Ma hema ical Biosciences 380 (2025) 109362
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O iginal esea ch a icle
Compe ing elas ic and iscous g adien s de e mine di ec ional cell mig a ion
Pablo Saez a,b,∗, Palla i U. Shi ke c, Jyo i R. Se h c, Jo ge Aleg e-Cebollada d, Abhiji Majumde c
aLaCàN, Uni e si a Poli ècnica de Ca alunya-Ba celonaTech, 08034 Ba celona, Spain
bIns i u e o Ma hema ics o UPC-Ba celonaTech.–IMTech, Ba celona, Spain
cDepa men o Chemical Enginee ing, Indian Ins i u e o Technology Bombay (IITB), 400076 Mumbai, India
dCen o Nacional de In es igaciones Ca dio ascula es (CNIC), 28029 Mad id, Spain
ARTICLE INFO
Keywo ds:
Clu ch model
Ac i e gel models
Visco axis
Du o axis
Cell adhesion
Mechano ansduc ion
ABSTRACT
Cell mig a ion egula es cen al li e p ocesses including emb yonic de elopmen , issue egene a ion, and umo
in asion. To es ablish he di ec ion o mig a ion, cells ollow exogenous cues. Du o axis, he di ec ed cell
mig a ion owa ds elas ic s i ness g adien s, is he classical example o mechanical axis. Howe e , whe he
g adien s o he elaxa ion p ope ies in he ex acellula ma ix may also induce ac ic esponses ( isco axis)
is no well unde s ood. Mo eo e , whe he and how du o axis and isco axis in e ac wi h each o he has
ne e been in es iga ed. He e, we in eg a e clu ch models o cell adhesions wi h an ac i e gel heo y o
cell mig a ion o e eal he mechanisms ha go e n isco axis. We show ha isco axis is enabled by an
asymme ic exp ession o cell adhesions ha u he pola ize he in acellula mo ili y o ces o es ablish he
cell on , simila o du o axis. Mo e impo an ly, when bo h elaxa ion and elas ic g adien s coexis , du o axis
appea s mo e e icien in con olling di ec ed cell mig a ion, which we con i m wi h expe imen al esul s.
Howe e , he p esence o opposing elaxa ion g adien s o an elas ic one can a es o shi he mig a ion
di ec ion. Ou model a ionalizes o he i s ime he mechanisms ha go e n isco axis and i s compe i ion
wi h du o axis h ough a ma hema ical model.
1. In oduc ion
Cell mig a ion de e mines key biological p ocesses such as em-
b yonic de elopmen , issue egene a ion, wound healing, and umo
in asion [1–3]. Fo decades, he e has been inc easing in e es in
unde s anding how cells o ganize hemsel es o mig a e. The e is also
g owing conce n abou con olling cell mig a ion, as i may allow us
o a es umo me as asis [4–6], enhance issue egene a ion, and
design be e biomime ic ma e ials [7,8]. Th ee o ces coope a e o
enable cell mig a ion on wo-dimensional (2D) subs a es (see Fig. 1).
Fi s , he F-ac in ne wo k, along wi h myosin mo o s ha pull on ac in
ilamen s, c ea es an ac i e iscoelas ic low, o e og ade low [9,
10]. Second, he F-ac in ne wo k con inuously polyme izes, pushing
he cell memb ane ou wa d [11,12], c ea ing a di e en ia ed ac in
ne wo k compa ed o he con ac ile ac omyosin ne wo k. These wo
compe ing ac in ne wo ks, he i s mo ing inwa ds and he la e
ou wa ds, de e mine he p o uding eloci y o he cell con ou and,
he e o e, cell mig a ion. Thi d, cells c ea e adhesions ha link he
in acellula ac omyosin ne wo k wi h he ex acellula ma ix (ECM).
Cell adhesion is media ed by molecules ha sense and espond o he
mechanical p ope ies o he ECM [13–15]. Th ough he o ma ion and
disassembling o adhesion complexes, cells une he ic ion be ween
he in acellula ne wo k and he ECM.
∗Co esponding au ho a : LaCàN, Uni e si a Poli ècnica de Ca alunya-Ba celonaTech, 08034 Ba celona, Spain.
E-mail add ess: [email p o ec ed] (P. Saez).
Bu hen, how does a cell o ganize i sel o de e mine he di ec ion
o mig a ion? Mos cells equi e s imuli o b eak he symme y in
he in acellula o ce gene a ion o ex acellula o ce ansmission o
enable di ec ed cell mig a ion. O he wise, cells would jus sp ead wi h
symme ic dis ibu ions o F-ac in e og ade low, polyme iza ion,
and adhesion o ma ion [16,17]. Mechanical [3,18–20] and chemical
signals [21,22] ha e been well s udied. In e ms o mechanical cues
o cell mig a ion, which we analyze he e, mos s udies ha e ocused on
du o axis, he mo ion o cells owa ds elas ic s i ness g adien s o he
ECM [18,19,23,24]. Du o axis is exp essed ac oss di e en cell ypes,
including ib oblas s [25], smoo h muscle cells [26], neu ons [24], and
cance cells [27]. When cells a e placed on ECM wi h an elas ic s i ness
g adien , hey es ablish an asymme ic adhesi e s a e [28,29], wi h
one side o he cell mo e adhesi e han he o he . In esponse o his
ic ion g adien be ween he cell and he subs a e, he e og ade low
also pola izes [29] and es ablishes a p ospec i e cell on and ea .
Mo eo e , he di ec ion o he adhesion g adien de e mines whe he
he cell on o ien s owa ds he posi i e o nega i e elas ic s i ness
g adien [29].
Howe e , ex acellula ma ices in i o a e no pu ely elas ic bu
also p esen iscous p ope ies [30]. The iscoelas ici y o he ECM
h ps://doi.o g/10.1016/j.mbs.2024.109362
Recei ed 16 June 2024; Recei ed in e ised o m 3 Decembe 2024; Accep ed 4 Decembe 2024
Ma hema ical Biosciences 380 (2025) 109362
2
P. Saez e al.
Fig. 1. (a) Ske ch o he main o ces
ac ing in cell mig a ion eloci y ( ). A
con ac ile ac omyosin low, (𝑣𝐹black
a ows), compe es wi h p o usi e o ces
a he on (𝑙𝑓(𝑡)) and ea (𝑙𝑟(𝑡)) o he
cell (𝑣𝑝, blue a ows). A he cell-ECM
in e ace, in acellula o ces a e ans-
mi ed ex acellula ly h ough a ic ion
o ce ha appea s due o adhesion mech-
anisms. (b) A s anda d linea model
desc ibes he iscoelas ic beha io o he
ECM, de e mined by 𝑘𝑒,𝑘𝑣, and 𝜇. On
he igh a e he o ce and displacemen
cu es o a elaxa ion es . (c) De ail o
an adhesion complex, made o adap o
p o eins ( op) and esul s o a clu ch
model (bo om). Resul s show he p ob-
abili y o bound binde s, 𝑃𝑏, showing
cycles o o ma ion and up u e o he
adhesion complex, he eloci y o he
ac in ilamen s, 𝑣, and ac ion o ces on
he subs a e, 𝑃.
media es se e al cell unc ions, including he o ma ion o adhesion
complexes [31–34], cell di e en ia ion [34,35], p oli e a ion [34,36],
lamellipodial and ilopodial p o usion [37], and cell sp eading [31–
33,38]. Mos s udies ha e analyzed he e ec o homogeneous iscous
p ope ies o he ECM on cell unc ion. How cells mig a e along g adi-
en s in he elaxa ion p ope ies o he ECM has only been p elimina ily
analyzed expe imen ally, wi hou explo ing he unde lying physical
p ocesses ha de e mine isco axis [39] o h ough physical models
ha neglec he he e ogeneous in e nal e og ade low and ac omyosin
dis ibu ion [40], which is cause and e ec o di ec ed cell mig a ion.
He e, based on he same p inciples ha d i e du o axis, we e eal
h ough physical models mechanisms o cell isco ac ic mig a ion in
2D. Mo eo e , because elas ic and elaxa ion g adien s may appea
simul aneously in i o and in i o, we analyze how du o axis and
isco axis compe e wi h each o he , which has no been add essed by
any p e ious wo k. A clea unde s anding o how elas ic and iscous
g adien s a ec each o he can allow us o be e enginee bioma e ials
o speci ic biological unc ions.
In his wo k, we i s desc ibe he ma hema ical models used o
cell adhesion ( he clu ch model) and o he cell mo ili y (an ac i e
gel model). Nex , we discuss how we p epa ed he gel and es ed
mechanically. We inish he me hods sec ion by desc ibing cell cul u e,
ime lapse mic oscopy and cell mig a ion analysis o ou wo k. The,
in ou esul s, we i s show he cell adhesion beha io as a unc ion
o he elas ici y and iscoelas ici y o he ECM, which we la e use
o p opose a ma hema ical model o single cell isco axis. Then, we
explain heo e ically how isco axis appea and show ha isco axis
can compe e wi h du o axis o con ol cell mig a ion. Finally, we
conclude discussing he main esul s o ou wo k.
2. Ma e ials and me hods
2.1. Clu ch model o cell adhesions based on Mon eca lo simula ions
We use p e ious clu ch models o unde s and he esponse o cell
adhesion o he iscoelas ici y o he ECM [29,41–43] (Fig. 1(b-c)). I
conside s se e al molecula clu ches ha a ach o he ECM on one side
and o he adap o alin od on he o he . A i s o he end, alin is
linked o ac in ilamen s pulled by myosin mo o s. The compu a ion
o he clu ch model elies on epea ed andom sampling, which we
sol e by a Mon e Ca lo (MC) simula ion, du ing which many e en s o
engagemen and disengagemen occu , esul ing in cycles o o ma ion
and up u e o adhesion complexes (see Fig. 1c). We use a cons an ime
s ep, 𝛥𝑡 =0.005 s, and a inal ime o 𝑡𝑓= 100 s, which was shown o
p o ide accu a e esul s [43]. Then, he MC simula ions a e a e aged
o e ime o ob ain he esul s a speci ic mechanical p ope ies o he
ECM.
Du ing he cycling p ocess o cell adhesion, he ac omyosin ne wo k
pulls on he adhesion molecules hanks o he con ac ile o ces o he
myosin mo o s. The ac in eloci y, 𝑣, is gi en by he o ce- eloci y
ela ion
𝑣 =𝑣𝑢(1 −𝐹∕𝐹𝑠𝑡𝑎𝑙 𝑙),(1)
whe e 𝑣𝑢is he ee eloci y o he ac in low. 𝐹𝑠𝑡𝑎𝑙 𝑙is he s all o ce o
he myosin mo o s, being 𝐹𝑠𝑡𝑎𝑙 𝑙=𝑛𝑚𝐹𝑚, whe e 𝐹𝑚is he o ce equi ed
o s all he ac i i y o one myosin mo o and 𝑛𝑚 he numbe o mo o s
in he sys em. 𝐹is he eac ion o ce in he subs a e, which balances
he o ce on all he molecula clu ches, 𝐹=∑𝑛𝑒𝑛𝑔
𝑖=1 𝐹𝑐 ,𝑖, being 𝑛𝑒𝑛𝑔 he
numbe o engaged clu ches. The displacemen o he engaged clu ches
is compu ed as 𝛥𝑧 =𝑣𝛥𝑡, and he displacemen o he subs a e a he
cu en ime s ep, 𝑧𝑛+1, is ob ained by sol ing he balance o o ces
be ween he 𝑛𝑒𝑛𝑔 engaged molecula clu ches and he subs a e, om
whe e we can la e ob ain 𝐹.
The o ce a each 𝑖 h clu ch, 𝐹𝑐 ,𝑖, is gi en by 𝐹𝑐 ,𝑖 =𝜅𝑐(𝑧𝑐 ,𝑖 −𝑧𝑠𝑢𝑏),
whe e 𝜅𝑐is he s i ness o each molecula clu ch (see [41–43] o
de ails) and 𝑧𝑐 ,𝑖 is he displacemen o he 𝑖 h molecula clu ch, 𝑧𝑠𝑢𝑏 is
he displacemen o he subs a e. Once 𝐹is compu ed, he cell ac ion
can be ob ained, 𝑃=𝐹∕𝜋 𝑎2, whe e (𝜋 𝑎2)is he a ea occupied by a
ci cula adhesion complex.
Mo eo e , once he o ce a each clu ch, 𝐹𝑐 ,𝑖, is compu ed, we can
calcula e he binding and unbinding e en s a each molecula clu ch.
The a achmen o he 𝑛𝑐molecula clu ches o he ECM, which ep-
esen s he in eg in-ECM link, ollows a ca ch-bond beha io [42,44].
Se e al molecula clu ches bind o he ECM wi h a cons an a e 𝑘𝑜𝑛,
and unbind wi h a dissocia ion a e 𝑘∗
𝑜𝑓 𝑓, ha depends on he o ce on
each molecula binde , 𝐹𝑐, as
𝑘∗
𝑜𝑓 𝑓=𝑘𝑜𝑓 𝑓
𝑠𝑒(𝐹𝑐∕𝐹𝑠𝑙 𝑖𝑝
𝑏)+𝑘𝑜𝑓 𝑓
𝑐𝑒(−𝐹𝑐∕𝐹𝑐 𝑎𝑡𝑐 ℎ
𝑏),(2)
whe e he i s and second e ms ep esen a slip and ca ch bond,
espec i ely. 𝑘𝑜𝑓 𝑓
𝑠and 𝑘𝑜𝑓 𝑓
𝑐and 𝐹𝑠𝑙 𝑖𝑝
𝑏and 𝐹𝑐 𝑎𝑡𝑐 ℎ
𝑏a e he unloaded
dissocia ion a es and he cha ac e is ic bond up u e o ces in he slip
and ca ch pa hways, espec i ely.
Mechanosensi i i y is p omo ed by alin un olding, which induces
inculin ec ui men and subsequen in eg in clus e ing [42]. This p o-
cess o in eg in c owding is usually e med adhesion ein o cemen . We
in oduce alin ein o cemen by conside ing alin un olding acco ding
o Bell’s model a a a e 𝑘∗
𝑢𝑛𝑓 [42]. Then, i can ei he e old again o
inculin can bind o i a a o ce-independen a e 𝑘𝑜𝑛𝑣. I alin un olds,
Ma hema ical Biosciences 380 (2025) 109362
3
P. Saez e al.
Table 1
Le : Pa ame e s adop ed o he clu ch model o cell adhesion. The model pa ame e s
o he ca ch bonds wi h alin ein o cemen a e ob ained om [42,43]. The unbinding
a e is conside ed o ollow a cycle o mechanical ein o cemen , ollowing 𝑘∗
𝑜𝑓 𝑓=
8.10−4 𝑒(𝐹𝑐∕8.16) + 10.14 𝑒(−𝐹𝑐∕6.24) + 900 𝑒(−𝐹𝑐∕0.01) [42]. Righ : Pa ame e s adop ed o he
cell mig a ion model and ela ed e e ences. * indica es ha he alue was adjus ed in
his wo k.
Clu chs pa ame e Mig a ion pa ame e s
𝑎(nm) 1700 𝜇𝐹(kPa s) 10 [48,49]
𝑛𝑐1200 𝜂0(kPa s/μm2) 0.05 [48,49]
𝜅𝑐(pN∕nm) 1000 𝜁(kPa) 0.05 [47,48]
𝑘𝑜𝑛𝑡(μm2∕s) 2.11 × 10−4 𝑘0.1 [49]
𝑑𝑖𝑛𝑡 (in /μm2) 300 𝐷𝐹(μ m∕s) 0.3 [48,50]
𝐹𝑚(pN) 2 𝐷(μ m∕s) 0.4 [48,50]
𝑛𝑚800 𝑘𝑑0.1 [51,52]
𝑣𝑢(nm∕s) 110 𝑘𝑝0.1 [51,52]
𝑘𝑜𝑛𝑣 (s−1)1 × 108𝑣𝑝
0(μ m∕s) 0.55 [53]
𝑖𝑛𝑡𝑎𝑑 𝑑(in /μm2) 24 𝐿𝑏(μm) 7 *
𝑚𝑟(in /μm2) 15 000 𝜏𝑠𝑡𝑎𝑙 𝑙(nN∕μm) 0.4 *
𝛥𝑡 (s) 0.005
𝑡𝑓(s) 100
i can ei he e old again o inculin can bind o i . Then, in eg in
densi y inc eases by 𝑖𝑛𝑡𝑎𝑑 𝑑= 24 in eg ins/𝜇 𝑚2and he binding a e o
in eg ins esul s in 𝑘𝑜𝑛 =𝑘𝑜𝑛𝑡𝑑𝑖𝑛𝑡, whe e 𝑘𝑜𝑛𝑡 is he ue binding a e
cha ac e izing each in eg in- ib onec in bond, and 𝑑𝑖𝑛𝑡 is he densi y o
in eg ins in he adhesion complex. We do no allow in eg in densi y
o go abo e a maximum in eg in densi y, 𝑚𝑟, de e mined by he sep-
a a ion be ween in eg ins (see [43] o de ails). I he clu ch unbinds
be o e inculin binds o alin, in eg in densi y is dec eased by 𝑖𝑛𝑡𝑎𝑑 𝑑,
e lec ing he ac ha adhesions lose in eg ins i o ce applica ion is
dec eased [45,46].
To model he iscoelas ic beha io o he ECM, we use he Maxwell
ep esen a ion o a S anda d Linea Solid (SLS) model such ha he
s ess–s ain ela ion is gi en as
𝐹+𝜇∕𝑘𝑣
𝐹=𝑘𝑒𝑧𝑠𝑢𝑏 +𝜇(𝑘𝑣+𝑘𝑒)𝑧𝑠𝑢𝑏∕𝑘𝑣.(3)
𝑘𝑒is he long- e m s i ness, 𝑘𝑣is he elaxa ion s i ness, which ep e-
sen s he d op o he long- e m s i ness as he ma e ial elaxes, 𝑧𝑠𝑢𝑏 =𝑣,
and 𝜇is he iscosi y o he ECM (see Fig. 1(b)). The ime de i a i es
a e disc e ized using a backwa d Eule me hod, which esul s in an
equa ion o 𝑧𝑛+1
𝑠𝑢𝑏 as a unc ion o 𝐹𝑛+1.
He e, we assume ha he ic ion be ween he cell and he ECM is
𝜂=𝑃∕𝑣 [29,47], so ic ion changes as a unc ion o he ECM s i ness
simila o how ac in eloci y and ac ions change. This ela ion as-
sumes ha s ong and s able adhesions, which a e ep esen ed by la ge
ac ion o ces, induce la ge ic ion in compa ison o small and weak
adhesion complexes.
All model pa ame e s ela ed o he clu ch model a e summa ized
in Table 1.
2.2. Ac i e gel model and nume ical solu ion o he sys em o equa ions
Recen ly, we and o he s ha e shown ha du o axis eme ges om an
asymme y in he cell adhesion s eng h [28,29]. Hence, ou esul s on
hese ic ion g adien s due o g adien s in he elaxa ion p ope ies o
he ECM s ongly sugges ha single cells may also exp ess isco axis.
To analyze he exposu e o single mo ile cells o elaxa ion g adi-
en s o he ECM, we use an ac i e gel model o cell mig a ion [29,
47,49,54]. We conside a 1D domain, 𝛺, wi h mo ing coo dina es
𝑥(𝑡) ∈ [𝑙𝑟(𝑡), 𝑙𝑓(𝑡)], whe e 𝑙𝑟(𝑡)and 𝑙𝑓(𝑡) ep esen s he ea and on
bounda ies o he cell and, he e o e, he cell leng h is de e mined
as 𝐿(𝑡) =𝑙𝑓(𝑡) −𝑙𝑟(𝑡)(Fig. 1). The e og ade low is modeled wi h
a con ac ile iscous gel in con ac wi h he ECM. The cons i u i e
ela ion o he s ess in he ac i e luid is gi en as
𝜎𝐹=𝜇𝐹𝜕𝑥𝑣𝐹+𝜁 𝜌𝐹𝜌𝑀.(4)
𝑣𝐹is he eloci y o he e og ade low, 𝜇𝐹is he shea iscosi y, 𝜁 he
ac i e con ac ion exe ed by he myosin mo o s and 𝜌𝑀and 𝜌𝐹a e he
densi ies o myosin mo o s and F-ac in, espec i ely (see below). The
balance o linea momen um o he ac i e gel is hen
𝜕𝑥𝜎𝐹=𝜂𝐹𝑣𝐹.(5)
We assume ze o s esses on he cell bounda ies. The igh -hand
side ep esen s he ic ion be ween he in acellula ne wo k and he
ECM, whe e he clu ch model o cell adhesion desc ibed abo e couples
wi h he ac i e gel model o cell mig a ion. The ic ion pa ame e is
𝜂𝐹=𝜂0+𝜂, whe e 𝜂is desc ibed abo e and 𝜂0=0.05 kPa⋅s∕μm2is
added o he e ec i e ic ion 𝜂as baseline ic ion o he e og ade
low wi h he su ounding cy oskele al s uc u es.
To compu e he eloci y o ac in ne wo k polyme iza ion agains
he memb ane, which we conside as a sepa a ed F-ac in s uc u e,
we use p e ious models (see [53,55,56] o de ails) whe e ac in poly-
me iza ion dec eases when he memb ane ension inc eases as 𝑣𝑝=
𝑣𝑝
0[1 −𝜏(𝐿(𝑡))∕𝜏𝑠𝑡𝑎𝑙 𝑙]8 om a ee-g ow h eloci y 𝑣𝑝
0.𝜏𝑠𝑡𝑎𝑙 𝑙is a ension
equi ed o s all he polyme iza ion o he ac in ne wo k. The mem-
b ane ension is calcula ed as a linea Hookean sp ing, 𝜏(𝐿(𝑡)) =𝑘(𝐿(𝑡) −
𝐿𝑏), 𝐿(𝑡)< 𝐿𝑏, whe e 𝑘is he sp ing cons an and 𝐿𝑏accoun s o he
es ing leng h and he bu e memb ane leng h o he cell memb ane.
The model assumes ze o comp essi e s esses. The ou wa d polyme iza-
ion eloci ies, 𝑣𝑝
𝑟,𝑓 , compe e wi h he inwa d e og ade low eloci y,
𝑣𝐹|𝑟,𝑓 , a he cell ea and on . This compe i ion makes he cell ends
expand o e ac wi h eloci y
𝑙𝑟,𝑓 (𝑡) =𝑣𝑝
𝑟,𝑓 −𝑣𝐹|𝑟,𝑓 , espec i ely (Fig. 1)
and he mig a ion eloci y is de ined as 𝑣= (
𝑙𝑟(𝑡) −
𝑙𝑓(𝑡))∕2, he mean
alue o he on and ea eloci y o he cell. The dis ibu ion o
he F-ac in densi y, 𝜌𝐹(𝑥, 𝑡)is modeled h ough he ollowing anspo
equa ion,
𝜕𝑡𝜌𝐹+𝜕𝑥(𝑤𝜌𝐹−𝐷𝐹𝜕𝑥𝜌𝐹)=𝑘𝑝−𝑘𝑑𝜌𝐹,(6)
whe e he igh -hand side includes he polyme iza ion and depolyme -
iza ion e ms wi h a es 𝑘𝑝and 𝑘𝑑, espec i ely [52], 𝐷𝐹is he di usi e
pa ame e o he F-ac in. Simila ly, he myosin mo o s bound o he
F-ac in ne wo k [57], 𝜌𝑀, as
𝜕𝑡𝜌𝑀+𝜕𝑥(𝑤𝜌𝑀−𝐷 𝜕𝑥𝜌𝑀) = 0,(7)
whe e 𝐷is he e ec i e di usion pa ame e [49,50].
F-ac in and bound myosin mo o s a e also d i ed in he cell ame
wi h eloci y 𝑤=𝑣𝐹−𝑣. We impose ze o luxes on he cell bounda ies
o e lec ha no F-ac in o myosin mo o s can en e o lea e he cell
domain.
To sol e he sys em o PDEs desc ibed abo e, we use a s agge ed
ini e elemen me hod o disc e ize he sys em in space, and an implici
second-o de C ank–Nicholson me hod o disc e ize he equa ions in
ime [58,59]. To a oid undesi ed oscilla ions om he nume ical so-
lu ion o he pa abolic equa ions i he p oblem becomes con ec i e
dominan , i.e. 𝑃 𝑒 >1, we include he S eam-Upwind P e o Gale kin
(SUPG) s abiliza ion e m. This s a egy allows us o use ini e elemen s
o cons an size ℎ=𝑙𝑓(𝑡)∕𝑁, whe e 𝑁is he numbe o elemen s and,
consequen ly, keep he numbe o elemen s o ou domain cons an . The
comple e ini e elemen and ime disc e iza ion ha e been p esen ed
p e iously [57]. All model pa ame e s ela ed o he mig a ion model
a e summa ized in Table 1.
2.3. Gel p epa a ion and mechanical es ing
All he expe imen al p ocedu es and expe imen al da a desc ibed in
his wo k o he gel mechanical p ope ies a e aken om a p e ious
publica ion [39]. Viscoelas ic p ope ies, ep esen ed by he s o age
modulus (G’) and loss modulus (G’’), we e measu ed o each gel. C eep
measu emen s we e also pe o med wi h bo h gels whe e cons an shea
s ess, 𝜎0, was applied and s ain 𝛾was measu ed o an hou .
Ma hema ical Biosciences 380 (2025) 109362
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P. Saez e al.
Fig. 2. (a) F ic ion o he cell-ECM adhesion, 𝜂, o 𝑘𝑒=1 pN/nm and a ange o
alues o he elaxa ion s i ness, 𝑘𝑣, and iscosi y, 𝜇. (b) F ic ion as a unc ion o he
iscosi y, 𝜇, o 𝑘𝑣=0.2 pN/nm (dash line) and 𝑘𝑣=20 pN/nm (solid line). (c) F ic ion
as a unc ion o he elaxa ion s i ness, 𝑘𝑣, o 𝜇=2 pN⋅s/nm (dash line) and 𝜇=20
pN⋅s/nm (solid line).
He e, we i he expe imen al compliance (J= 𝛾( )/𝜎0) wi h he
compliance exp ession o a SLS model,
𝐽(𝑡) =1
𝑘𝑒
(1 − [𝑘𝑣
𝑘𝑒+𝑘𝑣
]𝑒𝑥𝑝(− 𝑘𝑒𝑘𝑣
𝜂(𝑘𝑒+𝑘𝑣)𝑡)),(8)
using he non-linea leas squa es i ing algo i hm in MATLAB. Since
he ini ial compliance 𝐽(0) = 1∕(𝑘𝑒+𝑘𝑣)is known and is equal o bo h
high-loss and low-loss gels (as seen om he expe imen al da a), 𝑘𝑒+𝑘𝑣
was cons ained o emain cons an o he wo i s.
2.4. Cell cul u e, ime lapse mic oscopy and cell mig a ion analysis
Again, he expe imen al da a desc ibed in his wo k o he cell cul-
u e a e aken om a p e ious publica ion [39]. In sho , umbilical co d
human Mesenchymal s em cells (UC-hMSCs) be ween passages 4 and 7
we e used o he expe imen s. Then, he pa hs o he cells we e acked
using he ImageJ so wa e (Na ional Ins i u es o Heal h, Be hesda,
USA) plugin wi h manual acking de eloped by Fab ice Co deliè es a
Ins i u Cu ie, O say, F ance, and u he mig a ion analysis was done
using he ‘‘Chemo axis and Mig a ion Tool’’ plugin de eloped by IBIDI
So wa e. we e used o he expe imen s. We e e o ou p e ious wo k
o de ails [39].
3. Resul s
3.1. Adhesion beha io as a unc ion o he elas ici y and iscoelas ici y o
he ECM
We i s analyze he cell adhesion beha io o subs a es o di e en
long- e m s i nesses, elaxa ion s i nesses, and iscosi ies (Fig. 2).
Changes in long- e m s i ness a e ela ed o du o axis. In ag eemen
wi h p e ious models [29,42], he ic ion inc eases as we inc ease 𝑘𝑒
(see Fig. 2(a)). To analyze he iscoelas ic e ec s, we ix he long- e m
s i ness, 𝑘𝑒=1pN/nm, and analyze he e ec o he elaxa ion s i ness
𝑘𝑣and iscosi y 𝜇. We show an inc ease in he ic ion be ween he cell
and he subs a e as we inc ease 𝑘𝑣and 𝜇(Fig. 2(b)). Fo a elaxa ion
s i ness lowe han ≈1 pN/nm, ic ion anishes o all alues o ECM
iscosi ies, 𝜇 <0.5pN⋅s/nm (Fig. 2(c)). Fo alues o 𝑘𝑣bigge han
≈1 pN/nm, ic ion inc eases as he iscosi y o he ECM inc eases
(Fig. 2(c)). Fo cons an alues o iscosi y, we show a ansi ion o
ic ion alues. Fo iscosi ies la ge han ≈10 pN⋅s/nm, we obse e
mono onically inc easing alues o ic ion, ha each ≈2 kPa⋅s∕μm2.
Fo iscosi ies smalle han ≈4 pN⋅s/nm, we see a biphasic beha io .
F ic ion inc eases up o ≈0.7 kPa⋅s∕μm2a 𝑘𝑣≈10 pN/nm, and hen
i dec eases, desc ibing a nega i e g adien o he elaxa ion s i ness
(Fig. 2(c)).
Ou esul s con i m p e ious da a on how cell adhesion esponds
o he elas ic s i ness o he ECM [29,41,42], showing ha cells mi-
g a e owa ds posi i e g adien s o ic ion c ea ed by he asymme ic
s eng h in cell adhesions [28,29]. When cells exp ess alin ein o ce-
men , as in ou model, his mig a ion occu s owa ds he posi i e
g adien o he elas ic s i ness [29,42]. Fu he , we show ha an
inc ease in he iscosi y o cons an alues o elaxa ion s i nesses
beha es equi alen ly o a s i ness inc ease in pu e elas ic subs a es
wi h cells exp essing alin ein o cemen [42]. Howe e , an inc ease
in he elaxa ion s i ness induces, o inc easing alues o iscosi y,
a ansi ion om an adhesion beha io eminiscen o non- ein o ced
adhesion complexes o ein o ced ones (Fig. 2(c)).
3.2. Theo y p edic s isco axis in single cells
We simula e cells placed on subs a es o leng h 𝐿𝑠equal o 200 and
1000 μm (Fig. 3). To ob ain alues o ic ion in he mo ing coo dina e
sys em ha ep esen s he mig a ing cell, 𝑘𝑒,𝑘𝑣, and 𝜇become unc-
ions o space ha ollow a linea ela ion wi h posi i e (e.g. 𝑥𝑘𝑣∕𝐿𝑠)
o nega i e slope (e.g. 𝑘𝑣(𝐿𝑠−𝑥)∕𝐿𝑠). Because he absolu e iscoelas ic
p ope ies in which cells a e ini ially loca ed modi y he ic ion hey
expe ience, we also analyze h ee di e en ini ial loca ions. In hese
h ee samples, we ix he long- e m s i ness o 𝑘𝑒= 1 pN/nm and we
a y independen ly 𝑘𝑣and 𝜇 om 10−2 o 102(Fig. 2b-c). We conside
ha cells unde go isco axis i he mig a ion eloci y is la ge han 𝑣=
1nm/s, o he wise we assume ha hey sp ead and emain s a iona y.
Cells on he 200 μm sample exp ess he s onges isco ac ic esponse
compa ed o he 1000 μm sample, which did no exp ess isco axis
(Fig. 3). In all cases, he s onges isco ac ic esponse is ob ained o
cells loca ed a he la ges g adien o he sample. The e o e, o analyze
he di e ences be ween he iscous p ope ies o hese ma ices in
de ail, we ocus on he 200 μm sample Fig. 3(a).
Cells ini ially loca ed a he cen e o he sample show a pe sis en
posi i e isco axis wi h eloci ies o ∼5 nm/s o samples o a ying
𝑘𝑣and ixed 𝜇=2 and 20 pN⋅s/nm. Same beha io is obse ed o
samples o a ying 𝜇and 𝑘𝑣≈20 pN/nm. On he o he hand, cells
ini ially loca ed a he cen e o he sample bu o a cons an 𝜇= 0.02
pN⋅s/nm emained s a iona y because he e was no ic ion g adien
o he cells o pola ize. The same beha io is obse ed o samples o
a ying 𝜇and 𝑘𝑣≈0.02 pN/nm (see Fig. 2b-c). Howe e , when cells
a e ini ially loca ed a ha same middle spo bu o samples o a ying
𝑘𝑣and cons an 𝜇= 2 pN⋅s/nm, hey exp ess nega i e isco axis as hey
mo e owa ds nega i e g adien s o he iscosi y, which co espond o
egions whe e he ic ion g adien is nega i e (Fig. 2c).
Ou esul s show ha in mos cases cells mig a e owa ds he
posi i e g adien s o he elaxa ion pa ame e s 𝑘𝑣and 𝜇o he ECM
(Fig. 3), i.e. exp essed posi i e isco axis. Howe e , he e a e speci ic
combina ions o iscous p ope ies ha induce nega i e isco axis. Ou
esul s sugges , in ag eemen wi h p e ious esul s on du o axis [28,
29], ha g adien s in he elaxa ion s i ness o he ECM can exp ess
bo h posi i e and nega i e isco axis. Howe e , g adien s in 𝜇can
only induce posi i e isco axis. All hese esul s a e explained by he
adhesion s eng h shown in he p e ious sec ion (Fig. 2). In o he
wo ds, ou esul s indica e ha cells will always mig a e wi h he
cell on o ien ed wi h he s onges adhesion o , equi alen ly, o-
wa ds posi i e g adien s o ic ion [29]. The asymme ic adhesion
o ces induce asymme ic e og ade low which, gi en he cons an
polyme iza ion eloci y a bo h cell edges, es ablishes he cell on .
3.3. Visco axis is media ed by in acellula asymme ic mo ile o ces
To ge a deepe unde s anding o how isco axis eme ges and
p og esses, we analyze he in acellula mo ile o ces ha coo dina e
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P. Saez e al.
Fig. 3. Mig a ion eloci y o e ime o subs a es o 200 μm (a) and o 1000 μm (b) in leng h. Le and igh columns a e o cells loca ed a poin s ii and iii, espec i ely. Cells
loca ed a poin i did no mig a e (da a no shown). Fo each subs a e, esul s in he i s ow a e o ixed alues o 𝜇 o 0.02 (blue), 2 ( ed), and 20 (black) pN⋅𝑠/nm while cells
mig a e on a subs a e o a ying 𝑘𝑣. In he second aw, we ixed alues o 𝑘𝑣 o 0.02 (blue), 0.2 ( ed), and 20 (black) pN/nm while cells mig a e on a subs a e o a ying 𝜇.
Fig. 4. Fo he black line case (i in Fig. 3a, bo om-le ) we show he e olu ion o
in acellula a iables in space and ime. A s eady s a e, (a) ac in (black) and myosin
(blue) densi ies and (b) e og ade low, (c) polyme iza ion eloci y (solid) and he
e og ade eloci y a he cell memb ane (dash) in ime, and (d) kymog aph o he
ic ion 𝜂.
cell mig a ion in space and ime. We ocus on he case ha showed he
s onges isco ac ic esponse (sample wi h a iscous g adien o 200 μm
in leng h, ixed 𝑘𝑒=0.1 pN/nm and 𝑘𝑣= 20 pN/nm, and cells ini ially
loca ed a 𝜇= 20 pN⋅s/nm). As cells mig a e, hey a el h ough
egions o di e en iscosi ies and, he e o e, he ic ion compu ed
h ough he clu ch model changes in space and ime (Fig. 4d). This
he e ogeneous ic ion along he cell c ea es an asymme ic e og ade
low (Fig. 4b, which compe es wi h he cons an polyme iza ion e-
loci y 𝑣𝑝a he cell ends (Fig. 4c). The igh end p o udes as e
and, he e o e, ep esen s he cell on . The asymme ic e og ade
low also pola izes he ac omyosin ne wo k (Fig. 4a). All o he cases
o isco axis desc ibed abo e exhibi simila esul s, wi h he model
a iables pola ized owa ds he posi i e o nega i e di ec ion, and
a a smalle o la ge ex en , depending on he s eng h o he cell
ic ion. Because isco axis and du o axis eme ge om he same o ce
p inciples, he s eng h o he cell adhesion and, he e o e, he s eng h
o he asymme ic ic ion alues, ou esul s a e in ag eemen wi h he
beha io obse ed du ing du o axis [28,29].
3.4. Compe i ion be ween isco axis and du o axis
Then, we wonde how isco axis compe es wi h du o axis. G adien s
o he elas ic s i ness may coexis wi h g adien s o he elaxa ion
p ope ies o he ECM in i o. The e o e, i is impo an o unde s and
how hese wo di e en bu coexis ing mechanical signals may con ol
cell mig a ion. To do so, we ake ou p e ious model o du o axis [29]
and pu i in compe i ion wi h ou cu en model o isco axis. We name
his coope a i e esponse as iscoelas o axis.
We use again he s onges isco ac ic esponse desc ibed abo e.
Howe e , we le he long- e m s i ness 𝑘𝑒change o enable du o axis.
We also use he s onges du o ac ic esponse o a sample o 200 μm
in leng h whe e he long- e m s i ness changes be ween 10−2 o 102
pN/nm. We simula e ma ices whe e he long- e m and iscosi y s i -
ness change along he same di ec ion o opposi e o each o he . Fo
aligned iscoelas ic and elas ic g adien s, he ic ion p o ile is simila
o pu e isco axis (Fig. 5a), al hough he mig a o y abili y is enhanced
and cells each a maximum eloci y o 6 nm/s. Howe e , when he
wo mechanical g adien s o he ECM a e in opposi e di ec ions, cells
may exp ess posi i e, nega i e, o null iscoelas o axis. We show a
minimum in ic ion a ≈1 pN/nm elaxa ion s i ness (Fig. 5(a). We
e m his spo di e ging s i ness. Cells placed ini ially on he le o he
di e ging s i ness exp ess nega i e iscoelas o axis while cells on he
igh exp ess posi i e du o axis. Cells loca ed a he di e ging s i ness
would end o mig a e away om i con a y o du o axis, whe e an
op imum igidi y ma ks a goal loca ion o he cells o mig a e [29].
Finally, we use p e ious expe imen al esul s [39] o alida e ou
compu a ional model. Cells we e loca ed in subs a es o di e en
iscous and elas ic p ope ies. In summa y, wo d ops o p e-polyme
solu ions, which we also es ed mechanically, a e placed in con ac and
di use wi h each o he o o m a elaxa ion g adien (see [39] and
Ma e ials & Me hods o de ails). The equency sweep es showed ha
hese ma e ials ha e he same s o age modulus (G’) bu di e en loss
moduli (G’’). F om he small ampli ude oscilla ion egime, we show
ha he elas ic moduli we e nea ly he same o bo h gels, which
also ansla es o equal sho - e m o bo h gels (see Fig. 6a). These
wo polyme s we e also es ed in a c eep es o applied s ess o
𝜎0=10 Pa. High loss and low loss gels de o m equally a sho ime
scales ( <2 s). Fi ing o he ma e ial pa ame e s o he SLS model
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P. Saez e al.
Fig. 5. (a) E ec i e ic ion o he aligned elas ic and elaxa ion g adien s (solid
line) and opposing elas ic and elaxa ion g adien s (dash line). G adien di ec ion is
speci ied by a iangle. F ic ion inc eases mono onically o inc easing alues o he
elas ic s i ness, while i p esen s a conca e shape wi h a minimum a ≈ 1pN/nm,
whe e he di e ging igidi y localizes. (b) Mig a ion eloci ies o single cells in a sample
o 200 μm in leng h exposed o aligned and opposing du o ac ic and isco ac ic signals.
Black line co esponds o he case o aligned g adien s. Fo opposing cues, cells a e
ini ially loca ed a he ed c oss (𝑘𝑒=0.1 pN/nm and 𝑘𝑣=10 pN/nm), blue c oss (𝑘𝑒=1
pN/nm and 𝑘𝑣1=pN/nm) and yellow c oss (𝑘𝑒=10 pN/nm and 𝑘𝑣=0.1 pN/nm) in
panel a. Colo s in eloci ies in panel b co espond o hese colo s in panel a.
Fig. 6. (a) Expe imen al equency sweep G’ modulus (diamonds) and G’’ modulus
(ci cles) o Low Loss (blue) and High Loss ( ed) gels. (b) Compliance (J) o he Low
Loss (blue) and High Loss ( ed) gels. The expe imen al compliance (ci cles) was i ed
ollowing he SLS model (lines). (c) A collec ion o acks o many cells ob ained om
mul iple independen expe imen s o hMSCs mig a ing on he gels wi h g adien s in he
iscoelas ic p ope ies. The ma e ial a ies om 𝑘𝑒= 1.06 pN/nm, 𝑘𝑣= 0.28 pN/nm
and 𝜇=38.5 pN⋅s/nm on he le o 𝑘𝑒= 0.22 pN/nm, 𝑘𝑣= 1.11 pN/nm and 𝜇=46.2
pN⋅s/nm on he igh (scale ba 50 μm). (d) No malized eloci ies wi h espec o he
con ol case in (c) . Colo s ep esen he con ol case in black, ixed 𝜇in blue, ixed
𝑘𝑒in ed, ixed 𝑘𝑒and 𝑘𝑣in g een, and ixed 𝑘𝑣in magen a.
(Fig. 6b). showed ha he i s ma e ial has 𝑘𝑒= 1.06 pN/nm, 𝑘𝑣= 0.28
pN/nm and 𝜇=38.5 pN⋅s/nm and he second has 𝑘𝑒= 0.22 pN/nm,
𝑘𝑣= 1.11 pN/nm and 𝜇=46.2 pN⋅s/nm. This ep esen s a complex
isco ac ic s a e whe e g adien s o long- e m and iscous modulus and
iscosi y a e es ablished in opposi e di ec ions and he p edic ion o
wha di ec ion cells would mo e owa ds is no clea .
Then, cells we e seeded on he subs a e and he mig a ion di ec ion
was measu ed. 23 cells ou o 33 analyzed mig a ed owa ds he
posi i e g adien o he long- e m modulus, in opposi e di ec ions o he
iscous g adien s (Fig. 6c). Then, we simula e he same iscoelas o axis
case. Ou esul s ep oduce he mig a ion owa ds he posi i e g adien
o he long- e m modulus (see con ol case, Fig. 6d).
To u he explo e he model esul s, we pe o med se e al simu-
la ions o p edic how isco-elas o axis would beha e i he ma e ial
p ope ies changed. We se each o he iscoelas ic pa ame e s cons an
and lea e he o he s as he con ol case (Fig. 6d). Ou esul s show ha
i we keep 𝑘𝑒o 𝑘𝑒and 𝑘𝑣cons an mig a ion di ec ion is e e sed.
A close look in o he mig a ion eloci ies indica es ha he opposing
g adien s in 𝑘𝑒and 𝑘𝑣ac as compe ing mechanical s imuli. When one
o hese pa ame e s is kep cons an , he mig a ion di ec ion inc eases
by one o de o magni ude, indica ing ha in he con ol case, hey
we e ac ing agains each o he . When 𝜇was ixed, we obse ed a mild
change in mig a ion eloci y. These esul s indica e ha he elaxa ion
ime scale is a om he imescale o he adhesion cycle de e mined by
he clu ch model. Indeed, when 𝑘𝑒and 𝑘𝑣a e ixed, mig a ion eloci y
dec eases by one o de o magni ude.
4. Discussion
A deep unde s anding and p ecise con ol o cell mig a ion a e
key o medicine and bioenginee ing as i may allow us o no only
unde s and how umo cells in ade and me as asize [4,5] o how
cells o ches a e issue egene a ion [1,60], bu also o enginee cell
mig a ion o be e biomime ic issue designs [61,62].
In his wo k, we ha e shown h ough physical a gumen s how single
cells may exp ess isco axis, as p e iously shown by an expe imen al
s udy. Mos li ing issues a e iscoelas ic; he e o e, cells mig a ing
ac oss hese ma e ials eel a combina ion o iscoelas ic p ope ies. Un-
de s anding how g adien s in he elaxa ion p ope ies o he ECM may
impac cell mig a ion is a undamen al ques ion in biology, medicine,
and bioenginee ing.
He e, we show ha he same physical mechanisms ha enable
du o axis dic a e isco axis. Ou esul s show ha g adien s o he
elaxa ion p ope ies o he ECM may ac i a e an o ien ed cell mig a-
ion hanks o an asymme ic adhesion s eng h, o cell-ECM ic ion.
We also show ha speci ic combina ions o iscoelas ic p ope ies
may induce posi i e and nega i e isco axis. Speci ically, g adien s in
he iscosi y o he ECM always induce posi i e isco axis when he
elaxa ion s i ness is la ge enough compa ed o he long- e m s i ness,
o no mig a ion a all i he elaxa ion s i ness is negligible because
no elaxa ion o he ma e ial occu s a he ime scales o he cellula
adhesion. Howe e , g adien s in he elaxa ion s i ness show bo h
posi i e and nega i e isco axis. I he elaxa ion s i ness is smalle
o simila compa ed o he long- e m s i ness, cells will mig a e wi h
a weak posi i e isco axis, because he ma e ial’s elaxa ion will also
be weak. As he alues o he elaxa ion s i ness inc ease, posi i e
isco axis is enhanced when a la ge iscosi y o he ma e ial allows
o ma e ial elaxa ion. Howe e , i he iscosi y is low compa ed o
he ime scale o cell adhesion, he e ec o high elaxa ion s i ness is
hinde ed.
We also con on ed isco axis and du o axis and ound ha a di-
e ging minimum in he cell ic ion makes cells mig a e away om i ,
con a y o wha happens wi h he op imum s i ness in du o axis [28,
29]. O he ma hema ical models o du o axis ha e also been used o
explain how du o axis eme ged [63–66]. This di e ging poin appea s
due o he sum o he opposing g adien s in he long e m and elaxa ion
s i nesses, o ic ion. The poin a which he minimum o hese
added con ibu ions appea s co esponds o he di e ging minimum.
Ou esul s also indica ed ha du o axis seems o be mo e e icien in
con olling o ien ed cell mig a ion han isco axis. Howe e , a iscous
g adien could also shi he di ec ion o du o axis, o inhibi i . When
he elaxa ion imescale o he ma e ial is much la ge han he ad-
hesion imescale, o when he elaxa ion s i ness is low compa ed o
he long- e m s i ness, cells mos ly sense he nea ly elas ic long- e m
Ma hema ical Biosciences 380 (2025) 109362
7
P. Saez e al.
s i ness o he subs a e. In hese cases, he isco ac ic esponse is weak
and cell mig a ion would be due o du o axis o would no occu .
These opposing esponses o iscoelas o axis can be used o design-
ing elaxa ion g adien s o he ECM o ei he a es o shi an ini ially
o ien ed du o ac ic esponse (see Fig. 5). O , he o he way a ound,
elas ic s i ness g adien s can be manipula ed o change pu e isco ac ic
esponses, as shown in Fig. 5.
The e is also limi a ion in ou model ha should be add essed in
he u u e. The i s one is due o he 1D app oach we chose, which
limi s he analysis o shape changes in he cell o he s udy o mig a ion
in mo e complex en i onmen s. Well-es ablished nume ical me hods
could be used in he u u e o add ess mig a ion in highe dimen-
sions [67,68]. We also used a simple SLS model. Pe haps, ce ain ECMs
a e be e ep oduced wi h o he iscoelas ic models. Mo eo e , we
ocused exclusi ely on mechanical signals and, i hey a e no p esen ,
he cell would jus sp ead symme ically. Howe e , he e a e always
o he signals ha may also coexis wi h he mechanical g adien s,
and ha can be esponsible o he na u al pola iza ion o he cell.
These signals can igge downs eam pola iza ion o GTPases [69]
ha may sel -pola ize due o noisy s imuli, which would e en ually
pola ize he downs eam signals ha , e en ually, induce cell mig a ion.
Such in e ac ion be ween mechanical and chemical signals should be
add essed in he u u e.
In summa y, we ha e added he e ec o a iscous ma ix o
he no ion o du o axis and a ionalized how isco axis wo ks. Ou
model desc ibes o he i s ime he compe i ion be ween elas ic and
iscous g adien s o he ECM and es ablishes a new, o mo e complex
mechanical signal o con ol cell mig a ion.
CRediT au ho ship con ibu ion s a emen
Pablo Saez: W i ing – e iew & edi ing, W i ing – o iginal d a ,
So wa e, Me hodology, In es iga ion, Fo mal analysis, Concep ualiza-
ion. Palla i U. Shi ke: Da a cu a ion. Jyo i R. Se h: Fo mal analysis.
Jo ge Aleg e-Cebollada: W i ing – e iew & edi ing, Concep ualiza-
ion. Abhiji Majumde : W i ing – e iew & edi ing, Supe ision.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
in luence he wo k epo ed in his pape .
Acknowledgmen s
P.S acknowledges suppo om he Minis e io de Ciencia, Inno-
ación y Uni e sidades (MCIU), G an PID2019-11094GB-100 unded
by MICIU/ AEI /10.13039/501100011033. The CNIC is suppo ed by
he Ins i u o de Salud Ca los III (ISCIII), MCIU, and he P o CNIC Foun-
da ion and is a Se e o Ochoa Cen e o Excellence (g an CEX2020-
001041-S unded by MCIU).
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