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Resea ch A icle
Apico-basal in e cala ions enable he in eg i y o cu ed epi helia
Sami a Anba ia,1, Ped o Gómez-Gál ezb,c,e, ,g,1, Pablo Vicen e-Munue ad,,
Luis M. Escude oe, ,g, ,∗, Ja ie Buce ah, ,∗∗
aBiomedical Enginee ing Depa men , Johns Hopkins Uni e si y, Bal imo e, MD 21205, USA
bMRC Labo a o y o Molecula Biology, Camb idge Biomedical Campus, F ancis C ick A e., T umping on, Camb idge CB2 0QH, Camb idgeshi e, UK
cDepa men o Physiology, De elopmen and Neu oscience, Uni e si y o Camb idge, 4 Downing Pl., Camb idge CB2 3EL, Camb idgeshi e, UK
dLabo a o y o Molecula Cell Biology, Uni e si y College London, London, UK
eIns i u o de Biomedicina de Se illa (IBiS), Hospi al Uni e si a io Vi gen del Rocío/CSIC/Uni e sidad de Se illa, 41013 Se ille, Spain
Depa amen o de Biología Celula , Uni e sidad de Se illa, 41013 Se ille, Spain
gBiomedical Ne wo k Resea ch Cen e on Neu odegene a i e Diseases (CIBERNED), Mad id, Spain
hIns i u e o In eg a i e Sys ems Biology (I2SysBio), Theo e ical and Compu a ional Sys ems Biology P og am, CSIC-UV, Pa e na 46980, Spain
A B S T R A C T
Non-in asi e o ce in e ence based on imaging da a has significan ly ad anced ou unde s anding o he mechanical cues d i ing mo phogenesis. In 2D s udies o
confluen issues, hese me hods allow o he compu a ion o o ces ac ing on cells by analyzing hei geome ical ea u es. He e, we p esen a no el app oach o
3D o ce and ene gy in e ence in cu ed epi helia. Specifically, we ocus on ubula epi helia, which o m he ounda ion o many i al o gans, including he lungs,
kidneys, and ascula u e. Ou echnique analyzes he a e age mechanical beha io o cells along hei apico-basal axis and is based on an op imal pa ame iza ion o a
e ex model aimed a ob aining effec i e issue pa ame e s. We apply ou me hod o in silico da a o in es iga e he mechanical consequences o diffe en 3D cellula
packing scena ios. Ou esul s e eal ha in squamous epi helia, p isma ic cellula shapes a e mechanically s able. Howe e , in cubic/columna ubes, p isma ic
shapes a e incompa ible wi h he adhesion equi ed o main ain issue in eg i y. In conclusion, his s udy indica es ha in cubic/columna epi helia, s abili y can
only be achie ed i cells unde go apico-basal in e cala ions and adop an al e na i e shape: he scu oid.
1. In oduc ion
Du ing animal de elopmen , mul iple ac o s such as cell-cell in e -
ac ions, cell p oli e a ion, cell shape changes, and biophysical o ces
con ibu e o o gan o ma ion [1–3]. In confluen issues such as epi he-
lia, hese p ocesses de e mine cell packing p ope ies, and hei analysis
has significan ly ad anced ou unde s anding o mo phogenesis [4,5].
T adi ionally, mos s udies ha e been conduc ed in wo dimensions (2D)
by analyzing he apical su aces o monolaye epi helia. Howe e , ecen
esea ch has unde sco ed he need o in es iga e 3D cellula packing and
mo phology [6]. In his con ex , epi helial cells a e ypically classified
by hei wid h- o-heigh aspec a io as squamous (“fla ”), cuboidal, o
columna (“ all”) cells [7,8]. No ably, he ole o diffe en 3D cell ge-
ome ies, in conjunc ion wi h hei packing p ope ies, in main aining
issue in eg i y emains insufficien ly explo ed.
The impo ance o accu a ely analyzing 3D cell-cell con ac s has
been highligh ed by ecen s udies on a ious biological p ocesses, in-
*Co esponding au ho a : Ins i u o de Biomedicina de Se illa (IBiS), Hospi al Uni e si a io Vi gen del Rocío/CSIC/Uni e sidad de Se illa, 41013 Se ille, Spain.
** Co esponding au ho a : Ins i u e o In eg a i e Sys ems Biology (I2SysBio), CSIC-UV, 46980 Pa e na, Spain.
E-mail add esses: [email p o ec ed] (L.M. Escude o), [email p o ec ed] (J. Buce a).
1Equally con ibu ed.
cluding he g ow h o mouse emb yonic lung explan s [9], he ea ly
de elopmen o C. elegans [10] and ascidians [11], and he cellula
and mechanical basis o sel -o ganiza ion in cu ed issues wi hin a
confined geome y [12]. A majo b eak h ough in unde s anding ep-
i helial 3D packing and mo phogenesis came wi h he disco e y o a
no el geome ical shape, he scu oid, which plays a key ole in epi helial
mo phogenesis [6,13–20]. Scu oids, cha ac e ized by apico-basal in e -
cala ions o cells, acili a e epi helial packing when issues a e subjec ed
o cu a u e [14] and a e conside ed a gene al biophysical phenomenon
[21]. Addi ionally, scu oids ha e been shown o modula e local p essu e
inc eases due o cell di ision [22]. This cell shape has been iden i-
fied in epi helial issues ac oss a ious me azoans, including mammals
[6,19,22]. Howe e , key open ques ions emain: (1) how do apico-basal
in e cala ions con ibu e o he mechanical s abili y and in eg i y o an
epi helium? and (2) can a cu ed issue main ain i s in eg i y wi h-
ou scu oids? Expe imen ally add essing hese ques ions by emo ing
o modi ying apico-basal in e cala ions is cu en ly un easible. Thus, a
h ps://doi.o g/10.1016/j.csbj.2025.03.011
Recei ed 8 Janua y 2025; Recei ed in e ised o m 6 Ma ch 2025; Accep ed 8 Ma ch 2025
Compu a ional and S uc u al Bio echnology Jou nal 27 (2025) 1204–1214
1205
S. Anba i, P. Gómez-Gál ez, P. Vicen e-Munue a e al.
compu a ional app oach is equi ed o in e mechanical o ces in di -
e en 3D cellula packing configu a ions. Specifically, a me hodology
is needed o fi s simula e epi helial issues subjec ed o cu a u e and
composed o cuboidal/columna cells wi h ei he p isma ic o scu oidal
packing configu a ions, and second, o de elop a o ce/ene gy in e ence
echnique o analyze and pa ame e ize he esul s o hese simula ions.
A wide ange o expe imen al echniques a e a ailable o mea-
su e o es ima e o ces in cells and issues, including lase abla ion
[23–27], unc ionalized d ople s [28–30], op ical and magne ic weez-
e s [31–34], molecula senso s [35], and ac ion mic oscopy [36],
among o he s [37]. Al e na i ely, in epi helial monolaye s, in e ence
me hods p o ide non-in asi e means o es ima ing o ces using imag-
ing da a [38–46]. These echniques le e age he ac ha he apical
su aces o cells in hese issues exhibi polygonal-like shapes, allowing
hei geome ical ea u es o be co ela ed wi h mechanical equilib ium
condi ions. This polygonal-like packing cha ac e is ic led o he de el-
opmen o he e ex model, which aims o desc ibe he o ces ac ing
on epi helial cells in bo h 2D and 3D en i onmen s [47–51]. In a 3D
con ex , only ecen ly ha e some s udies a emp ed o quan i y o ces
( ensions and p essu es) using Young’s o mula [52,53]. No ably, o
da e, no o ce in e ence s udy has e alua ed he impac o epi helial
cell shape on he mechanical s abili y o issues.
He e, we implemen a no el app oach o 3D o ce in e ence o exam-
ine he ole o cellula shapes and hei packing in epi helial issues. Con-
en ional in e ence me hods ely on o ce balance wi hou inco po a ing
assump ions abou cell mechanics, such as elas ici y o adhesion. In con-
as , ou app oach assumes he alidi y o he e ex model’s desc ip-
ion o cellula mechanics and in e s he pa ame e alues ha sa is y
o ce balance. Fu he mo e, we employ a s a is ical me hod ha quan i-
fies he a e age cell beha io in e ms o ene gy and o ce componen s,
p o iding insigh s in o he ele ance o 3D cellula shapes. Specifically,
we aim o de e mine he biophysical pa ame e s ha bes desc ibe he
mechanical p ope ies and s abili y o issues depending on packing con-
figu a ions. We es ou in e ence me hod on simula ed ubula issues.
These epi helia o m he ounda ion o many i al o gans, including
he lungs, kidneys, and ascula u e, and ha e been shown o exhibi
he highes p e alence o scu oidal cells due o cu a u e effec s com-
pa ed o o he epi helial geome ies [14] (see Discussion). We gene -
a ed ubula epi helia h ough a compu a ional geome y app oach ha
p oduces wo possible cellula shapes: scu oidal shapes, which exhibi
apico-basal in e cala ions, and p isma ic-like shapes, whe e apico-basal
in e cala ions a e absen . Addi ionally, we compa e hese wo scena -
ios in diffe en issue ypes by employing compu a ional ubula models
composed o ei he squamous o cuboidal/columna cells. The biophysi-
cal pa ame iza ion o hese models e eals ha issue in eg i y depends
on a balance be ween packing a chi ec u e and he wid h- o-heigh as-
pec a io o cells. Ou findings sugges ha he adhesi e p ope ies
cha ac e is ic o epi helia equi e he p esence o apico-basal in e ca-
la ions o s abilize ubula s uc u es composed o cuboidal/columna
cells. In o he wo ds, he physical p ope ies o epi helial issues p e en
he exis ence o cuboidal/columna epi helial ubes composed solely o
p isma ic-like shapes.
2. Me hods
2.1. Ve ex model: o ce balance
Ou o ce in e ence me hodology is based on a pa ame iza ion o
he e ex model [54] and assumes he s anda d ene gy unc ional o
each o he e exes, 𝑖, ha define he polygonal-like shape o a cell a
a gi en plana su ace:
𝐸𝑖=∑
𝛼
𝐾𝛼
2 (𝐴𝛼−𝐴0
𝛼(𝑡))2+∑
𝛼
Γ𝛼
2
𝐿2
𝛼+∑
⟨𝑖𝑗⟩
Λ𝑖𝑗𝑙𝑖𝑗
=𝐸elas ic
𝑖+𝐸con ac .
𝑖+𝐸adh.
𝑖
(1)
whe e he sums indexed by 𝛼and ⟨𝑖𝑗⟩ un, espec i ely, o e he cells,
𝛼, and he e exes, 𝑗, ha sha e e ex 𝑖. The fi s e m, 𝐸elas ic
𝑖=
∑𝛼
𝐾𝛼
2 (𝐴𝛼−𝐴0
𝛼(𝑡))2accoun s o he elas ic ene gy o cells (𝐾𝛼be-
ing p opo ional o he Young modulus) due o he diffe ence be ween
he ac ual cell a ea, 𝐴𝛼, and he a ge a ea 𝐴0
𝛼(𝑡). The second e m,
𝐸con ac .
𝑖=∑𝛼
Γ𝛼
2
𝐿2
𝛼, models con ibu ions om he ension associa ed
o he con ac ion ac i i y o he ac omyosin co ical ing, 𝐿𝛼being he
cell pe ime e . Finally, he hi d e m, 𝐸adh.
𝑖=∑⟨𝑖𝑗⟩Λ𝑖𝑗𝑙𝑖𝑗 , ep esen s
he line ension (adhesion o ce), wi h 𝑙𝑖𝑗 being he leng h o he edge
connec ing neighbo ing e exes 𝑖and 𝑗.
By neglec ing ine ia (low Reynolds numbe ) and including dissipa-
ion, he ollowing o ce balance equa ion holds,
𝟎=−∇𝐸𝑖−𝛾
𝐫𝑖
whe e 𝐫𝑖is he posi ion ec o o e ex 𝑖and 𝛾is he coefficien ha
de e mines he cha ac e is ic ime scale linked o he dissipa ion o he
mechanical ene gy, 𝑡𝑐=𝛾∕(𝐾𝛼𝐴0
𝛼). Fu he , i he cha ac e is ic ime
scale o cell a ea changes, ei he due o g ow h o mechanical inpu s, is
slow compa ed o 𝑡𝑐(o equi alen ly i he issue is a a s eady s a e),
he balance o he conse a i e o ces a each cell e ex de e mines he
equilib ium condi ion:
𝟎≃−∇𝐸𝑖=𝐅elas ic
𝑖+𝐅con ac .
𝑖+𝐅adh.
𝑖(2)
In he Euclidian plane, he a ea, 𝐴𝛼, and he pe ime e , 𝐿𝛼, o a
polygon (i.e., cell 𝛼) wi h 𝑛clockwise-o de ed e exes a e gi en by:
𝐴𝛼=−1
2
𝑛
∑
𝑘=1 (𝑥𝛼
𝑘𝑦𝛼
𝑘+1 −𝑥𝛼
𝑘+1𝑦𝛼
𝑘)
𝐿𝛼=
𝑛
∑
𝑘=1
𝑙𝛼
𝑘,𝑘+1 =
𝑛
∑
𝑘=1 √(𝑥𝛼
𝑘+1 −𝑥𝛼
𝑘)2
+(𝑦𝛼
𝑘+1 −𝑦𝛼
𝑘)2
whe e 𝐫𝛼
𝑖=(𝑥𝛼
𝑖,𝑦𝛼
𝑖) ep esen s he Ca esian coo dina es o e ex 𝑖o
cell 𝛼, and he ollowing pe iodic bounda y condi ions o he polygon
ha desc ibe a cell apply: 𝐫𝛼
𝑛+1 =𝐫𝛼
1and 𝐫𝛼
0=𝐫𝛼
𝑛. Thus, he diffe en
o ce e ms in Eq. (2) ead,
𝐅elas ic
𝑖=− 𝜕
𝜕𝐫𝑖∑
𝛼
𝐾𝛼
2 (𝐴𝛼−𝐴0
𝛼(𝑡))2
=1
2∑
𝛼
𝐾𝛼(𝐴𝛼−𝐴0
𝛼(𝑡))(𝑦𝛼
𝑖+1 −𝑦𝛼
𝑖−1,𝑥𝛼
𝑖−1 −𝑥𝛼
𝑖+1)
𝐅con ac .
𝑖=− 𝜕
𝜕𝐫𝑖∑
𝛼
Γ𝛼
2
𝐿2
𝛼
=−∑
𝛼
Γ𝛼𝐿𝛼(𝑥𝛼
𝑖−𝑥𝛼
𝑖−1
𝑙𝛼
𝑖−1,𝑖
+
𝑥𝛼
𝑖−𝑥𝛼
𝑖+1
𝑙𝛼
𝑖,𝑖+1
,𝑦𝛼
𝑖−𝑦𝛼
𝑖−1
𝑙𝛼
𝑖−1,𝑖
+
𝑦𝛼
𝑖−𝑦𝛼
𝑖+1
𝑙𝛼
𝑖,𝑖+1 )
𝐅adh.
𝑖=− 𝜕
𝜕𝐫𝑖∑
⟨𝑖𝑗⟩
Λ𝑖𝑗𝑙𝑖𝑗
=−∑
⟨𝑖𝑗⟩
Λ𝑖𝑗 (𝑥𝑗−𝑥𝑖
𝑙𝑖𝑗
,𝑦𝑗−𝑦𝑖
𝑙𝑖𝑗 )
2.2. Geome ical decomposi ion o o ces: es ima ion o no mal and shea
s esses
In o de o p o ide an in ui i e in e p e a ion o he di ec ionali y
o he o ces, we implemen a geome ical decomposi ion ha es ima es
no mal and shea s esses. Since o ces in a e ex model a e applied
a cell e exes, he no mal and pe pendicula di ec ions a e no well-
defined. To add ess his, we define mock no mal and shea o ces based
on he angle o med by he edges adjacen o a e ex (Fig. 1A). Fo
a gi en o ce ype applied o a e ex 𝑖, 𝐅𝑖=(𝐹𝑖,𝑥,𝐹𝑖,𝑦), we compu e
he no mal and shea componen s, 𝐅′
𝑖=(𝐹𝑖,n,𝐹𝑖,s)by using he ge-
ome ic o a ional ans o ma ion 𝐅′
𝑖=𝑅𝜃𝐅𝑖whe e 𝜃is defined as a
Compu a ional and S uc u al Bio echnology Jou nal 27 (2025) 1204–1214
1206
S. Anba i, P. Gómez-Gál ez, P. Vicen e-Munue a e al.
Fig. 1. A: No mal and shea o ces. To es ima e he no mal and shea componen s o a o ce 𝐅𝑖ac ing upon e ex 𝑖, we implemen a geome ical decomposi ion
ha defines he no mal di ec ion along he bisec o o he angle 𝛼 o med by he adjacen edges 𝐫𝑖+1 −𝐫𝑖and 𝐫𝑖−1 −𝐫𝑖(a o a ion o an angle 𝜃wi h espec o he
ex e nal coo dina e sys em). B: Simula ion o Vo onoi and us a ubes. (Le ) Side iew o a ube. F om he apical (inne ) adius, 𝑅𝑎, o he basal (ou e ) adius,
𝑅𝑏, he simula ion app oach is based on he adial p ojec ion o geome ical ea u es o he cells (a ow). (Righ ) In Vo onoi ubes, he cell seeds a e p ojec ed, while
in us a ubes, he cell e exes a e p ojec ed (see ex ). C: Simula ion Examples. In ou in silico expe imen s we conside ubes ha mimic squamous (𝑠𝑏≃1.5)
and cuboidal/columna (𝑠𝑏=4) cells ha pack ollowing ei he a us a-like (Le ) o a scu oidal-like (Righ ) geome y. The g een and ed ci cles indica e he apical
(inne ) and basal (ou e ) adii, espec i ely. The shapes and packing o highligh ed cells (whi e con ou s in ubes) a e shown a diffe en apico-basal coo dina es
( om bo om o op, 𝑠=1,2,4). Since each ealiza ion o us a and Vo onoi ubes was gene a ed om he same ini ial cell dis ibu ion a he apical su ace, 𝑠=1
(apical su ace), he cells ha e iden ical shapes in bo h cases. No ice ha in Vo onoi ubes, a he apical su ace, he ed and yellow cells a e nea es neighbo s while
he g een and blue cells a e no , whe eas a he basal su ace, he nea es -neighbo ela ionship is e e sed (blue and g een cells become nea es neighbo s while
ed and yellow a e no ). In us a ubes, he neighbo ing ela ionships among cells emain unchanged along he apico-basal axis as in e cala ions a e p ecluded. D:
Quan ifica ion o cellula packing. 3D his og ams o cells neighbo s, 𝑃(𝑛𝑎,𝑛𝑏), o Vo onoi (𝑠𝑏=1.5, Cen e ; 𝑠𝑏=4, Righ ) and us a (Le ) ubes, sample size o
10 ubes. The absence o apico-basal in e cala ions in us a ubes esul s in iden ical polygonal dis ibu ions a he apical and basal su aces. In Vo onoi ubes, as he
su ace a io inc eases, he dis ibu ion widens, indica ing an inc ease in he numbe o apico-basal in e cala ions (scu oid-like cell shapes). E: Tubula geome y:
o ce in e ence. Gi en an epi helial ube o leng h 𝐿and apical and basal adii 𝑅𝑎and 𝑅𝑏, ou omog aphic app oach o o ce in e ence assumes equilib ium is
achie ed a each su ace o cons an adius be ween 𝑅𝑎 o 𝑅𝑏. F: Elas ic null-s ess plane. In a plana epi helial monolaye cells acqui e a p ism-like shape wi h
a cha ac e is ic a ea in each plane om apical o basal 𝐴0(g ey mesh). When he issue is subjec ed o bending ( ubula geome y) he cells de o m and acqui e a
us um-like shape. Cell olume conse a ion implies ha cells a e unde comp ession in he apical plane and unde ension in he basal plane. The elas ic null-s ess
plane is defined by he alue o he apico-basal coo dina e o which he a ea in ha plane is 𝐴0.
unc ion o he bisec o o he angle o med by he edges (𝐫𝛼
𝑖+1 −𝐫𝛼
𝑖)
and
(𝐫𝛼
𝑖−1 −𝐫𝛼
𝑖)o e ex 𝑖:
(𝐹𝑖,n
𝐹𝑖,s)=[𝑐𝑜𝑠(𝜃)𝑠𝑖𝑛(𝜃)
−𝑠𝑖𝑛(𝜃)𝑐𝑜𝑠(𝜃)](𝐹𝑖,𝑥
𝐹𝑖,𝑦)(3)
Finally, he magni ude o he no mal/shea o ces ac ing on each cell
is calcula ed by a e aging he magni udes o he no mal/shea o ces
applied o all i s e exes, 𝑀, such ha 𝐹n=1
𝑀∑𝑀
𝑖=1 𝐹𝑖,nand 𝐹s=
1
𝑀∑𝑀
𝑖=1 𝐹𝑖,s. The a e age cellula no mal and shea o ces a e hen com-
pu ed by a e aging o e all cells, 𝑁: ⟨𝐹𝑛⟩=1
𝑁∑𝐹nand ⟨𝐹𝑠⟩=1
𝑁∑𝐹𝑠.
2.3. Simula ions o Vo onoi and us a ubula models
An in-house Ma lab code was used o model ubula epi helia ol-
lowing wo diffe en app oaches ha lead o dis inc cellula packing
configu a ions: Vo onoi and us a ubes. Vo onoi ubes we e gene a ed
ollowing a simila app oach as desc ibed in [14]. The inne (apical) su -
ace o he ubes (i.e., hollow cylinde s) was popula ed wi h a numbe o
seeds (i.e., poin s) andomly dis ibu ed. Subsequen ly, he Vo onoi al-
go i hm was applied o he seeds o gene a e he co esponding Vo onoi
cells. To mimic he ac ual apical o ganiza ion o epi helial ubes (see
[6,15]), he Lloyd algo i hm was i e a i ely applied 7 imes [55]. The
Lloyd algo i hm upda es he loca ion o he Vo onoi seeds by eloca -
ing hem o he cen e o mass o each Vo onoi cell; subsequen ly, he
Vo onoi algo i hm is eapplied o he upda ed Vo onoi seeds, esul ing
in a mo e homogeneous Vo onoi essella ion in e ms o cellula size.
Once he 2D Vo onoi cells o he apical su ace we e gene a ed, he
3D cellula shape was buil as ollows. The ou e (basal) su ace o he
ubes was gene a ed wi h a adius 𝑅𝑏=𝑠𝑏𝑅𝑎and he Vo onoi cell seeds
we e adially p ojec ed om he inne o he ou e su ace (Fig. 1B).
A each in e media e adius, 𝑅∈[𝑅𝑎,𝑅𝑏], he in e sec ion o he adial
p ojec ions wi h he adial su ace (su ace wi h cons an adius) defines
he loca ion o he Vo onoi seeds, and he esul ing Vo onoi essella ion
ende s he shape o he cells a ha su ace. Fo a gi en seed, he se
o 2D Vo onoi cells ac oss all adial su aces, om 𝑅𝑎 o 𝑅𝑏, defines he
3D cellula shape (Fig. 1B).
Fo us a ubes, we p oceed as p e iously desc ibed o define he
apical shape o he cells. Howe e , ins ead o p ojec ing he cell seeds,
we p ojec ed he cell e exes (Fig. 1B). Thus, he cell shape o a gi en
cell in a pa icula adial su ace is desc ibed by he in e sec ion o he
p ojec ed e exes. To c ea e he 3D shape o an indi idual cell, we
p oceed as p e iously desc ibed: combining he se o 2D cells ac oss
all adial su aces, om 𝑅𝑎 o 𝑅𝑏. Fig. 1C shows examples o ubula
epi helia simula ed using hese al e na i e app oaches. Ou esul s a e
ypically based on he analysis o 10 ealiza ions o each ubula config-
u a ion wi h 𝑁= 200 cells and he ollowing p ope ies: 𝑠𝑏=𝑅𝑏∕𝑅𝑎=
1.46875 ≃ 1.5( e e ed o as squamous cells), 𝑠𝑏=4 ( e e ed o as
cuboidal/columna cells), see also Table 1.
The exis ence and effec o apico-basal in e cala ions in Vo onoi
ubes is shown by he 3D his og am o cell neighbo s in Fig. 1D, i.e.,
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Table 1
Cellula packing p ope ies in us a and Vo onoi ubes. A e age numbe
o neighbo s (in-plane polygonal o de , ⟨𝑛⟩), s anda d de ia ion (𝜎𝑛), and coe -
ficien o a ia ion (𝐶𝑉 ) in he apical and basal su aces o us a and Vo onoi
ubes. Apico-basal in e cala ions lead o g ea e diso de in he polygonal dis-
ibu ion o cuboidal/columna cells.
F us a ubes Vo onoi ubes
Apical /Basal Apical (𝑠=1) Basal (𝑠≃1.5) Basal (𝑠=4)
⟨𝑛⟩6.0 6.0 6.0 6.0
𝜎𝑛0.8 0.8 0.8 1
𝐶𝑉 =𝜎𝑛
⟨𝑛⟩0.13 0.13 0.13 0.17
he p obabili y 𝑃(𝑛𝑎,𝑛𝑏)o cells ha ing 𝑛𝑎neighbo s in he apical
su ace and 𝑛𝑏neighbo s in he basal su ace [15]. In us a ubes,
𝑃(𝑛𝑎,𝑛𝑏)=𝑃(𝑛𝑎)𝛿𝑛𝑏,𝑛𝑎=𝑃(𝑛𝑏)𝛿𝑛𝑎,𝑛𝑏(𝛿𝑖,𝑗 being he K onecke del a).
Tha is, he his og am only con ains en ies along he diagonal since
he e a e no changes in he nea es -neighbo ela ionship om apical o
basal. Howe e , in Vo onoi ubes, he en ies o he 3D his og am ou -
side he diagonal indica e apico-basal in e cala ions: some cells ei he
gained o los neighbo s be ween he apical and basal su aces.
2.4. Dimensionless uni s
In e ms o he cha ac e is ic leng h scale based on he cellula size,
𝑙𝑐=√⟨𝛼𝑎⟩, whe e ⟨𝛼𝑎⟩is he a e age a ea o cells a he apical
su ace (⟨𝛼𝑎⟩=⟨𝛼0⟩when conside ing only 2D cellula packing,
as in con ol simula ions; see Appendix), and he cha ac e is ic elax-
a ion/dissipa ion ime o he mechanical ene gy, 𝑡𝑐=𝛾∕(𝐾𝛼⟨𝑎
𝛼⟩), we
implemen a dimensionless o m o Eq. (1).
Using hese defini ions, he dimensionless elas ic pa ame e is se as
𝐾𝛼=1, while he adhesion and con ac ili y pa ame e s a e gi en by
Λ𝑖𝑗 =Λ𝑖𝑗∕(𝐾𝛼⟨𝛼𝑎⟩3∕2)and
Γ𝛼=Γ
𝛼∕(𝐾𝛼⟨𝑎
𝛼⟩), espec i ely.
Fu he mo e, when applying he omog aphic app oach owa d o ce
in e ence (see below), he dimensionless a ge cell a ea in he elas ic
null-s ess su ace (see below) is gi en by
𝛼0=𝛼0∕⟨𝛼𝑎⟩. Thus,
we pe o m an addi ional change o uni s such ha
𝛼0=1 (i.e.,
𝑙′
𝑐=√
𝛼0), leading o
𝐾𝛼 =1
,
Λ𝑖𝑗 =
Λ𝑖𝑗∕(
𝐴0
𝛼
3∕2), and
Γ𝛼=
Γ𝛼∕
𝐴0
𝛼.
Fo he sake o simplici y, we assume ha all quan i ies a e exp essed
in dimensionless uni s and omi he single and double o e -ha symbols.
2.5. Linea p og amming: omog aphic o ce in e ence
Dis ega ding bounda y effec s, in a issue comp ising 𝑁cells, he
equilib ium condi ion, Eq. (2), p o ides 2×2𝑁scala equa ions (on
a e age, each cell has six e exes sha ed by h ee cells) a any gi en
ime. Rega ding he numbe o unknowns, in a issue composed o cells
o he same ype (i.e., wi h supposedly iden ical mechanical p ope ies),
he numbe is jus ou (𝐴0, 𝐾, Γ, and Λ). Mo eo e , hese pa ame e s
can be educed o h ee (see Dimensionless uni s abo e), making he
sys em o equa ions in Eq. (2) o e de e mined and sol able.
Gi en ha he esul ing o ce balance equa ions desc ibed by Eq. (2)
a e linea in he unknowns (i.e., he pa ame e s Λ, Γ, e c.), we sol e
he o e de e mined sys em using a linea p og amming op imiza-
ion app oach and employ an 𝑙1-minimiza ion me hod [56]. The 𝑙1-
minimiza ion aims o find he minimum 𝑙1-no m solu ion o a linea
sys em 𝔸𝐗=𝔹, whe e 𝐗∈ℝ𝑚(i.e., 𝑚unknowns), 𝔹∈ℝ𝑝(𝑚<𝑝,
whe e 𝑝 ep esen s he o al numbe o e exes), and 𝔸∈ℝ𝑝×𝑚[57].
In ou s udy, we implemen he “L1-No m Minimiza ion” unc ion and
he “simplex” algo i hm in Wol am Ma hema ica [58] o de e mine he
op imal alues o he unknowns.
The omog aphic o ce in e ence app oach o 3D ubula issues
(Fig. 1E) assumes equilib ium a e e y alue o he apico-basal coo -
dina e (i.e., adial su ace) and sol es Eq. (2) simul aneously o all su -
aces, om 𝑅𝑎 o 𝑅𝑏. The adhesion pa ame e (ene gy pe uni leng h)
and 𝐴0( a ge cell a ea in he elas ic null-s ess plane, see below) a e
assumed o be he same o all alues along he apico-basal coo dina e,
while he con ac ili y pa ame e , Γ, is allowed o a y along he apico-
basal axis.
Thus, in a 3D con ex , he numbe o equa ions and unknowns a e
gi en by 2×2𝑁×𝑁su aces and 2+𝑁su aces (i.e., 𝐴0+Λ+𝑁su aces ×Γ),
espec i ely, whe e 𝑁su aces is he numbe o apico-basal su aces (i.e.,
adial “slices”) conside ed.
In he analyses p esen ed in his s udy o squamous and cuboidal/
columna issues, 𝑁su aces is 6and 7, espec i ely. As shown in he Ap-
pendix, ou con ol simula ions indica e ha he esul s emain obus
o diffe en alues o 𝑁su aces.
2.6. Elas ic null-s ess plane: ene gy as a unc ion o he apico-basal
coo dina e
Gi en an epi helial monolaye wi h 𝑁cells and issue dep h (i.e.,
cell heigh ) ℎ, i he issue is in a plana configu a ion (lacking ension-
comp ession s esses due o issue bending/cu a u e), hen he a e age
cell olume is ⟨𝑉⟩=⟨𝐴0⟩ℎ, whe e ⟨𝐴0⟩is he a ge a ea (Fig. 1F). On
he o he hand, i he same issue is shaped in o a ubula configu a ion,
wi h leng h 𝐿and apical and basal adii 𝑅𝑎and 𝑅𝑏, espec i ely, hen
he a e age cell olume is gi en by
⟨𝑉⟩=𝜋𝐿
𝑁(𝑅2
𝑏−𝑅2
𝑎).
I he cell heigh emains cons an , 𝑅𝑏−𝑅𝑎=ℎ, and he cell olume
is conse ed, i ollows ha
⟨𝐴0⟩=𝜋𝐿(𝑅𝑏+𝑅𝑎)
𝑁=𝑅𝑎𝜋𝐿(𝑠𝑏+1
)
𝑁=1
2⟨𝐴𝑎⟩(𝑠𝑏+1
),
whe e ⟨𝐴𝑎⟩is he cell apical a ea, and we define he dimensionless
apico-basal coo dina e 𝑠=𝑅∕𝑅𝑎, e e ed o as he su ace a io (𝑠𝑏=
𝑅𝑏∕𝑅𝑎).
The a e age elas ic ene gy o a gi en su ace a io eads:
⟨𝐸⟩𝐴=𝐾
2 ⟨(𝐴−𝐴0)2⟩=𝐸⟨𝐴⟩+𝐾
2
𝜎2
𝐴,
whe e 𝐸⟨𝐴⟩=𝐾
2 (⟨𝐴⟩−⟨𝐴0⟩)2is he elas ic ene gy o a cell wi h a e -
age a ea ⟨𝐴⟩, and 𝜎2
𝐴=⟨𝐴2⟩−⟨𝐴⟩2is he cellula a ea a iance. Cell
numbe conse a ion implies ha o a gi en su ace a io:
⟨𝐴⟩=2𝜋𝑅𝐿
𝑁=⟨𝐴𝑎⟩𝑠.
Consequen ly,
𝐸⟨𝐴⟩=𝐾⟨𝐴𝑎⟩2
2 (𝑠−1
2(𝑠𝑏+1
))2
.
The elas ic null-s ess plane, whe e 𝐸⟨𝐴⟩=0, is loca ed a :
𝑠∗=1
2(𝑠𝑏+1
)=⟨𝐴0⟩
⟨𝐴𝑎⟩.
F om he pe spec i e o plana elas ic de o ma ion, i 1≤𝑠<𝑠
∗, cells
a e unde comp ession, whe eas i 𝑠𝑏≥𝑠>𝑠
∗, cells a e unde ension.
Fu he mo e, since ⟨𝐿⟩∼⟨𝐴⟩1∕2 =(⟨𝐴𝑎⟩𝑠)1∕2, he a e age adhesion
and con ac ile ene gies, ⟨𝐸⟩𝐿and ⟨𝐸⟩𝐿2, espec i ely, scale as a unc-
ion o 𝑠as:
⟨𝐸⟩𝐿=𝐸⟨𝐿⟩=Λ⟨𝐿⟩∼Λ⟨𝐴𝑎⟩𝑠1∕2,
⟨𝐸⟩𝐿2=Γ
2
⟨𝐿2⟩=𝐸⟨𝐿2⟩+Γ
2
𝜎2
𝐿∼Γ
2
𝑠.
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Fig. 2. Con ac ili y in us a and Vo onoi ubes depending on he cellula
packing geome ies. Con ac ili y pa ame e as a unc ion o he su ace a io
(apico-basal coo dina e) as es ima ed by he in e ence me hod o us a (cyan)
and Vo onoi (yellow) geome ies in squamous (le ) and cuboidal/columna
( igh ) cells. The e o band accoun s o he s anda d de ia ion (10 samples).
3. Resul s
3.1. Fo ce in e ence e eals mechanical diffe ences be ween cell geome ies
We de eloped and implemen ed a o ce in e ence app oach based
on a e ex model pa ame iza ion (Me hods). Using con ol simula-
ions, we showed ha he p oposed in e ence me hodology is obus
o fini e-size and bounda y effec s and is also capable o cap u ing ime-
dependen mechanical pa ame e s accu a ely as long as hei empo al
a ia ion is slow compa ed o he ene gy dissipa ion ime scale (Ap-
pendix). We le e aged his la e ac o implemen a omog aphic ap-
p oach (i.e., plane by plane om he apical o he basal su ace) o 3D
o ce in e ence by exchanging he concep s o space and ime. This ool
was applied o in es iga e he mechanical cha ac e iza ion o epi helial
ubes based on hei cellula packing o ganiza ion and mo phological
p ope ies (Me hods). Specifically, we simula ed ubes composed o ei-
he cuboidal/columna o squamous epi helial cells and a ied hei
packing such ha apico-basal in e cala ions we e ei he allowed (scu-
oidal cell geome ies) o p ecluded ( us a cell geome ies).
We used dimensionless uni s such ha 𝐴0=1a he null-s ess plane
(𝑠∗=(𝑠𝑏+1)∕2) and assumed ha he pa ame e Λ(ene gy “cos ” pe
uni leng h) emains cons an along he apico-basal coo dina e, 𝑠(Me h-
ods). Howe e , we allowed he cell con ac ili y pa ame e o a y along
he apico-basal axis, i.e., Γ(𝑠). This assumes ha while cells adhe e o
each o he wi h he same “in ensi y,” co ical ac i i y may change as a
unc ion o he apico-basal coo dina e [59]. Fig. 2shows he es ima ed
con ac ili y pa ame e as a unc ion o he apico-basal coo dina e, 𝑠, o
us a and Vo onoi ubes using ei he squamous o cuboidal/columna
cells.
Fo a gi en ubula configu a ion (ei he squamous o cuboidal/colum-
na ), bo h cellula geome ies ( us a o Vo onoi) yielded simila con-
ac ili y alues and ends: Γdec eases as 𝑠inc eases (i.e., om apical
o basal) and app oaches ze o a he basal su ace. Howe e , diffe -
ences eme ged be ween ubula configu a ions, as cuboidal/columna
cells exhibi ed highe con ac ili y alues. In his ega d, we no e ha
he alue o ⟨
𝐴0
𝛼⟩diffe s be ween squamous and cuboidal/columna
cells (see Me hods, Dimensionless uni s):
⟨
𝐴0
𝛼⟩||||𝑠𝑏=4
⟨
𝐴0
𝛼⟩||||𝑠𝑏=1.5
≃2.
Consequen ly, using he same dimensionless uni s o cuboidal/colum-
na and squamous cells, he maximum con ac ili y pa ame e (apical
su ace) is app oxima ely ou imes la ge in cuboidal/columna cells
han in squamous cells.
Rega ding he line ension (adhesi eness), squamous cells displayed
simila alues o Vo onoi and us a geome ies:
ΛVo onoi =0.04 ± 0.01, Λ us a =0.03 ± 0.01.
Howe e , cuboidal/columna cells showed la ge diffe ences be ween
packing shapes:
ΛVo onoi =0.023 ± 0.009, Λ us a =0.0023 ± 0.002.
Fo he same dimensionless uni s, he adhesion pa ame e in Vo onoi
cuboidal/columna cells is ac ually la ge han in squamous cells (by
a ac o o ∼1.85), whe eas us a cuboidal/columna cells exhibi ed
significan ly lowe adhesion alues (∼0.185 o he alue obse ed in
squamous cells).
In summa y, he in e ence me hod e ealed ha co ical ac i i y,
as cha ac e ized by he con ac ili y pa ame e , is simila in us a and
Vo onoi ubes bu significan ly g ea e in cuboidal/columna cells com-
pa ed o squamous cells. In con as , o he line ension pa ame e ,
scu oid- ee ( us a) cuboidal/columna cells exhibi ed mechanical s a-
bili y ha depended on ex emely low adhesion alues —an o de o
magni ude smalle han ei he Vo onoi o us a cells in squamous ubes
o Vo onoi cells in cuboidal/columna ubes.
3.2. Ene gy p ofiles e eal he ole played by apico-basal in e cala ions in
cuboidal/columna epi helia
Once he mechanical pa ame e s o he ubula models we e cali-
b a ed, we compu ed he a e age cellula ene gy p ofiles (Me hods),
Fig. 3. These p ofiles p o ide a “map” om he apical o he basal
su ace, ep esen ing he cha ac e is ic alues o diffe en ene gy com-
ponen s and illus a ing he effec o apico-basal in e cala ions (cellu-
la geome y) in squamous (Fig. 3A-B) and cuboidal/columna cells
(Fig. 3C-D).
Fo squamous cells, we ob ained simila ene gy p ofiles o us a and
Vo onoi geome ies. Tha is, in squamous epi helia, modi ying he cellu-
la geome y om us a o Vo onoi —and consequen ly al e ing cellula
connec i i y [15]— does no con e an ene ge ic ad an age. We p opose
ha his could explain why he scu oidal shape has no ye been epo ed
in squamous cells (Discussion). Addi ionally, he unc ional beha io o
he diffe en ene gy componen s aligns wi h he heo e ical expec a ions
(Me hods). In pa icula , he elas ic ene gy eaches a minimum a he
apico-basal coo dina e:
𝑠∗=1
2(𝑠𝑏+1)= ⟨𝐴0⟩
⟨𝐴𝑎⟩,
which defines he elas ic null-s ess plane.
Fo he cuboidal/columna ubula model, he unc ional beha io
o he ene gy componen s as a unc ion o 𝑠is also in ag eemen wi h
he heo e ical expec a ions. Howe e , we obse ed significan quan i a-
i e and quali a i e diffe ences be ween packing geome ies, o igina ing
om he dissimila alues o he line- ension pa ame e . In us a ge-
ome ies, elas ic ene gy domina es om he apical o he basal su ace,
whe eas in Vo onoi cells, adhesion becomes dominan a ound he elas-
ic null-s ess plane. Addi ionally, ega ding o al ene gy, he a e age
cellula ene gy is highe in Vo onoi han in us a ubes. Howe e , his
obse a ion mus be conside ed in he con ex o issue s uc u al s a-
bili y. The low adhesi eness equi ed o balance mechanical o ces in
us a-columna ubes is a guably incompa ible wi h he unc ionali y
o eal issues (Discussion).
3.3. Apico-basal in e cala ions buffe shea s esses in cuboidal/columna
epi helia
We u he analyzed he a e age cell o ce p ofile o elucida e how
apico-basal in e cala ions shape he s abili y o he ubes. To ha end,
we compu ed he diffe en o ce e ms exe ed by he cells and pe -
o med a decomposi ion in o no mal and shea s esses (Me hods),
Fig. 4. No mal o ces exe ed by cells exhibi ed wo egimes as a unc-
ion o he apico-basal coo dina e, 𝑠: ei he an expansi e (⟨𝐹𝑛⟩>0) o
comp essi e (⟨𝐹𝑛⟩<0) beha io . As expec ed, he ac i e cellula con-
ac ile o ce is always comp essi e (i.e., poin ing inwa d, ⟨𝐹𝑛⟩<0) and
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Fig. 3. Ene gy in e ence in squamous and cuboidal/columna ubula models depending on he cell packing geome y. A-B: A e age ene gy pe cell as a
unc ion o he su ace a io (apico-basal coo dina e) (Le ) and indi idual cell ene gies a he apical, null-s ess, and basal su aces (Righ ) in ep esen a i e us a
(A) and Vo onoi (B) squamous ubes sha ing he same packing configu a ion a he apical su ace. In he le panels, he e o bands co espond o he s anda d
de ia ion (10 samples), and he g een, yellow, and ed iangles indica e he alues o 𝑠a he apical, null-s ess, and basal planes, espec i ely. Colo scales in he
igh panels ange om minimum (whi e) o maximum alues. C-D: Same in o ma ion as in A-B panels o ubes o med by cuboidal/columna cells. No ice ha in
us a ubes, he lack o apico-basal in e cala ions o ces cells o s e ch.
is mo e p onounced a he apical su ace. Addi ionally, since he line-
ension pa ame e is posi i e, Λ>0, he adhesion o ce a o s cellula
comp ession h oughou he apico-basal axis.
Rega ding diffe ences in no mal o ces be ween packing config-
u a ions o squamous (Fig. 4A-B) and cuboidal/columna epi helia
(Fig. 4C-D), he no mal o ce is domina ed by elas ic e ms (i.e., cell
olume conse a ion) in all cases. Howe e , in cuboidal/columna ep-
i helia, us a packing esul s in weake no mal o ces nea he basal
su ace compa ed o Vo onoi packing.
Fo shea s ess, we fi s no e ha i s sign does no ha e a pa icu-
la physical meaning (Me hods). In squamous epi helia, he diffe ences
be ween us a and Vo onoi packing a e minimal, simila o he case o
no mal s esses, and he elas ic componen domina es o e adhesion and
con ac ile o ces. Addi ionally, shea s esses a e a mos wo o de s o
magni ude smalle han no mal s esses. In con as , in cuboidal/colum-
na epi helia, he elas ic componen emains dominan , bu quan i a i e
diffe ences a ise depending on cellula packing.
In us a ubes, he maximum shea o ce (a he basal su ace) is ap-
p oxima ely h ee imes la ge han in Vo onoi ubes. This esul s om
inc eased cell s e ching in us a ubes compa ed o scu oidal cells (see
Fig. 3C-D). Consequen ly, he balance be ween shea and no mal o ces,
||⟨𝐹𝑠⟩∕⟨𝐹𝑛⟩||, is la ge in us a packing a he basal su ace, whe e shea
s esses each a maximum:
||⟨𝐹𝑠⟩∕⟨𝐹𝑛⟩|| us a ≃(101)×||⟨𝐹𝑠⟩∕⟨𝐹𝑛⟩||Vo onoi
Finally, es ima ion o he ne o ces exe ed a cell e exes (Fig. 4,
bo om ow) e eals ha o ce balance is achie ed independen ly o
cellula packing and geome y.
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S. Anba i, P. Gómez-Gál ez, P. Vicen e-Munue a e al.
Fig. 4. Fo ce in e ence in squamous and cuboidal/columna ubula models depending on he cell packing geome y. A/B: A e age no mal ( op) and shea
(middle) o ces pe cell as a unc ion o he su ace a io (apico-basal coo dina e) in us a (A) and Vo onoi (B) ubes. The e o bands co espond o he s anda d
de ia ion om 10 samples, and he g een, yellow, and ed iangles indica e he alues o 𝑠a he apical, null-s ess, and basal su aces, espec i ely. C/D: Same
esul s as in A and B o ubes ep esen ing cuboidal/columna epi helia. In all cases, he plo s in he bo om ow depic he dis ibu ion (PDF) o ne o ces exe ed
on all cell e exes, demons a ing ha o ce balance is achie ed, 𝐅=(𝐹𝑥,𝐹𝑦)≃𝟎, ega dless o cellula packing.
4. Discussion
He ein, we ha e in oduced a no el 3D o ce/ene gy in e ence ap-
p oach ha ocuses on wo main aspec s: ob aining he a e age cellula
beha io in issues and pa ame e izing a e ex model o de e mine he
effec i e biophysical pa ame e s o cells. Ou me hodology employs a o-
mog aphic app oach ha de e mines o ce equilib ium plane by plane
along he apico-basal axis. The pa ame iza ion, based on mapping o
he e ex model, allows us o elucida e elas ic, adhesi e, and con ac ile
o ce componen s. In his con ex , we exploi ed he ac ha he o ce
equilib ium condi ion is linea in hese o ce pa ame e s o implemen
a linea p og amming op imiza ion app oach.
While cells and issues a e inhe en ly 3D s uc u es, echnical di -
ficul ies in ob aining accu a e imaging da a ha e long hinde ed he
de elopmen o o ce in e ence me hods in 3D. Recen ad ances in mi-
c oscopy and machine-lea ning-assis ed segmen a ion [19] ha e signi -
ican ly p og essed he field o 3D o ce in e ence [60]. Howe e , all
cu en me hods s ill ace limi a ions. Recen app oaches assume he
emb yo as a “ oam” in equilib ium [52], wi h u he imp o emen s
achie able using simula ion-based in e ence [61]. S ill, hese me hods
p ima ily p o ide ela i e alues o cellula p essu e and su ace en-
sion.
Ou app oach assumes ha cell shape de o ma ion is d i en by in-
plane o ces while o ces along he apico-basal axis a e negligible. This
app oxima ion is well jus ified by he expec ed beha io o elas ic ma e-
ials. Howe e , applying ou me hodology o eal expe imen al ubula
epi helia equi es cau ion. The p oposed analysis elies on he assump-
ion ha when un olling he cylinde (see Fig. 1F), adial planes p ese e
e ex- e ex dis ances and cell a eas. I his condi ion is no me , o ce
es ima ion a cell e exes may in oduce a i ac s. Consequen ly, ou
me hodology is bes sui ed o compu a ional s udies, whe e i p o ides
insigh s in o a key biological p oblem: how apico-basal in e cala ions
con ibu e o he s abili y and in eg i y o ubula epi helia. Along hese
lines, while he me hodology could, in p inciple, be adap ed o o he
epi helial s uc u es, such as sphe ical o o al epi helia, doing so would
equi e subs an ial modifica ions. Specifically, defining o ce balance in
a ully enclosed 3D s uc u e a he e exes whe e apico-basal in e cala-
ions occu would necessi a e he de elopmen o a no el ene gy/ o ce
unc ional o cu ed su aces whe e s able scu oids de elop spon a-
neously —one ha , o he bes o ou knowledge, has no ye been ei he
p oposed o es ed in epi helial mechanics. O e all, while ou app oach
is heo e ically ex endable o o he epi helial geome ies, i is specifi-
cally op imized and well-sui ed o analyzing ubula epi helia.
Impo an ly, modeling is he only easible way o in es iga e ou
ques ions o in e es , as cu en expe imen al echniques canno selec-
i ely modi y issue packing a chi ec u es. Since hei disco e y, scu-
oids ha e been ecu en ly ound in cuboidal/columna epi helia ha
ha e been analyzed in 3D. Examples include issues in mice, zeb afish,
D osophila, cell cul u es, and o ganoids, suppo ing he gene ali y o his
packing shape. Howe e , o he bes o ou knowledge no s udy has e-
po ed he exis ence o scu oidal cell shapes in squamous epi helia. He e,
we explo ed he biophysical basis o his phenomenon by compa ing
ubula issues wi h wo diffe en su ace a ios, 𝑠𝑏≃1.5and 𝑠𝑏=4.
The logic unde lying his analysis is as ollows. I has been epo ed
ha a lowe alue o 𝑠𝑏co ela es wi h a educed numbe o scu oidal
cells [14], as he sho e heigh o hese cells limi s neighbo exchanges
along he apico-basal axis. This sugges s ha ubula epi helia wi h ei-
he us a o scu oidal cell packing can be ene ge ically compa ible and
main ain in eg i y a low alues o 𝑠𝑏. To simpli y, we use 𝑠𝑏≃1.5 ubes
as a model o squamous issue o p o ide a baseline o unde s anding
he esul s ob ained o 𝑠𝑏=4(cuboidal/columna model). Howe e , in
eal biological issues, squamous cells ypically exhibi e en lowe al-
ues o 𝑠𝑏. Indeed, ou simula ions e eal ha in 𝑠𝑏≃1.5Vo onoi ubes,
scu oids de elop in app oxima ely 35 ± 6% o cells. Taken oge he , ou
findings indica e ha in squamous issues subjec ed o cu a u e, us a
and scu oidal cell packing esul in simila ene gy and o ce p ofiles
(Figs. 3A-B and 4A-B). Thus, we p opose ha in eal issues, squamous
cells do no de i e any biophysical ad an age (ene ge ically speaking)
om emodeling hei mo phology in o a scu oidal shape.
Howe e , in cuboidal/columna issues, when apico-basal in e ca-
la ions a e supp essed (i.e., us a cell shapes), o ce equilib ium can
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S. Anba i, P. Gómez-Gál ez, P. Vicen e-Munue a e al.
only be achie ed wi h ex emely low le els o cellula adhesion—
app oxima ely an o de o magni ude lowe han in sho squamous
cells. Such low adhesion is bo h un ealis ic and incompa ible wi h ep-
i helial in eg i y, whe e igh cell packing is essen ial. Fu he mo e, p e-
en ing apico-basal in e cala ions in cuboidal/columna cells dis up s
he balance be ween no mal and shea o ces, u he challenging issue
in eg i y. This imbalance is linked o basal su ace de o ma ion in us a
cells when 𝑠𝑏=4(Fig. 3C) bu is absen in squamous epi helia (𝑠𝑏≃1.5,
Fig. 3A). Con e sely, when apico-basal in e cala ions a e allowed in
cuboidal/columna issues, cellula adhesion alues a e highe han in
squamous cells, and he balance be ween shea and no mal o ces is
mo e compa ible wi h issue homeos asis, as no mal o ces clea ly dom-
ina e o e shea o ces (Fig. 4D).
In a b oade con ex , ecen s udies on 2D epi helial packing dynam-
ics ha e shown ha an inc ease in cell junc ion ension upon con ac ion
and a educ ion in ension upon ex ension can s abilize highe -o de
(e.g., ou - old) e ices [62]. Ex ending his o a 3D con ex , apico-basal
in e cala ions, es ablished by ou -cellula junc ions along he apico-
basal axis, may play a key ole in esol ing uns able cell geome ies
esul ing om cons ain s du ing issue packing while main aining is-
sue in eg i y, as demons a ed in ou s udy. Al hough ou wo k sheds
ligh on he ole o apico-basal in e cala ions in s a ic 3D epi helial
ubes, he mechanics unde lying cell dynamics du ing mo phogenesis—
while simul aneously accoun ing o s a iona y apico-basal in e cala-
ions and cell di ision— emain la gely unknown. Despi e ecen effo s
[20,22,63], u he esea ch is needed o elucida e he in e play be ween
mechanics and scu oid dynamics o be e unde s and complex p ocesses
such as issue g ow h and ea angemen , wound healing, cell ex usion,
and cell mig a ion.
CRediT au ho ship con ibu ion s a emen
Sami a Anba i: So wa e, Me hodology, In es iga ion. Ped o
Gómez-Gál ez: W i ing – e iew & edi ing, W i ing – o iginal d a ,
Visualiza ion, Valida ion, So wa e, Me hodology, In es iga ion. Pablo
Table 2
Pa ame e in e ence: fini e size effec s. G ound u h and es ima ed e ex
model pa ame e s alues as a unc ion o he a io 𝜌=𝑁𝑝∕𝑁.
Pa ame e G ound
T u h
𝜌=0.7𝜌=0.57 𝜌=0.51
(𝑁=20, 𝑁𝑝=14) (𝑁=30, 𝑁𝑝=17) (𝑁=45, 𝑁𝑝=23)
𝐴01.000 1.023 (𝛿=2.3%) 1.015 (𝛿=1.5%) 1.001 (𝛿=0.1%)
Λ 0.040 0.043 (𝛿=5.5%) 0.042 (𝛿=5%) 0.041 (𝛿=2.5%)
Γ 0.02 0.022 (𝛿= 10%) 0.021 (𝛿=5%) 0.020 (𝛿=0%)
Table 3
Pa ame e in e ence: diffe en ene gy egimes. G ound u h and es ima ed
e ex model pa ame e s alues in diffe en scena ios wi h espec o he domi-
nan ene ge ic con ibu ion.
Pa ame e Case A (elas ic dominan ) Case B (con ac ile dominan )
G ound T u h Es ima ion G ound T u h Es ima ion
𝐴02 2.001 (𝛿=0.05%) 1 1.002 (𝛿=0.2%)
Λ 0.04 0.041 (𝛿=2.5%) −0.02 −0.02 (𝛿=0%)
Γ 0.02 0.02 (𝛿=0%) 0.02 0.02 (𝛿=0%)
Fig. 5. In e ence o cell ene gy componen s: fini e-size effec s. A: Compa ison o cellula ene gies be ween es ima ed and g ound u h alues as a unc ion o
he a io 𝜌(𝑁and 𝑁𝑝a e indica ed in Table 2). Ci cles co espond o indi idual cells. A sp eading away om he diagonal indica es a misma ch be ween es ima ed
and g ound u h alues. B: Cell ene gy componen s in simula ed issues (colo code as in panel A); columns co esponds o he alues o 𝜌indica ed in panel A. The
scale ba s ange be ween he obse ed minimum and maximum o he ene gy alues in all cases.
Compu a ional and S uc u al Bio echnology Jou nal 27 (2025) 1204–1214
1212
S. Anba i, P. Gómez-Gál ez, P. Vicen e-Munue a e al.
Vicen e-Munue a: Visualiza ion, Me hodology. Luis M. Escude o:
W i ing – e iew & edi ing, W i ing – o iginal d a , Valida ion, Supe i-
sion, Funding acquisi ion, Concep ualiza ion. Ja ie Buce a: W i ing –
e iew & edi ing, W i ing – o iginal d a , Visualiza ion, Valida ion, Su-
pe ision, So wa e, Me hodology, In es iga ion, Funding acquisi ion,
Fo mal analysis, Concep ualiza ion.
Decla a ion o compe ing in e es
Decla e ha none o he au ho s: Ha e an undisclosed ela ionship
ha may pose a compe ing in e es ; Ha e an undisclosed unding sou ce
ha may pose a compe ing in e es .
Acknowledgemen s
This wo k was suppo ed by he Minis e io de Ciencia e Inno-
ación o Spain h ough g an s PID2019-103900GB-I00 AEI/10.13039/
501100011033 (L.M.E.), PID2022-137101NB-I00/AEI/10.13039/
501100011033/FEDER UE (L.M.E.), PID2022-137436NB-I00 (J.B.),
PID2019-105566GB-I00 (J.B.) and om he Li eHUB Resea ch Ne -
wo k h ough g an PIE-202120E047-ConexionesLi e (CSIC). P.G.-G.
has been unded by he Ma ga i a Salas p og am – Nex Gene a-
ion E.U. J.B. also ecei ed unding om he esea ch ne wo k
RED2022-134573-T unded by Minis e io de Ciencia e Inno ación
(MCIN/AEI/10.13039/501100011033) and by ‘ERDF: A way o mak-
ing Eu ope’, by he Eu opean Union. L.M.E. and J.B. ecei ed addi ional
suppo om he E.U. COST ac ion CA22153 ‘Eu opean Cu a u e and
Biology Ne wo k’ (Eu oCu oBioNe ).
Appendix A. Con ol simula ions
We an con ol simula ions using he e ex model o es he eli-
abili y o he o ce in e ence me hod. To ha end, we used he TiFoSi
package [64,65]. In addi ion o he pa ame e s ha desc ibe he cell me-
chanical p ope ies (i.e., he pa ame e s o be in e ed), we in oduced
some le el o s ochas ici y in he du a ion o he cell cycle o achie e di -
e en cellula sizes a a gi en ime. Thus, he du a ion o he cell cycle,
𝜏, is defined as,
𝜏=𝜖𝑡𝑑𝑒𝑡. +(1−𝜖)𝑡𝑠𝑡𝑜.
whe e 𝑡𝑑𝑒𝑡. is a de e minis ic ime scale ha accoun s o a mean cell
cycle du a ion and 𝑡𝑠𝑡𝑜. is a andom a iable ha accoun s o he a i-
abili y o cell cycle du a ion and ha is assumed o ollow an exponen ial
dis ibu ion:
𝜌(𝑡𝑠𝑡𝑜.)=𝑒−𝑡𝑠𝑡𝑜.
𝑡𝑑𝑒𝑡.
𝑡𝑑𝑒𝑡.
We se a alue o 𝜖=0.8 o he pa ame e ha weigh s he s ochas-
ici y o he cell-cycle du a ion (see [64,65] o de ails), and we se a
dimensionless a e age cell cycle du a ion o ⟨𝜏⟩=1.5⋅103(∼20hou s)
in all simula ions.
Con ol simula ions we e pe o med in wo s ages. Fi s , we allowed
cells o g ow and di ide un il he issue eached a gi en numbe o
cells, 𝑁. Du ing he second s age, we s opped cellula g ow h and ei-
he allowed he issue o mechanically elax o a s able configu a ion
o modula ed some mechanical p ope ies as a unc ion o ime o d i e
he sys em ou o equilib ium. Since cell e exes a he issue pe iphe y
a e sha ed by ei he one o wo cells, in con as o e exes in he issue
bulk ha a e sha ed by h ee cells, we fi s es ed he sensi i i y o ou
me hod o fini e-size (i.e., bounda y) effec s. To do so, we used he same
alues o he 𝐴0, Λ, and Γpa ame e s bu a ied he a io 𝜌=𝑁𝑝∕𝑁,
whe e 𝑁𝑝is he numbe o pe iphe al cells. Since 𝜌∼1∕
√𝑁, be e in-
e ence esul s a e expec ed as 𝑁inc eases (i.e., as 𝜌dec eases). The
esul s (Table 3) indica e ha as 𝜌app oaches ∼0.5, he obse ed e -
o , 𝛿, in he in e ed pa ame e s emains below 3% and s ays unde
10% e en in issues whe e bounda y effec s a e dominan (𝜌=0.7).
Once he pa ame e s 𝐴0, Λ, and Γwe e compu ed, we es ima ed in
each simula ion he ene gy componen s o each cell 𝑘:
𝐸elas ic
𝑘=1
2(𝐴𝑘−𝐴0)2
𝐸con ac .
𝑘=Γ
2
𝐿2
𝑘(A.1)
𝐸adh.
𝑘=Λ𝐿𝑘
Fig. 5shows ha con e gence o g ound u h alues is achie ed as 𝜌
app oaches ∼0.5.
We u he checked he obus ness o he in e ence me hod agains
he ela i e impo ance o diffe en ene gy con ibu ions. To ha end,
we pe o med con ol simula ions using diffe en pa ame e se s while
keeping 𝑁cons an . Table 3and Fig. 6show wo ep esen a i e cases
whe e Γis kep he same bu 𝐴0and Λchange such ha ei he he
con ac ile o he elas ic ene gy becomes he dominan ene ge ic con-
ibu ion (in con as o he simula ions shown in Fig. 5whe e he
adhesion ene gy is dominan ). The in e ence esul s emain in excellen
ag eemen wi h g ound u h alues ega dless o he dominan ene gy
componen .
Finally, we e alua e he eliabili y o he o ce in e ence app oach
when he mechanical pa ame e s a e modula ed in ime. In his ega d,
he ime scale o mechanical ene gy elaxa ion is 𝑡𝑟≃1(dimension-
less ime uni s). Thus, o ce equilib ium is eached a imes 𝑇≫1(i.e.,
𝑡𝑟∕𝑇≪1). This sugges s ha a dynamic modula ion o mechanical pa-
Fig. 6. In e ence o cell ene gy componen s: diffe en dominan con ibu-
ions. A/B: Elas ic/con ac ile ene gy dominan issues. In bo h cases 𝑁=45
and e ex model pa ame e s as shown in Table 3. In A ci cles co espond o in-
di idual cells. In B he colo code o cell ene gy componen s as in panel A and
he columns co esponds o he cases shown in panel A. The scale ba s ange be-
ween he obse ed minimum and maximum alues o he ene gy in all cases.
We no ice ha nega i e alues o he adhesion ene gy ob ained when Λ<0a e
because he alue o he line ension pa ame e is aken in o accoun in he cal-
cula ion o 𝐸adh. as s a ed in Eq. (A.1).
