no ine une he pi ch in his way.
5 Conclusions
We ha e s udied he possibili ies and limi a ions o a shape mo phing mechanism o hin
su aces, inspi ed by he ac i e en elope o euglenids, called pellicle. This mechanism consis s
o locally shea ing in-plane he su ace along p ede ined di ec ions. Non-uni o m ac ua ion
leads o cu a u e and shape changes, as p esc ibed by Gauss’ Eg egium heo em. Unde
he hypo hesis o axisymme y, and ocusing on a cylind ical pellicle modeling he cell’s
elonga ed body, we ha e p o ided simple equa ions o ind isome ic embeddings, i.e. su -
aces ha exac ly ealize he p esc ibed shea wi hou s e ching, and cha ac e ized he
embeddabili y es ic ions. We ha e hen examined he mos elemen a y shapes o ze o and
cons an Gaussian cu a u e, as well as localized de o ma ion. These shapes a e seen in
expe imen al obse a ions, as euglenids ound o execu e me aboly. We ha e ound ha
he e is a la ge amily o con inuously accessible shapes, highligh ing he e sa ili y o his
ac ua ion mechanism. Some o hese canonical shapes ha e been also achie ed by o he
me hods o p esc ibe a a ge me ic (Ma de and Papanicolaou,2006;Ama e al.,2012).
By modeling he pellicle as a non-Euclidean pla e, we ha e examined he mechanics o
he euglenoid su ace mo phing s a egy when i is b ough beyond he geome ic singula -
i y o embeddabili y, esul ing in buckling pa e ns ha y o accommoda e he geome ic
us a ion by de eloping con olu ed shapes. We ha e ela ed some o hese con igu a ions
o he shapes adop ed by an anomalous euglenid, wi h a s ong endency o buckling. Be-
yond axisymme ic ac ua ion, we ha e p oposed helical ac ua ion pa e ns o shea , and
p o ided nume ical e idence ha hey can lead o exac isome ic embeddings, i.e. p oduce
non-axisymme ic de o ma ions wi hou s e ch. We ha e shown ha his mechanism can
accomplish la ge ampli ude bending and wis ing o an elonga ed ubula pellicle, as seen in
in i o obse a ions o euglenids, e.g. Figu e 1B. I is also possible o combine he di↵e en
elemen a y de o ma ions examined he e in di↵e en pa s o a ubula pellicle o access a
la ge epe oi e o shapes, which sugges s a possible concep o so obo ic a m o which
he analysis o mo ion planning p oblem would be e y in e es ing.
In he NEP model conside ed he e, he unde lying ma e ial model is iso opic, while i is
clea ha he pellicle is mos likely mechanically aniso opic, see Figu e 2. A mo e ealis ic
ma e ial model, oge he wi h quan i a i e mechanical es ing o euglenids, could p o ide
22
u he in o ma ion abou he mechanics o me aboly, including he o ce gene a ion by
molecula mo o s, o whe he some cell shapes can be a ibu ed o geome ic ins abili ies.
The esul s p esen ed he e could could help in es iga e he biophysical basis o beha io ,
e.g. he link be ween senso y s imuli and ac ua ion, on a single cell minimal model sys em.
Besides helping unde s and he mo ili y o euglenids, hese esul s may p o ide he back-
g ound o man-made so machines abiding by hese p inciples. These could be made o
chi al smec ic C elas ome s (Hi aoka e al.,2005;Adams e al.,2007), which when p epa ed
in hin ilms ha e been shown o p oduce signi ican in-plane shea upon hea ing. Fo his
pu pose, i may be ins uc i e o analyze he possibili ies and limi a ions o di↵e en e e -
ence pellicle con igu a ions, such as la pa ches, o su aces wi h a mo e in ica e pellicle
ex u e.
Acknowledgemen s
We hank F ancisco Pujan e o p o iding mo ies o euglenids. M.A. acknowledges he sup-
po o he Eu opean Resea ch Council (FP7/2007- 2013)/ERC G an Ag eemen n 240487,
and o he Gene ali a de Ca alunya hough he p ize “ICREA Academia” o excellence in
esea ch. This wo k was comple ed while A.D.S. was pa icipa ing in he p og amme “The
Ma hema ics o Liquid C ys als” as a isi ing ellow o he Isaac New on Ins i u e o Ma h-
ema ical Sciences o he Uni e si y o Camb idge, whose hospi ali y and inancial suppo is
g a e ully acknowledged.
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