Systemic cellular migration: The forces driving the directed locomotion movement of cells

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). 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