Sys emic cellula mig a ion: The o ces d i ing he
di ec ed locomo ion mo emen o cells
Ilde onso M. De la Fuen e
a,b,
*, Jose Ca asco-Pujan e
c
, Bo ja Camino-Pon es
d
, Ma ia Fede z
e
, Ca los B ingas
c
,
Albe o Pé ez-Sama ín
, Go ka Pé ez-Ya za
c
, José I. López
d
, Ike Malaina
a,†
and Jesus M. Co es
c,d,g,†
a
Depa men o Ma hema ics, Facul y o Science and Technology, Uni e si y o he Basque Coun y, UPV/EHU, Leioa 48940, Spain
b
Depa men o Nu i ion, CEBAS-CSIC Ins i u e, Espina do Uni e si y Campus, Mu cia 30100, Spain
c
Depa men o Cell Biology and His ology, Facul y o Medicine and Nu sing, Uni e si y o he Basque Coun y, UPV/EHU, Leioa 48940, Spain
d
Biobizkaia Heal h Resea ch Ins i u e, Ba akaldo 48903, Spain
e
Depa men o Cell Biology and Immunology, Ins i u e o Pa asi ology and Biomedicine “López-Ney a”, CSIC, G anada 18016, Spain
Depa men o Neu osciences, Facul y o Medicine and Nu sing, Uni e si y o he Basque Coun y, UPV/EHU, Leioa 48940, Spain
g
IKERBASQUE: The Basque Founda ion o Science, Bilbao 48009, Spain
*To whom co espondence should be add essed: Email: [email p o ec ed]; [email p o ec ed]
†
I.M. and J.M.C. con ibu ed equally o his wo k.
Edi ed By: Ho acio Espinosa
Abs ac
Di ec ional mo ili y is an essen ial p ope y o cells. Despi e i s eno mous ele ance in many undamen al physiological and pa hological
p ocesses, how cells con ol hei locomo ion mo emen s emains an un esol ed ques ion. He e, we ha e add essed he sys emic
p ocesses d i ing he di ec ed locomo ion o cells. Speci ically, we ha e pe o med an exhaus i e s udy analyzing he ajec o ies o
700 indi idual cells belonging o h ee di e en species (Amoeba p o eus, Me amoeba lening adensis, and Amoeba bo okensis) in ou
di e en scena ios: in absence o s imuli, unde an elec ic ield (gal ano axis), in a chemo ac ic g adien (chemo axis), and unde
simul aneous gal ano ac ic and chemo ac ic s imuli. All mo emen s we e analyzed using ad anced quan i a i e ools. The esul s
show ha he ajec o ies a e mainly cha ac e ized by cohe en in eg a i e esponses ha ope a e a he global cellula scale. These
sys emic mig a o y mo emen s depend on he coope a i e nonlinea in e ac ion o mos , i no all, molecula componen s o cells.
Keywo ds: cellula mig a ion, sys emic beha io , sel -o ganiza ion, quan i a i e analysis, amoebae
Signi icance S a emen
Cellula mig a ion is a co ne s one issue in many human physiological and pa hological p ocesses. Fo yea s, he scien i ic a en ion
has been ocused on he indi idualized s udy o he di e se molecula pa s in ol ed in di ec ional mo ili y; howe e , locomo ion
mo emen s ha e ne e been ega ded as a sys emic p ocess ha ope a es a a global cellula scale. In ou quan i a i e expe imen al
analysis, essen ial sys emic p ope ies unde lying locomo ion mo emen s we e de ec ed. Such eme gen sys emic p ope ies a e no
ound speci ically in any o he molecula pa s, pa ial mechanisms, o indi idual p ocesses o he cell. Cellula displacemen s seem
o be egula ed by in eg a i e p ocesses ope a ing a sys emic le el.
Compe ing In e es : The au ho s decla e no compe ing in e es .
Recei ed: Decembe 21, 2023. Accep ed: Ap il 11, 2024
© The Au ho (s) 2024. Published by Ox o d Uni e si y P ess on behal o Na ional Academy o Sciences. This is an Open Access a icle
dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License (h ps://c ea i ecommons.o g/licenses/by/4.0/), which pe mi s
un es ic ed euse, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
In oduc ion
Sel -locomo ion is one o he mos impo an complex beha io s
o cells endowed wi h mig a o y esponses. In he pe manen
s uggle o su i al, ee cells mo e e icien ly o ind ood ollow-
ing adequa e di ec ion and speed, a oiding p eda o s and ad e se
condi ions. Cell mo ili y is c ucial o li e in Me azoan o ganisms
and undamen al o es ablish he app op ia e o ganiza ion o all
mul icellula o ganisms, playing a cen al ole in a ple ho a o essen-
ial biological phenomena such as emb yogenesis, mo phogenesis,
o ganogenesis, neu al de elopmen , adul issue emodeling, wound
healing, immune esponses, angiogenesis, issue egene a ion
and epai , cell di e en ia ion, e c. (1). Mo eo e , he de egula ion
o cell mo emen s in humans is in ol ed in many pa hological p oc-
esses such as me as a ic umo p og ession (2–5), a he oscle osis
and o he ascula diseases (6), congeni al b ain pa hologies (7),
os eoa h i is (8), heuma oid a h i is (8, 9), hea ing diso de s (10),
as hma (11–13), ch onic obs uc i e pulmona y disease (14), mul iple
scle osis (15, 16), pso iasis (17, 18), C ohn’s disease (19, 20), and
immune- ela ed ac inopa hies (21). Al hough cell locomo ion was
al eady obse ed in 1675 by an Leeuwenhoek in his espec ed pion-
ee mic oscopic s udies (22), esea che s and schola s ha e no ye
come o un eil how cells mig a e in he p esence o complex cues.
Gi en i s impo ance, g ea a en ion has been ocused on he
s udy o he di e se molecula pa s in ol ed in di ec ional mo il-
i y. A ele an numbe o hese expe imen al s udies ha e
PNAS Nexus, 2024, 3, pgae171
h ps://doi.o g/10.1093/pnasnexus/pgae171
Ad ance access publica ion 20 Ap il 2024
Resea ch Repo
Downloaded om h ps://academic.oup.com/pnasnexus/a icle/3/5/pgae171/7655426 by Cen o Medi e aneo de In es igaciones Ma inas y Ambien ales use on 21 No embe 2024
unequi ocally shown ha locomo ion mo emen s a e complex
p ocesses ha in ol e p ac ically all cellula componen s. So, di-
ec ed mo emen s a e p ima ily d i en by he cy oskele on ( he
essen ial pa o he locomo ion sys em) which is a sophis ica ed
dynamic s uc u e o med by h ee main molecula componen s:
ac in mic o ilamen s, mic o ubules, and in e media e ilamen s,
all o hem in e ac ing in complex dynamic ne wo ks (23).
In pa icula , he ac i i y o ac in cy oskele on ne wo ks is
la gely dependen on a wide a ie y o egula o y molecules
such as small GTPases (24), in eg ins (25), and many pos ansla-
ional modi ica ions such as phospho yla ion, ace yla ion, a giny-
la ion, oxida ion, and o he s (26). In addi ion, he dynamic
u no e o he ac in ilamen ne wo ks is essen ial o egula e
cell mig a ion (27). Cy oskele on ne wo ks a e coupled wi h o he
complex egula ed sys ems such as memb ane su ace ecep o s
and signal ansduc ion pa hways which also pa icipa e in he
con ol o locomo ion mo emen s (28). Ene gy is ano he essen ial
elemen in cell mo ili y; when cells mo e he cy oskele on ans-
o ms chemical ene gy in o mechanical o ces (dynein cy oskel-
e al mo o p o eins) en ailing conside able bioene ge ic
demands; o such a pu pose he mi ochond ial ac i i y and he
adenyla e ene gy sys em a e impo an egula o s o di ec ional-
i y mo ion (29). Cell memb ane ac i i ies a e also necessa y o im-
plemen an adequa e mig a ion (30, 31).
Recen s udies ha e e ealed he impo ance o au ophagy (an
in acellula p ocess ha con ols p o ein and o ganelle deg ad-
a ion and ecycling) in he con ol o locomo ion (32). The u n-
o e o ocal adhesions also egula es cell sp eading and
mig a ion (33). Calcium ions (Ca
2+
), which impac globally on al-
mos e e y aspec o cellula li e, play an impo an ole in he
con ol o di ec ed mo emen s (34). In his sense, he endoplasmic
e iculum, a mul i unc ional signaling o ganelle which con ols a
wide ange o cellula p ocesses such as he en y and elease o
calcium ions, also pa icipa e in he egula ion o cell locomo ion
(35, 36). Cell pola i y is equi ed o an adequa e di ec ionali y mo-
ion and he e a e a lo o molecula p ocesses ha ha e been im-
plica ed in he in insic pola i y s a us o cells; in his ega d, he
cen osomes posi ioning se es as a s ee ing de ice o he di ec-
ional mo emen (37, 38) and dynein oge he wi h o he mole-
cules egula es cen osomal o ien a ion o es ablish and
main ain cell pola i y (39).
The Golgi appa a us (ano he impo an molecula p ocess-
ing cen e o modi ied p o eins ecei ed om he endoplasmic
e iculum) allows he emodeling o in acellula a ic p oc-
esses owa d he di ec ion o mo emen ; he e o e, signals
om he Golgi ma ix play an impo an ole in cell mo ili y
(40). The nucleus is e y impo an o de eloping app op ia e
mechanical esponses du ing cell mig a ion, in ac his o gan-
elle beha es as a cen al mechanosenso y s uc u e, and i s
physical p ope ies s ongly connec ed o he cy oskele on gua -
an ee a p ope cell mig a ion (41, 42). Recen ly, i has been de-
sc ibed ha s uc u al ch oma in o ganiza ion also has a key
ole in he cellula mig a ion p ocess (43). In addi ion, many mol-
ecules and co esponding p ocesses a e in ol ed in di ec ional
mo emen o cells as, o ins ance, ocal adhesion p o eins ( a-
lin, paxillin, inculin, and o he s) (28), SCAR/WAVE p o eins
(44), ac in-binding p o eins (45), p21-ac i a ed kinases (a amily
o se ine/ h eonine kinases) (46), TORC2/PKB pa hway (47),
mi ogen-ac i a ed p o ein kinases (48), A p2/3 complexes (49),
WASP amily p o eins (50), Nck amily o adap o p o eins
(51), e c. In addi ion, nume ous s udies p o ide ha in eg a i e
unc ional esponses unde lie in cell unc ionali y which allow he
eme gence o sys emic beha io s (52–57).
All his e idence sugges s ha cell mig a ion is no a me e
me abolic-molecula p ocess which can be egula ed by any o
i s indi idually conside ed componen s. Mos , i no all, unda-
men al cellula physiological p ocesses appea o be in ol ed in
cellula locomo ion, which is indica i e o he eme gence o a glo-
bal unc ional phenomenon in he cell. Howe e , con i ming he
sys emic na u e o cellula locomo ion ep esen s a scien i ic
challenge o g ea di icul y. Such e i ica ion equi es mul idis-
ciplina y app oaches ha combine complex expe imen al s udies
wi h ad anced quan i a i e me hods.
He e, we ha e add essed he key ques ion: Is cell mig a ion a
highly coo dina ed and in eg a ed eme gen p ocess a he global
cellula le el? To answe his ques ion, we ha e designed a la ge
quan i a i e s udy o analyze he sys emic ajec o ies o 700 indi-
idual cells belonging o h ee di e en species: Amoeba p o eus,
Me amoeba lening adensis, and Amoeba bo okensis. Such analysis
has been pe o med unde ou di e en scena ios: in absence
o s imuli, unde chemo ac ic g adien (we ha e used an nFMLP
pep ide, which indica es o he amoebae he possible p esence
o ood in hei immedia e en i onmen ), in an elec ic ield ( he
elec ic memb ane po en ial o cells enables p eda o s like amoe-
bas he de ec ion o p eys), and unde complex ex e nal condi-
ions such as simul aneous gal ano ac ic and chemo ac ic
g adien s imuli.
To unde s and he o ces d i ing he locomo ion mo emen o
he cell, all ajec o ies we e analyzed using compu a ional me h-
ods and ad anced nonlinea physical–ma hema ical ools oo ed
in S a is ical Physics (S a is ical Mechanics). These quan i a i e
s udies ocused on some essen ial cha ac e is ics o he sys emic
dynamics unde lying locomo ion mo emen s. The esul s indi-
ca e ha a e y complex dynamic s uc u e eme ges in he mig a-
o y mo emen s o all he cells analyzed. Such s uc u e is mainly
cha ac e ized by highly o ganized mo e-s ep sequences wi h e y
low en opy and high in o ma ion, ma ked in e dependence in
he mo e s eps wi h powe -law au oco ela ion decays, s ong
anomalous supe di usion dynamics, pe sis ence e ec s wi h
end- ein o cing beha io , and e icien mo emen s o explo e
he ex acellula medium.
This ou s anding cellula dynamic s uc u e is a consequence
o he eme gen sys emic dynamics occu ing in he cell. The loco-
mo ion mo emen s seem o depend on a complex in eg a ed sel -
o ganized sys em ca e ully egula ed a global le el, a ising om
he coope a i e nonlinea in e ac ion o mos , i no all, cellula
componen s. Such eme gen sys emic p ope ies a e no ound
speci ically in any o he molecula pa s, pa ial mechanisms,
o indi idual p ocesses o he cell.
Resul s
The mig a o y ajec o ies o 700 indi idual cells belonging o he
h ee species, A. p o eus, M. lening adensis, and A. bo okensis, we e
eco ded in ou di e en scena ios: in absence o s imuli, unde
chemo ac ic g adien , in an elec ic ield, and unde simul aneous
gal ano ac ic and chemo ac ic s imuli. Amoebae show obus
mo emen in esponse o an elec ic ield in a ange be ween
300 and 600 mV/mm (gal ano axis). Unde such condi ions, p ac-
ically all amoebae mig a e owa d he ca hode (58). Likewise,
hese cells also exhibi chemo ac ic mo emen s. Mo e speci ical-
ly, he pep ide nFMLP (N- o mylme hionyl-leucyl-phenylalanine)
sec e ed by bac e ia may indica e ha ood migh be in he nea
en i onmen , p o oking a s ong chemo ac ic esponse (59).
All ou expe imen s we e pe o med in a speci ic se up consis -
ing o wo s anda d elec opho esis blocks (17.5 cm long), wo aga
2 | PNAS Nexus, 2024, Vol. 3, No. 5
Downloaded om h ps://academic.oup.com/pnasnexus/a icle/3/5/pgae171/7655426 by Cen o Medi e aneo de In es igaciones Ma inas y Ambien ales use on 21 No embe 2024
b idges, a powe supply, and in he middle o he expe imen al
pla o m, a s uc u e o s anda d glass slide and co e s whe e
he cells we e loca ed (see Fig. 1A–E and supplemen a y
ma e ial). One elec opho esis block was di ec ly plugged in o a
no mal powe supply and he o he was connec ed o he i s
one h ough wo aga b idges, hus p e en ing he di ec con ac
o he anode and ca hode wi h he medium (Chalkley’s simpli ied
medium (58)) whe e he cells we e placed. Speci ically, he amoe-
bae we e a anged in he cen e o he s uc u e o s anda d glass
slide and co e s (expe imen al chambe ) and hei mig a o y dis-
placemen s we e moni o ed. The glass expe imen al s uc u e
enabled he gene a ion o a lamina lux allowing he elec ic cu -
en o pass h ough, on one hand, and gene a ing an nFMLP pep-
ide g adien , on he o he (supplemen a y ma e ial).
P io o each expe imen , all cells we e s a ed o 24 h. The in-
di idual mig a o y mo emen s o each cell we e eco ded o e pe-
iods o 30 min using a digi al came a a ached o a s e eo
mic oscope. The expe imen s on la 2D su aces we e always
made wi h small g oups o cells (no mo e han nine cells pe ep-
lica ion). The ollowing basic expe imen al in o ma ion da a
(BEID) is p o ided o each scena io: “N ” is he numbe o cells
pe eplica ion, “E ” is he numbe o expe imen al eplica ions,
and “N” is he o al numbe o cells. Finally, he eco ded ajec o -
ies we e analyzed in he o m o ime se ies using ad anced non-
linea dynamic ools.
Cellula mig a o y mo emen s wi hou ex e nal
s imulus
Fi s , we eco ded he locomo ion ajec o ies o 153 indi idual
cells belonging o he 3 species conside ed in a medium wi hou
any ex e nal in luence (BEID: A. p o eus: n = 50, E = 7, N = 7–8;
M. lening adensis: n = 51, E = 7, N = 5–8; A. bo okensis: n = 52, E = 7,
N = 6–8). In Fig. 2A, a ep esen a i e example o hese amoebae
mig a o y mo emen s in absence o s imuli is depic ed ( o cla i y
only 60 cells we e andomly aken om he o al). I can be ob-
se ed ha a e 30 min, cells ha e explo ed p ac ically all he di-
ec ions o he expe imen a ion chambe . To quan i a i ely
analyze cell di ec ionali y, we calcula ed he displacemen cosine
o each ajec o y, 153 cells in o al (Fig. 2A′). Values close o −1
indica e a p e e ence owa d he le , while alues close o 1 sug-
ges a p e e ence owa d he igh . Ou analysis showed ha
alues anged be ween −1 and 1, wi h a median/IQR (in e qua ile
ange) o 0.05/1.40. Median/IQR alues o each species we e
0.42/1.24 (A. p o eus), −0.25/1.21 (M. lening adensis), and 0.13/1.23
(A. bo okensis). These esul s indica e ha in absence o s imuli cells
mo ed andomly wi hou any de ined guidance.
Cell mig a ion unde gal ano axis condi ions
The mig a o y ajec o ies o 147 cells belonging o he h ee spe-
cies we e eco ded unde an ex e nal con olled di ec -cu en
elec ic ield o abou 300–600 mV/mm. In Fig. 2B, a ep esen a i e
example o he mig a o y mo emen s o 60 cells is depic ed. They
show an unequi ocal sys emic esponse consis ing o he mig a-
ion o he ca hode which has been placed on he igh side o
he se up. The o e all median/IQR alue o he displacemen co-
sines o all 147 cells (Fig. 2B′) was 0.99/0.07. This inding con i med
ha a undamen al beha io cha ac e ized by an unequi ocal di -
ec ionali y owa d he ca hode had eme ged unde hese gal ano-
ac ic condi ions. The median/IQR alues o each species we e
0.99/0.02 (A. p o eus), 0.98/0.10 (M. lening adensis), and 0.99/0.08
(A. bo okensis). We compa ed he dis ibu ions o he alues o
he displacemen cosines unde gal ano axis wi h he alues
ob ained in he expe imen wi hou s imuli using he Wilcoxon
ank-sum es . The esul s indica ed ha bo h beha io s we e sig-
ni ican ly di e en o he h ee species and ha he gal ano ac ic
cellula beha io is highly unlikely o be ob ained by chance
(P- alues: 10
−9
, 10
−15
, and 10
−12
; Z: −5.85, −7.79, and −6.85 o
A. p o eus, M. lening adensis, and A. bo okensis, espec i ely). BEID:
A. p o eus: n = 49, E = 7, N = 6–8; M. lening adensis: n = 48, E = 7,
N = 6–8; A. bo okensis: n = 50, E = 8, N = 3–9.
Cell locomo ion unde chemo axis condi ions
The mig a o y beha io o 166 cells belonging o he h ee species
conside ed was eco ded unde condi ions o chemo ac ic g adi-
en . All he amoebae we e exposed o 30 min o an nFMLP pep ide
g adien which had been placed on he le side o he se up. In
Fig. 2C, a ep esen a i e example o hese mig a o y ajec o ies
wi h 60 cells is depic ed. Unde hese condi ions 78.31% o all
s udied amoebae showed locomo ion mo emen s owa d he a -
ac an pep ide.
The displacemen angle cosines o he 166 indi idual ajec o -
ies anged om −1 o 1, wi h a median/IQR alue o −0.67/0.87.
Median/IQR alues o each species we e −0.65/0.75 (A. p o eus),
−0.77/0.68 (M. lening adensis), and −0.51/1.17 (A. bo okensis), indi-
ca ing ha hey exhibi ed a single undamen al beha io o mo e-
men owa d he pep ide (Fig. 2C′). The Wilcoxon ank-sum es
showed signi ican di e ences be ween he cosine alues ob-
ained wi h and wi hou chemo ac ic s imulus (P- alues: 10
−7
,
0.00, and 0.02; Z: 5.08, 2.94, and 2.40 o A. p o eus, M. lening adensis,
and A. bo okensis, espec i ely), and be ween he cosine alues
wi h chemo ac ic g adien and wi h he p esence o an elec ic
ield (P- alues: 10
−15
, 10
−17
, and 10
−14
; Z: 8.00, 8.55, and 7.68
o A. p o eus, M. lening adensis, and A. bo okensis, espec i ely).
This con i med ha he sys emic locomo ion beha io unde he
chemo ac ic g adien was comple ely di e en om bo h
he absence o s imuli and he p esence o an elec ic ield.
BEID: A. p o eus: n = 51, E = 8, N = 5–7; M. lening adensis: n = 60,
E = 9, N = 5–8; A. bo okensis: n = 55, E = 10, N = 5–7.
Cellula displacemen unde simul aneous
gal ano ac ic and chemo ac ic s imuli
Once he locomo ion mo emen s o he cells we e eco ded unde
he h ee p e ious independen expe imen al scena ios (wi hou
s imuli, unde gal ano axis, and unde chemo axis), we s udied
he ajec o ies o 234 cells unde simul aneous gal ano ac ic
and chemo ac ic s imuli. Fo such a pu pose, he nFMLP pep ide
was a anged on he le o he se up (in he anode a ea) and he
ca hode was placed on he igh . In Fig. 2D, a ep esen a i e ex-
ample o hese locomo ion mo emen s (60 cells in o al) is de-
pic ed. Unde hese complex ex e nal condi ions, he esul s
showed ha 42% o he amoebae mig a ed owa d he ca hode
while he emaining 58% mo ed owa d he pep ide (anode).
The displacemen cosines o he 234 cells had an o e all me-
dian/IQR alue o −0.29/1.66. Mo e speci ically, he alues o
each species we e (−0.32/1.59, median/IQR) o A. p o eus,
(−0.54/1.81, median/IQR) o M. lening adensis, and (−0.14/1.45,
median/IQR) o A. bo okensis. This analysis quan i a i ely e i ied
ha wo main cellula mig a o y beha io s had eme ged in he
expe imen , one owa d he anode and ano he owa d he ca h-
ode (Fig. 2D′). The s a is ical analysis (Wilcoxon ank-sum es )
con i med he p esence o hese wo di e en beha io s o
A. p o eus (P- alue = 10
−14
; Z = 7.67), M. lening andensis (P- alue =
10
−13
; Z = 7.23), and A. bo okensis (P- alue = 10
−14
; Z = 7.55). BEID:
De la Fuen e e al. | 3
Downloaded om h ps://academic.oup.com/pnasnexus/a icle/3/5/pgae171/7655426 by Cen o Medi e aneo de In es igaciones Ma inas y Ambien ales use on 21 No embe 2024
A. p o eus: n = 83, E = 12, N = 6–8; M. lening adensis: n = 73, E = 11,
N = 5–8; A. bo okensis: n = 78, E = 12, N = 4–8.
Long- ange in e dependence in he mo e s eps o
cellula mig a o y displacemen s
An essen ial cha ac e is ic o sys emic beha io in complex sys-
ems is he p esence o dynamics wi h s ong long- ange co ela-
ions (60), “long- ange” e e s o decay in he au oco ela ion
unc ion slowe han exponen ial decay, whe e a single scale
domina es he decay. One o he mos ecognized ools o analyze
he p esence o hese co ela ions in ime se ies (mig a o y ajec-
o ies he e) is he “ oo mean squa e luc ua ion” (“ ms ” analysis),
a classical me hod in S a is ical Mechanics based on he ideas
aised by Gibbs (61) and Eins ein (62).
Long- ange in e dependence can be de ec ed by a powe -law
ela ion such ha F(l)∼lα, whe e l is he numbe o s eps. Fo un-
co ela ed da a, he luc ua ion exponen α is abou 0.5, whe eas
α>0.5 o α<0.5 indica e, espec i ely, he p esence o posi i e
o nega i e long- ange co ela ions (supplemen a y ma e ial). In
Fig. 3A, an illus a i e “ ms ” analysis o he locomo ion mo e-
men s o h ee ep esen a i e cells belonging o each species con-
side ed unde simul aneous chemo ac ic and gal ano ac ic
s imuli (A. p o eus and M. lening adensis) and unde chemo ac ic
condi ions (A. bo okensis) is depic ed.
The esul s o “ ms ” analysis o he 700 expe imen al cell a-
jec o ies a e shown in Fig. 3B ( o mo e de ails, see Table S1). All
he mig a o y ajec o ies exhibi long- ange co ela ions in hei
cellula mo e-s ep mig a o y luc ua ions. Speci ically, we ound
ha he scaling exponen α o he “ ms ” had a median/IQR alue
o 0.72/0.08 o A. p o eus, 0.73/0.08 o M. lening adensis, and 0.71/
0.08 o A. bo okensis. The alues o he “ ms ” analysis o he
o al expe imen al mig a o y ajec o ies analyzed anged om
0.56 o 0.87, wi h a median/IQR o 0.72/0.08, whe eas he alues
o he scaling exponen α o all shu led ajec o ies anged om
0.35 o 0.64, wi h a median/IQR alue o 0.47/0.07 (see Table S2
o mo e de ails). Mo eo e , a Wilcoxon es compa ing measu ed
exponen s o hose om shu led ajec o ies e ealed highly
Fig. 1. Expe imen al se up layou . A and D) Top and la e al iews o he expe imen al se up. 1: anode; 2: ca hode; 3: aga + KCl b idges; 4: chemo ac ic
pep ide; 5: elec ode used o p obe he elec ic ield; 6: s i e used o p ope ly mix he pep ide; 7: expe imen al glass chambe , he a ow signals he
ajec o y and di ec ion o he lamina low. B) Top iew o he glass pieces ha compose he expe imen al glass chambe . 8: 75 × 25 mm s anda d glass
slide; 9: longi udinal immed glasses; 10: sliding la e al glasses; 11: cen al piece o glass unde nea h which he cells a e placed. C) Top iew o he
expe imen al glass chambe . E) Axial sec ion o he expe imen al chambe . 12: low sec ional a ea. The expe imen al chambe can be opened and closed
by longi udinally displacing #10, allowing o place o emo e cells when open and es ablishing a lamina low o medium h ough #12 when closed (see
supplemen a y ma e ial o u he de ails). F) Close-up o he expe imen al se up be o e expe imen a ion, de oid o medium and cells. G) All elemen s
ske ched and desc ibed in A–E a e shown in eal condi ions.
4 | PNAS Nexus, 2024, Vol. 3, No. 5
Downloaded om h ps://academic.oup.com/pnasnexus/a icle/3/5/pgae171/7655426 by Cen o Medi e aneo de In es igaciones Ma inas y Ambien ales use on 21 No embe 2024
signi ican long- ange co ela ions in ou da a (P- alue ≅ 0,
Z = −32.31), indica ing he imp obabili y o chance occu ence.
We also calcula ed he ime du a ion o he co ela ions egime
and ound ha all cells exhibi ed long- ange co ela ions o e pe-
iods anging om 1.04 o 16.67 min wi h a median/IQR alue o
9.38/7.29 min. These indings indica e s ong dependences o
pas mo emen s las ing app oxima ely 1,125/875 (median/IQR)
mo e s eps (Fig. 3C and Table S3). A. p o eus cells exhibi ed long-
ange co ela ions up o a median/IQR du a ion o 10.42/
6.25 min, M. lening adensis cells showed 9.38/7.29 min, and
A. bo okensis cells 8.33/6.25 min, hus highligh ing he in luence
o p e ious ajec o y alues on each cellula mo e s ep. These e-
sul s show he p esence o powe -law au oco ela ions decays in
all mig a ion ajec o ies.
S ong anomalous mig a o y dynamics in cellula
locomo ion
Ano he cha ac e is ic o he mig a o y mo emen o cells is hei
s ong anomalous dynamics. This p ope y is di ec ly ela ed o
anomalous supe di usion, a complex p ocess wi h a high non-
linea ela ionship o ime which also co esponds o e icien sys-
emic di ec ional ajec o ies (63, 64).
One o he bes me hods o de e mine such dynamic p ope y is
he mean squa e displacemen (MSD), a me hod p oposed by
Eins ein (65) and la e by on Smoluchowski (66). This S a is ical
Mechanics ool allows o quan i y he amoun o space explo ed
by he amoebae du ing hei locomo ion. Acco ding o his p o-
cedu e (see supplemen a y ma e ial), he anomalous di usion ex-
ponen β is commonly used o e e o whe he no mal (B ownian,
β=1) o anomalous di usion (β≠1) is obse ed. The dynamics o
subdi usion and supe di usion co espond o 0 <β<1 and
β>1, espec i ely.
In Fig. 4A, we depic ed an MSD analysis o he locomo ion mo e-
men s o h ee ep esen a i e cells belonging o each species unde
absence o s imuli. The esul s o MSD analysis o he 700 expe i-
men al cells (see Fig. 4B and C and Table S4) show ha p ac ically
all ajec o ies exhibi s ong anomalous mig a o y dynamics.
Fo expe imen al ajec o ies, he a iable β, which cha ac e izes
he beha io o he di usion p ocess, had a median/IQR alue o
4%
10% 8%
10%
36%
6%
16%
10%
9.8%
11.8% 9.8%
7.8%
5.9%
19.6%
15.7%
19.6%
N=60(20-20 20)
=30' =30' =30'
=30'
18.5 19 19.5 20 20.5 21 21.5
x(mm)
14.2
14.4
14.6
14.8
15
15.2
15.4
15.6
15.8
16
16.2
y(mm)
20.4 20.6
x(mm)
15.4
15.45
15.5
15.55
y(mm)
20.44 20.46 20.48
x(mm)
15.54
15.56
15.58
15.6
15.62
y(mm)
15.6
20.45 20.5 20.55
+
D'
+
-
-
p
p
-1 0 1
0
21
-1 0 1
1
29
-1 0 1
1
20
-1 0 1
1
22
-1 0 1
0
27
-1 0 1
1
19
Scena io 1
Scena io 2 Scena io 3
Scena io 4
A. p o eus ajec o y Sc1
Amoeba p o eus
Me amoeba lening adensis
Amoeba bo okensis
11.5%
11.5% 7.7%
13.5%
15.4%
15.4%
17.3%
7.7%
-
-1 0 1
0
43
-1 0 1
0
41
-1 0 1
0
45
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-10 -5 0 5 10
-10
-5
0
5
10
0% 100%
0% 100%
0% 100%
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
75% 25%
80% 20%
80% 20%
-10 -5 0 5 10
-10
-5
0
5
10
55% 45%
55% 45%
55% 45%
N=60(20-20 20)
-
N=60(20-20 20)
-
N=60(20-20 20)
-
A B C
A' B'
D E
C'
Fig. 2. Rep esen a i e mig a ion ajec o ies o he h ee species unde ou expe imen al scena ios. A–D) Mig a ion beha io o he h ee species
(A. p o eus, M. lening adensis, and A. bo okensis) in he ou expe imen al scena ios (“Scena io 1” absence o s imuli, “Scena io 2” p esence o an elec ic ield,
“Scena io 3” p esence o a chemo ac ic pep ide g adien , and “Scena io 4” simul aneous gal ano ac ic and chemo ac ic s imuli). A′) he pe cen age o
cells mo ing in any speci ic sec o o he expe imen al chambe is ep esen ed as a pola his og am di ided in o eigh a eas o π/4 angle ampli ude each.
B′–D′) his og ams o he displacemen cosines om panels B–D), espec i ely. E) Digi ized cell ajec o y wi h wo inse s highligh ing displacemen s
egions o in e es . “N” is he o al numbe o cells; “ ” is he expe imen du a ion; “p” is he chemo ac ic pep ide (nFMLP); “+” ep esen s he anode; “−”
ep esen s he ca hode; “Sc1” Scena io One (absence o s imuli). Bo h he x- and y-axes show he dis ance in mm, and he ini ial loca ion o each cell has
been placed a he cen e o he diag am.
De la Fuen e e al. | 5
Downloaded om h ps://academic.oup.com/pnasnexus/a icle/3/5/pgae171/7655426 by Cen o Medi e aneo de In es igaciones Ma inas y Ambien ales use on 21 No embe 2024

1.90/0.13 o A. p o eus cells, o 1.88/0.17 o M. lening adensis cells,
and o 1.85/0.15 o A. bo okensis cells. These alues sugges an
anomalous supe di usi e p ocess, a complex beha io which
appea s o go e n he h ee g oups o cell ajec o ies. The al-
ues o expe imen al ajec o ies o he anomalous di usion ex-
ponen β anged om 1.09 o 2.02, median/IQR alue o 1.87/
0.15, whe eas he alues o shu led ajec o ies anged om
−0.01 o 0.01, wi h a median/IQR alue o 10
−4
/0.00 (see Fig. 4B
and Table S5). A Wilcoxon es compa ing anomalous di usion
exponen s om shu led showed ha ou esul s a e ex emely
unlikely o be ob ained by chance (P- alue ≅ 0, Z = 32.39).
Complexi y and in o ma ion in cellula mig a ion
To assess he in o ma ion con en wi hin locomo ion ajec o ies,
we implemen ed he app oxima e en opy (ApEn), a obus ap-
p oxima ion o he Kolmogo o –Sinai (K–S) en opy (67, 68), p o-
iding insigh in o he complex mig a o y beha io ha eme ges
om he cellula sys em.
In Fig. 5A (Tables S6 and S7), he esul s o he ApEn es ima ion
o he 700 cellula ajec o ies a e shown. The hea maps display
he app oxima e K–S en opy o all expe imen al (uppe ow) and
shu led ajec o ies (bo om ow) om each species, calcula ed
o 72 di e en ime windows (in e als) o inc easing leng h
Sc1 Sc2 Sc3 Sc4 Sc1 Sc2 Sc3 Sc4 Sc1 Sc2 Sc3 Sc4
A
B
C
Amoeba p o eus
Sc4
Me amoeba lening adensis
Sc4
Amoeba bo okensis
Sc3
Median
Mean
Sc1
Sc2
Sc3
Sc4
Amoeba p o eus Me amoeba lening adensis Amoeba bo okensis
Amoeba p o eus Me amoeba lening adensis Amoeba bo okensis
Fig. 3. Long- ange in e dependence in he mo e s eps o cellula mig a o y displacemen s. A) Log–log plo o RMSF F s. l s ep o a ep esen a i e cell o
each species. The slope was α = 0.84 o A. p o eus, α = 0.85 o M. lening adensis, and α = 0.84 o A. bo okensis, indica ing he p esence o s ong long- ange
in e dependence in he mo e s eps o all o hem. B) Diag am ep esen ing he alues (and he o e all a e age ± SD) o all he scaling exponen s α om
cells belonging o each species (A. p o eus, M. lening adensis, and A. bo okensis) unde each expe imen al scena io (Sc1–Sc4). C) Violin plo s showing he
es ima ed dis ibu ion, median and a e age memo y pe sis ence alues om cellula ajec o ies. “Sc1” Scena io One, absence o s imuli; “Sc2” Scena io
Two, p esence o an elec ic ield; “Sc3” Scena io Th ee, p esence o a chemo ac ic pep ide g adien ; and “Sc4” Scena io Fou , simul aneous gal ano ac ic
and chemo ac ic s imuli.
6 | PNAS Nexus, 2024, Vol. 3, No. 5
Downloaded om h ps://academic.oup.com/pnasnexus/a icle/3/5/pgae171/7655426 by Cen o Medi e aneo de In es igaciones Ma inas y Ambien ales use on 21 No embe 2024
A
B
C
Sc1
Sc2
Sc3
Sc4
Amoeba p o eus
Sc1
Me amoeba lening adensis
Sc1
Amoeba bo okensis
Sc1
Amoeba p o eus Me amoeba lening adensis Amoeba bo okensis
Amoeba p o eus Me amoeba lening adensis Amoeba bo okensis
Median
Mean
Sc1 Sc2 Sc3 Sc4 Sc1 Sc2 Sc3 Sc4 Sc1 Sc2 Sc3 Sc4
Fig. 4. S ong anomalous mig a o y dynamics in cellula locomo ion. A) G aphics showing he alue o he exponen β by i ing log–log plo s o MSD as a
unc ion o he ime in e al τ, o eigh p o o ypic cells o each species (A. p o eus, M. lening adensis, and A. bo okensis). β = 1 indica es no mal di usion,
while β = 2 indica es ballis ic di usion. The g ay egion de ines he a ea o supe di usion, which is a complex p ocess wi h a high nonlinea ela ionship
o ime, wi hin which all he expe imen al alues all. B) Diag am ep esen ing all alues o he β exponen s (and he o e all a e age ± SD) o all cells o
he h ee species in each expe imen al scena io (Sc1–Sc4) expe imen al alues in ed and shu led alues in blue. The shu ling s ep ex inguished he
long- e m co ela ion s uc u e, causing he sha p di ision be ween he expe imen al and shu led alue dis ibu ions (P ≅ 0, Z = −32.39) o all species
and expe imen al condi ions. C) Es ima ed dis ibu ion, median, and mean MSD β exponen alues om expe imen al ajec o ies a e illus a ed using
iolin plo s. “Sc1” Scena io One, absence o s imuli; “Sc2” Scena io Two, p esence o an elec ic ield; “Sc3” Scena io Th ee, p esence o a chemo ac ic
pep ide g adien ; and “Sc4” Scena io Fou , simul aneous gal ano ac ic and chemo ac ic s imuli.
De la Fuen e e al. | 7
Downloaded om h ps://academic.oup.com/pnasnexus/a icle/3/5/pgae171/7655426 by Cen o Medi e aneo de In es igaciones Ma inas y Ambien ales use on 21 No embe 2024
(in e al du a ion was inc eased by 25 s a e e y i e a ion).
In e als p esen ApEn alues ha a y om 10
−4
o 0.52 o ex-
pe imen al ajec o ies and om 0.14 o 2.13 o shu led ajec-
o ies. These indings allow o obse e ha p ac ically all he
expe imen al se ies exhibi ex emely low en opy.
In Fig. 5B (Table S6), he dec ease in en opy ha occu s om
SC1 o SC2–SC4 deno es how in absence o s imuli (SC1) cells
maximize en opy, bu when he e is a mo e de ined di ec ionali y
cellula ajec o ies become mo e ocused (less en opic displace-
men s). Speci ically, we ound ha he ApEn alues o expe imen-
al ajec o ies exhibi ed a na ow ange o low alues displaying a
median/IQR o 0.00/0.00 o A. p o eus, 0.00/0.00 o M. lening aden-
sis, and 0.00/0.00 o A. bo okensis. The ApEn analysis o all expe i-
men al mig a o y ajec o ies ob ained unde he ou scena ios
showed a median/IQR ApEn alue o 0.00/0.00 wi h alues anging
om 10
−4
o 0.02. The ApEn alues o shu led ajec o ies dis-
played a ange o e y high alues ( om 1.25 o 2.13, median/
IQR equal o 1.97/0.16) ela i e o he alues o expe imen al a-
jec o ies (Table S7).
The whole analysis con i ms he p esence o a complex
s uc u e cha ac e ized by high in o ma ion in he mo e-s ep se-
quences in he mig a ion ajec o ies o all cells. Fu he mo e, he
s a is ical analysis e ealed ha his complex dynamic s uc u e
obse ed in he mo e-s ep ajec o ies was highly unlikely o oc-
cu by chance, as indica ed by a P- alue ≅ 0 and Z = −32.39, esul s
o a Wilcoxon es compa ing he espec i e ApEn alue dis ibu-
ions o expe imen al and shu led ajec o ies.
Pe sis ence in cellula mig a o y mo emen s
Pe sis ence is ano he main cha ac e is ic o he sys emic cellula
mig a o y mo emen s in unicellula o ganisms (69, 70). The de-
ended luc ua ion analysis (DFA) (see supplemen a y ma e ial)
is a well-known echnique o measu ing pe sis en e ec s in
physiological ime se ies.
Fo a gi en obse a ion scale ℓ, DFA calcula es he unc ion F(ℓ)
o quan i y he luc ua ions o he ime se ies a ound he local
end. I he ime se ies displays scaling p ope ies, hen F(ℓ)∼
ℓγ asymp o ically, whe e γ ep esen s he scaling exponen .
This exponen is commonly es ima ed as he slope o a linea i
in he log(F(n)) s. log(ℓ) plo . Thus, γ se es as a measu e pe sis -
ence and helps o cha ac e ize he unde lying dynamical sys em.
Speci ically, alues close o 0.5 indica e he absence o long- ange
co ela ions, while when 1.5 < γ < 2, he p ocess exhibi s posi i e
long- ange pe sis ence (71) (Fig. 6A). Th ough he applica ion o
his quan i a i e me hod, we iden i ied he p esence o long- ange
pe sis ence in all expe imen al ajec o ies (Table S8), wi h a γ
o e all median/IQR alue o 1.78/0.11. Speci ically, he median/
IQR DFA scaling pa ame e γ was ound o be 1.80/0.09 o A. p o-
eus, 1.79/0.14 o M. lening adensis, and 1.78/0.10 o A. bo okensis
(Fig. 6B and Table S8), hus indica ing ha all he mo e-s ep
ajec o ies exhibi “ end- ein o cing beha io ” (signi ican
pe sis ence).
In o de o assess he eliabili y o he DFA analysis, we con-
duc ed a andom shu ling p ocedu e on 700 ime se ies. The e-
sul s demons a ed ha he s ong co ela ion alues obse ed
in he expe imen al mig a ion se ies anished a e shu ling ( e-
e o Fig. 6B and C and Table S9 o mo e in o ma ion), wi h γ o e -
all median/IQR o 0.48/0.14. This inding con i ms ha he
complex locomo ion s uc u e, cha ac e ized by well-o ganized
mo e-s ep sequences and pe sis en dynamics obse ed in he
mig a ion ajec o ies o he h ee cell g oups, is no a ibu able
o a andom chance (P- alue ≅0, Z = 32.39).
Kinema ic p ope ies in cellula locomo ion
ajec o ies
To quan i y some kinema ic p ope ies o he cell mig a ion a-
jec o ies, we s udied he In ensi y o he esponse (IR), he di ec-
ionali y a io (DR), and he a e age speed (AS) o amoebae
(Fig. 7A–C).
The IR is associa ed o he space explo ed by he cell, and in
pa icula , we quan i ied he module o he ajec o ies o e-
p esen he s eng h o he esponse. In his case, he median/
IQR IR was 5.32/3.5 o A. p o eus, 4.96/5.05 o M. lening adensis,
and 3.58/3.07 o A. bo okensis. Nex , we s udied he DR, which
quan i ies he ajec o y s aigh ness, anging be ween 0 ( o ully
cu ed ajec o ies) and 1 ( o ully s aigh ajec o ies), by con-
side ing he s a and end poin o he ajec o y. The alues
anged be ween 0.052 and 0.88 (median/IQR 0.53/0.31) o
A. p o eus, 0.04 and 0.87 (median/IQR 0.51/0.32) o M. lening adensis,
and 0.01 and 0.95 o A. bo okensis (median/IQR 0.47/0.30).
Finally, we calcula ed he AS o he ajec o ies, which anged
be ween 0.00 and 0.01 mm/s (median/IQR 0.01/0.00) o A. p o eus,
0.00 and 0.01 mm/s (median/IQR 0.01/0.00) o M. lening adensis,
and 0.00 and 0.01 mm/s (median/IQR 0.01/0.00) o A. bo okensis.
As i can be obse ed om he P- alues de i ed om K uskal–
Wallis analyses he e is a ema kable a iabili y ega ding kine ic
p ope ies, bo h be ween species ( o example, he P- alues com-
pa ing he IR, DR, and AS in Scena io 4 we e 10
−13
, 10
−4
, and 10
−16
,
espec i ely) and be ween scena ios ( o example, he P- alues o
A. p o eus o IR, DR, and AS compa ed among all ou scena ios
we e 10
−6
, 10
−9
, and 10
−10
, espec i ely); o mo e in o ma ion,
see Fig. 7and Table S10.
Figu e 7D–G shows a clus e ing analysis pe o med on all kine-
ma ic p ope ies conside ed. Each clus e was cha ac e ized by
he p opo ion o cell ypes and expe imen al condi ion p esen
in each g oup. The pe o mance o he ob ained clus e ing solu-
ion was assessed using he Silhoue e coe icien , which es i-
ma es all he di e ences be ween in aclus e poin s minus he
dis ances be ween in e clus e poin s. A highe Silhoue e index
indica es a model wi h be e de ined clus e s. The implemen a-
ion was achie ed using he silhoue e sco e implemen ed in
Sciki -Lea n. The h ee clus e s iden i ied in Fig. 7yielded a
Silhoue e coe icien o 0.37. Simila ly, he ou clus e s iden i ied
p o ided a Silhoue e coe icien o 0.37. Ano he al e na i e clus-
e ing analysis by means o a hie a chical agglome a i e me hod
(Fig. S3) showed ha he dis inc ion be ween cell ypes o expe i-
men al condi ions emains unchanged when a ying he clus e -
ing s a egy, which indica es he obus ness o he indings.
The high a iabili y o he s a is ics o he kinema ic pa ame-
e s, combined wi h he clus e analysis, indica es a high le el o
he e ogenei y in any o he h ee me ics used. This he e ogenei y
exis s bo h among cell ypes and among di e en expe imen al
condi ions and sugges s ha he beha io is indi idual in each
one and hence hey canno be ca ego ized in o g oups.
Dynamic s uc u e in mig a o y mo emen s
Finally, we ha e ep esen ed all he main me ics conside ed in
ou s udy, such as RMSF Alpha, RMSF co ela ion ime (measu ed
in mo e s eps o in minu es), DFA Gamma, MSD Be a, and ap-
p oxima e en opy by compa ing hem wi h hose da a ob ained
in he co esponding shu ling p ocedu es (Fig. 8A–C). As e iden
in each panel, he me ics’ shu led and nonshu led alues could
be dis inc ly g ouped and di e en ia ed. This sugges s ha he in-
he en sys emic in o ma ion s uc u e was dis up ed du ing he
shu ling p ocess. The ajec o ies o he expe imen ally obse ed
8 | PNAS Nexus, 2024, Vol. 3, No. 5
Downloaded om h ps://academic.oup.com/pnasnexus/a icle/3/5/pgae171/7655426 by Cen o Medi e aneo de In es igaciones Ma inas y Ambien ales use on 21 No embe 2024
cells a e comple ely di e en ia ed om he cells whose ajec o -
ies los sys emic p ope ies.
In Fig. 8G–J, a clus e ing analysis o he main me ics was pe -
o med. The h ee clus e s ela ed o cell ype iden i ied in Fig. 8
yielded a Silhoue e coe icien o 0.35. Simila ly, he ou clus e s
iden i ied ela ed o expe imen al condi ions p o ided a Silhoue e co-
e icien o 0.31. This unsupe ised clus e ing analysis combined o
he small a iabili y o he esul s, appea s o be qui e homogeneous,
ega dless he cell ype o scena io conside ed, since i s quan i a i e
aspec s show no dependency on ei he cell ype o expe imen al con-
di ion. Mo eo e , we u ilized an al e na i e app oach, speci ically hie -
a chical agglome a i e clus e ing (Fig. S4), and he esul ed p o ile
emained consis en when he di e en clus e ing me hods we e ap-
plied, highligh ing he obus ness o he esul s.
These indings sugges he eme gence o a highly in ica e dy-
namic s uc u e wi hin he mig a o y pa e ns o all examined
cell ajec o ies. Fu he mo e, his s uc u e appea s o be an in-
he en aspec o cell locomo ion, i espec i e o species o en i -
onmen al condi ions. Indeed, clus e analysis o all quan i a i e
pa ame e s e ealed no eliance on ei he cell ype o expe imen-
al con ex , sugges ing he po en ial uni e sali y o his beha io .
Discussion
Cellula mig a ion is a co ne s one issue in many essen ial
physiological and pa hological p ocesses. He e, we ha e ad-
d essed he in eg a i e sys emic dynamics in ol ed in he egula-
ion o di ec ional mo ili y. To his end, we ha e s udied he
AAmoeba p o eus Me amoeba lening adensis Amoeba bo okensis
Amoeba p o eus Me amoeba lening adensis Amoeba bo okensis
Median
Mean
BSc1 Sc2 Sc3 Sc4 Sc1 Sc2 Sc3 Sc4 Sc1 Sc2 Sc3 Sc4
Fig. 5. Complexi y and in o ma ion in cellula mig a ion. A) Hea maps o he app oxima e en opy alues o all 700 expe imen al (uppe ow panels) and
shu led (bo om ow panels) cell ajec o ies om each species (A. p o eus, M. lening adensis, and A. bo okensis). Each ow in e e y panel co esponds o a
single cell, while in he 72 columns he endpoin o he app oxima e en opy calcula ion is ep esen ed, inc eased in 25 s a e e y i e a ion. B) Violin plo s
illus a e he es ima ed dis ibu ion, mean, and median app oxima e en opy alues o all expe imen al cell ajec o ies. “Sc1” Scena io One, absence o
s imuli; “Sc2” Scena io Two, p esence o an elec ic ield; “Sc3” Scena io Th ee, p esence o a chemo ac ic pep ide g adien ; and “Sc4” Scena io Fou ,
simul aneous gal ano ac ic and chemo ac ic s imuli.
De la Fuen e e al. | 9
Downloaded om h ps://academic.oup.com/pnasnexus/a icle/3/5/pgae171/7655426 by Cen o Medi e aneo de In es igaciones Ma inas y Ambien ales use on 21 No embe 2024
87 Goldbe e A. 2002. Compu a ional app oaches o cellula
hy hms. Na u e. 420:238–245.
88 Halley JD, Winkle DA. 2008. Consis en concep s o sel -
o ganiza ion and sel -assembly. Complexi y. 14:10–17.
89 Mis eli T. 2009. Sel -o ganiza ion in he genome. P oc Na l Acad
Sci. 106:6885–6886.
90 Kondepudi DK, De Ba i B, Dixon JA. 2020. Dissipa i e s uc u es,
o ganisms and e olu ion. En opy. 22:1305.
91 Nicolis G. 1993. Physics o a - om-equilib ium sys ems and
sel -o ganiza ion. In: Da ies PCW, Baliba F, edi o s. The new
physics. Pa is (F ance): Flamma ion. p. 316–347.
92 De la Fuen e IM. 2010. Quan i a i e analysis o cellula me a-
bolic dissipa i e, sel -o ganized s uc u es. In J Mol Sci. 11:
3540–3599.
93 Nicolis G. 1977. I. P igogine, sel -o ganiza ion in nonequilib ium sys-
ems: om dissipa i e s uc u es o o de h ough luc ua ions. New
Yo k (NY): Wiley.
94 Klimon o ich YL. 1999. En opy and in o ma ion o open sys-
ems. Phys Uspekhi. 42:375–384.
95 P igogine I, S enge s I. 1984. O de ou o chaos: man’s New dialogue
wi h na u e. 1s ed. New Yo k (NY): Ban am P ess.
96 B ooks DR, Collie J, Mau e BA, Smi h JDH, Wiley EO. 1989.
En opy and in o ma ion in e ol ing biological sys ems. Biol
Philos. 4:407–432.
97 Kondepudi D, Kay B, Dixon J. 2017. Dissipa i e s uc u es, ma-
chines, and o ganisms: a pe spec i e. Chaos In e discip J
Nonlinea Sci. 27:104607.
98 Alla d J, Mogilne A. 2013. T a eling wa es in ac in dynamics
and cell mo ili y. Cu Opin Cell Biol. 25:107–115.
99 Be ni E, Döbe eine H-G, Go NS, Yochelis A. 2017. F on s and
wa es o ac in polyme iza ion in a bis abili y-based mechan-
ism o ci cula do sal u les. Na Commun. 8:15863.
100 Ca damone L, Laio A, To e V, Shahapu e R, DeSimone A. 2011.
Cy oskele al ac in ne wo ks in mo ile cells a e c i ically sel -
o ganized sys ems synch onized by mechanical in e ac ions.
P oc Na l Acad Sci. 108:13978–13983.
101 Balland M, Riche A, Galle F. 2005. The dissipa i e con ibu-
ion o myosin II in he cy oskele on dynamics o myoblas s.
Eu Biophys J. 34:255–261.
102 Mandelkow E, Mandelkow E-M, Ho ani H, Hess B, Mülle SC.
1989. Spa ial pa e ns om oscilla ing mic o ubules. Science.
246:1291–1293.
103 Ishii K, Hi ose K, Iino M. 2006. Ca2+ shu ling be ween endoplas-
mic e iculum and mi ochond ia unde lying Ca2+ oscilla ions.
EMBO Rep. 7:390–396.
104 Kondepudi DK, P igogine I. 1998. Mode n he modynamics: om
hea engines o dissipa i e s uc u es. Chiches e (England): John
Wiley & Sons.
105 Lema chand H, Nicolis G. 1976. Long ange co ela ions and he
onse o chemical ins abili ies. Phys S a Mech i s Appl. 82:
521–542.
106 Kondepudi DK, P igogine I. 1981. Sensi i i y o nonequilib ium
sys ems. Phys S a Mech i s Appl. 107:1–24.
107 Kondepudi DK. 1982. Sensi i i y o chemical dissipa i e s uc-
u es o ex e nal ields: o ma ion o p opaga ing bands. Phys
S a Mech i s Appl. 115:552–566.
16 | PNAS Nexus, 2024, Vol. 3, No. 5
Downloaded om h ps://academic.oup.com/pnasnexus/a icle/3/5/pgae171/7655426 by Cen o Medi e aneo de In es igaciones Ma inas y Ambien ales use on 21 No embe 2024