Super-Gaussian, super-diffusive transport of multi-mode active matter

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1 Super-Guin, uper-diffuive rnpor of muli-mode ive mer Seungoo Hhn,, Snggeun Song -3, De Hyun Kim -3, Gil-Suk Yng,3, Kng Tek Lee 4 * Jeyoung Sung -3 * Creive Reerh Iniiive Cener for Chemil Dynmi in Living Cell, Chung-Ang Univeriy, Seoul 6974, Kore. Deprmen of Chemiry, Chung-Ang Univeriy, Seoul 6974, Kore. 3 Nionl Iniue of Innovive Funionl Imging, Chung-Ang Univeriy, Seoul 6974, Kore. 4 Deprmen of Chemiry, Gwngju Iniue of Siene nd Tehnology, Gwngju 65, Kore Correponding Auhor: J. Sung (jeyoung@u..kr); K. T. Lee (klee@gi..kr) Abr Living ell exhibi muli-mode rnpor h wihe beween n ive, elf-propelled moion nd eemingly pive, rndom moion. Cellulr deiion-mking over rnpor mode wihing i ohi proe h depend on he dynmi of he inrellulr hemil nework reguling he ell migrion proe. Here, we propoe heory nd n exly olvble model of muli-mode ive mer. Our ex model udy how h he reverible rniion beween pive mode nd n ive mode i he origin of he nomlou, uper-guin rnpor dynmi, whih h been oberved in vriou experimen for mulimode ive mer. We lo preen he generlizion of our model o enomp omplex muli-mode mer wih rbirry inernl e hemil dynmi nd inernl e dependen rnpor dynmi.

2 Living ell in migrion regule heir onumpion of inrellulr hemil energy ording o he inruion enoded in heir gene; hey exhibi muliple rnpor mode during rnpor, oniing of wo hrerii moion: elf-propelled, bllii moion when he mer i in n ive mode nd n undireed, rndom moion when he mer i in pive mode. Depending on he regulory e of he ellulr reion nework underlying ell migrion, he rnpor mode of living ell wihe repeedly beween he ive nd he pive mode. Thi feure in living ell rjeorie pper imilr o h of Lévy wlk [,]. Anoher inereing feure of living ell moion i h hey repeedly revere heir direion. Thi run-nd-revere moion h been repored in vriou beril yem [3-7]. Thee feure hve lo been oberved in he rnpor of vriou ype of rgo nd veile in living ell [8-]. Aive mer, uh living ell nd inrellulr ive prile, generlly exhibi n nomlou, non-guin rnpor dynmi, whih nno be deribed by Einein heory of Brownin moion [,] or more reen heorie for nomlou rnpor in diordered environmen [3-7]. There re model of pively moving prile h hve been ued o explin he long ime behvior of he men qure diplemen (SD) of muli-mode ive mer oberved in experimen [8,9]. Alhough hee model ume h he ohi dynmi of mulimode ive mer i quliively he me h of pive mer, hey re ble o provide ifory explnion of experimenl reul for he long ime behvior of he SD in mny e [,]. However, experimenl d wih higher ime reoluion reveled h muli-mode ive mer h quliively differen ohi dynmi from pive mer; he SD of muli-mode ive mer how hor ime-diffuive moion, inermedie uperdiffuive moion, nd long-ime diffuive moion wih greer diffuion oeffiien [,3],

3 whih nno be explined by he pive mer model [4,5]. An lernive model o oun for he nomlou rnpor dynmi of ive mer i he ive Brownin prile model. In hi model, veloiy-dependen friion in he Lngevin equion i ued o deribe he elf-propelled moion of ive prile [7,6]. The ive Brownin prile model doe provide n enhned explnion for he nomlou SD of ive prile; however, hi model nd he model menioned bove nno explin he nomlou diplemen diribuion of ive mer, whoe pil diribuion i non-guin wih poiive exe kuroi [6,7]. A number of oher inereing model hve been propoed for elf-propelled prile [8-34]. However, o he be of our knowledge, none of hem repreen muli-mode ive mer, whih wihe beween n ive, elf-propelled rnpor mode nd eemingly pive, rndom mode depending on i inernl e dynmi. In hi Leer, we preen n exly olvble model for he ohi rnpor of mulimode ive mer. In he high friion regime, where we n fely negle he ineril erm in he Lngevin equion, he veloiy x () of muli-mode ive mer wih friion onn, γ, n be wrien he um of wo omponen: x v, () () = ( Γ ()) γ ξ() where v ( Γ ) nd γ ξ() repreen he veloiy omponen of elf-propelled, bllii moion, whih i dependen on he inernl e Γ nd he veloiy omponen ued by he rndom fluuing fore. Auming h he dynmi of he rndom fluuing fore our in ime le fr horer hn he inernl e dynmi, we model ξ () Guin whie noie, whoe ime orrelion, ξ( ) ξ( ), i proporionl o he Dir del funion, 3

4 δ (). On he oher hnd, he relxion of v( Γ ( )) v( Γ ( )) from he iniil vlue, v, o he finl vlue, v, our in he ime le of he inernl e dynmi. We ume h he ell e dynmi i n rbirry ohi proe h n be repreened by mulidimenionl rkov proe. The Fokker-Plnk equion orreponding o Eq. () i given by P x D v P x L P x x x where P(, x, ) ( Γ,, ) = ( Γ) ( Γ,, ) ( Γ) ( Γ,, ), () Γ denoe he probbiliy deniy funion (PDF) of ive mer wih he poiion x nd inernl e Γ ime [35,36]. In Eq. (), D nd for he diffuion oeffiien for pive moion origining from he rndom fluuing fore, whih i defined by D d = γ ξ ξ(). L( Γ ) denoe he mhemil operor deribing he inernl e dynmi of he yem. Our model yield nlyi reul for he SD nd he non- Guin prmeer [37]. Here, we ompre wo imple, exly olvble model of ive mer: one for inglemode ive mer nd he oher for muli-mode ive mer. Thee model re hown in Fig.. For he ingle-mode model, whih only exhibi n ive mode, hown in Fig. (), Eq. () yield P x, v P x, k k P x, D P x, = I x x v P x, k k P x,. (3) For he muli-mode model, whih exhibi boh ive nd pive mode, hown in Fig. (b), Eq. () yield 4

5 P x, v P x, k k P x, P x, D P x, k k k P x, = x I x. (4) P x, v P x, k k P x, In Eq. (3) nd (4), (, ) P x deigne he probbiliy deniy of he mer e Γ i i ( i,,) nd poiion x ime. ± v, k, nd k denoe, repeively, he veloiy ( Γ ) of he elf-propelled moion of he mer e Γ ±, he rniion re o eiher e, v ± Γ or Γ, of he mer in ive mode, nd he rniion re o he e, Γ, of he mer in pive mode, whih v ( Γ ) =. Typil ime re re diplyed for he wo differen model in Fig. nd Supplemenl eril [38]. Ex nlyi oluion of Eq. (3) nd (4) n be obined in he Fourier domin [37]. From he ex oluion, we obin he diribuion fv (,) of he men veloiy, defined by x () ( v ()) ; i i hown in Fig. for eh model. In he hor-ime limi, for boh model, he men veloiy diribuion i found o be liner ombinion of Guin enered he e-dependen elf-propelled veloiy, v ( Γ ), h i, i eq i ( i ) ( ) fv (, ) p Gvv( Γ ), D i <<, (5) wih eq eq p i being he equilibrium probbiliy of e Γ i, given by p ± = / for he ingle- eq mode model nd by p = k ( k ± k ) nd eq p = k ( k k ) for he muli-mode model. In Eq. (5), G( ν, σ ) denoe he Guin diribuion of ν wih he men nd vrine given by nd σ, repeively. Equion (5) n lo be obined from he diribuion of he 5

6 innneou veloiy given in Eq. (), beue he men veloiy, x /, i he me he innneou veloiy in he hor-ime limi [37]. Equion (5) erve good pproximion of he men veloiy diribuion moderely hor ime, where rniion beween e h ye o our, or ime erlier hn he hrerii relxion ime, dφ () [ ] v, where φ () denoe he normlized ime orrelion funion, v v() v() v φv (), of he inernl e dependen elf-propelled veloiy, v ( Γ ). The long-ime diribuion of he men veloiy beome Guin for boh ingle-mode nd muli-mode model of ive mer, while he hor-ime diribuion n vry depending S on he model. A hor ime ( ), he men veloiy diribuion, f( v,,) of he inglemode model h wo Guin pek enered v nd v, whih re he wo veloiie of elf-propelled moion. In omprion, he men veloiy diribuion, f ( v,), of he mulimode model hor ime h n ddiionl Guin pek enered, reuling from he e, Γ, of he ive mer in pive mode. The vrine of eh Guin pek, whih origine from he rndom fluuing fore, i pproximely given by D. However, hown in Fig., boh f ( v,) nd fs ( v,) onverge o Guin wih men of zero nd vrine proporionl o / long ime [37]. Th i o y, for boh model, he diribuion of x () pprohe Guin ble diribuion long ime, in ordne wih he Guin enrl limi heorem [37]. The relxion dynmi of he men veloiy diribuion i highly dependen on he 6

7 hrerii relxion ime,, of inernl e dependen elf-propelled veloiy, v ( Γ ). The men veloiy diribuion pprohe he long-ime ympoi Guin fer he vlue of deree (ee Fig. () nd (d)). The nlyi expreion of i dependen on he model in queion. For he ingle-mode model, i given by hlf he lifeime, k, of he e, Γ ±, of he ive mer in ive mode, i.e., = ( k). For he muli-mode model, i he me he lifeime, k, of he e Γ ± [37]. In he mll limi, he men veloiy diribuion i Guin ny finie ime. Noe lo h he vrine in he men veloiy, or he men qured veloiy, ny given ime deree wih he relxion peed,, of he fluuion in he elf-propelled veloiy, hown in Fig. (e). Thi i ommon feure of dynmilly diordered yem; in peroopy, i h been ermed moionl nrrowing. The men veloiy diribuion, f ( v,), of he muli-mode model i dependen on he lifeime, ( k ), of pive mode, Γ, well on he lifeime, ( k ) =, of he e in ive mode, Γ ±. A hown in Fig. (f), when he populion rio, eq eq eq ( ( ) ( ) ) R p p p = = k k, of he e in pive mode o he e in ive mode deree, f ( v,) pprohe f( v,) [37]. However, he vlue of R inree, S he pek enered v = in f ( v,) grow lrge, o h he SD of he muli-mode model i mller hn he SD of he ingle-mode model. For boh he model, he SD h hree differen kinei phe: he hor-ime diffuion 7

8 phe, n inermedie uper-diffuive phe, nd he long-ime diffuive phe wih greer diffuion oeffiien, in greemen wih he previou experimenl reul [,3]. Ex nlyi expreion of he SD for boh model n be wrien in he me formul, x ( ) = ( D D) D ( e ), (6) where D i he effeive diffuion oeffiien omponen onribued from he elf-propelled moion, defined by () = D d v v v p. Here, p deigne he probbiliy of he e in ive mode, whih i given by uniy for he ingle-mode model nd by p p p R eq eq = = ( ) for he muli-mode model. D nd hve he me mening bove. A hown in Fig. 3(), he SD i given by x D hor ime ( ) nd dominnly onribued from he eemingly pive, rndom moion. On he oher hnd, long ime ( ), he SD i given by D x ( D ), wih he diffuion oeffiien inreed by D. In inermedie ime ( D v < ), he SD how uper-diffuive behvior ( SD α wih < α ). In he erly ge of he inermedie region, he SD, hown by he i pproximely qudri funion of ime, i.e., x ( ) D D ( ) green line in Fig. 3(b), whih origine from he bllii, elf-propelled moion of ive mer. While boh he ingle-mode nd muli-mode model yield quliively he me nlyi reul for he SD, he reul hey yield for he diplemen diribuion n be quie differen from eh oher. The diplemen diribuion, P (,) x, of he muli-mode model n be 8

9 uper-guin, in ordne wih he experimenl d repored in Ref. [34,39], where PS (,) x of he ingle-mode model i lwy ub-guin. For he muli-mode model, he deviion of P (,) x of he muli-mode model from Guin meured by he non-guin prmeer, 4 R x ( x ) α () () 3 (), i eniive o he populion rio, R, of he e in pive mode o he e in ive mode, whih i hown in Fig. 3(). The ex nlyi expreion of α () i preened in he Supplemenl eril [37]. The impler ympoi expreion of ( ) R α boh hor ime nd long ime i given by R α R ( ) ( R ) ( R ) () D ( ),, D () R R D,, 3 ( R ) D D (7) where () D deigne v, or he vlue of D in he limi where he e of ive mer i lwy in he ive mode. Aording o Eq. (7), he diplemen diribuion, P (,) x, of muli-mode ive mer i uper-guin when R >, bu ub-guin when R < ( 5).6 ll ime [37]. However, when 5 < R <, he diplemen diribuion, P (,) x, of muli-mode ive mer wihe from ub-guin o uper- Guin over ime [37]. Noe h α () vnihe in he lrge R limi, where he e of R muli-mode mer i lwy in pive mode. Thi men h, in our model, i i he elfpropelled, bllii moion h ue he diplemen diribuion o be non-guin. In he oppoie, mll R limi, P (,) x h exly he me hpe PS (,) x [37]. Thu, he 9

10 non-guin prmeer, α R (), of he muli-mode model redue o α() = lim αr () R of he ingle-mode model, whoe ympoi behvior i given by α ( ) () D 6 D ( ) () D,. () D D,. (8) Equion (8) how h he diplemen diribuion, PS (,) x, of ingle-mode ive mer i ub-guin only [37]. Boh P (,) x nd PS (,) x pproh Guin long ime; however, heir deviion from Guin, whih i meured by he non-guin prmeer, lowly deree wih ime, following long ime ( ), ording o Eq. (7) nd (8). A hown in Fig. 3(), he deviion of he diplemen diribuion from Guin n be izble even long ime where he SD, given in Eq. (6), i linerly proporionl o ime. Thi h been oberved, for exmple, in lipoome diffuion in nemi oluion of in filmen [4]. The muli-mode ive mer model diued bove n be exended o more omplex model in he higher pil dimenion, d. For he generlized model, he ohi differenil equion orreponding o Eq. () i given by r v ξ, (9) () = ( Γ ()) γ () where eh bold ymbol denoe he d-dimenionl veor orreponding o eh lr quniy in Eq. (). The generl expreion of he SD obined from Eq. (9) i given by

11 d d D p ( D r ( ) = φ ) φ ( ) ξ v, () where D, p, D, nd re, repeively, defined by D d γ d = ξ ξ (), p dφξ (), = D d d v ( ) v (), nd dφv (). Here, φ x () denoe he normlized ime orrelion funion, x( ) x() x (), of veor () x. The funionl form of φv ( ) vrie depending on he inernl e dynmi nd i oupling o he elfpropelled veloiy. Given h he relxion ime of rndom fluuion fore ξ () i fr horer hn he obervion ime, Eq. () redue o ( ) r ( ) dd dd d φv ( ). Thi reul i he generlizion of equion (6) for muli-dimenionl yem wih rbirry φ ( ) ; i redue o equion (6) for he onedimenionl model wih φ ( ) = exp( ). In ddiion, he generl expreion of he non- v Guin prmeer n lo be obined from equion (9) follow: v α R ( ) = r v r ( ) ( ) α v ( ). () 4 α v i defined by α ( ) r ( ) r ( ) wih ( ) Here () ( ) 4 d d v v v r nd r v defined d d v( ) v ( ) nd v d 4 d 3 d d v( 4) v( 3) v( ) v ( ), repeively. The non-guin prmeer given in Eq. () vnihe in boh he hor ime nd he long ime limi. A ime

12 fr horer hn he relxion ime le, ( v ( ) ( ) ), of he elf-propelled veloiy, r r, nd hene he non-guin prmeer given in Eq. (), vnih [37]. ( v ) On he oher hnd, in he long ime limi, ( ) ( ) r r pprohe D ( D D) bu α v (), or he non-guin prmeer of he elf-propelled diplemen, dv ( ), vnihe beue he diribuion of he elf-propelled diplemen beome Guin ording o he Guin enrl limi heorem. However, he non-guin prmeer h non-zero vlue beween he wo limi. In he imple one-dimenionl muli-mode ive mer model wih he Poion e wihing dynmi, we n how h equion () redue o equion (7) [37]. Equion () nd () enble u o lule he SD nd non-guin prmeer for generl muli-mode ive mer wih poibly non-poion e wihing dynmi. In ummry, we preen n nlyi heory nd n exly olvble model of muli-mode ive mer, whih wihe beween n ive, elf-propelled rnpor mode nd eemingly pive, rndom mode depending on i inernl e hemil dynmi. Our ex model udy lerly how h he reverible rniion beween eemingly pive, rndom moion nd he elf-propelled, bllii moion i n imporn oure of he uper-guin diplemen diribuion ommonly oberved for muli-mode ive mer. Thi model i uffiienly flexible o h i n be eily generlized o enomp muli-e, muli-mode ive mer wih rbirry inernl e hemil dynmi nd inernl e oupled rnpor dynmi. The ppliion of he preen pproh o he quniive explnion of experimenl reul for exmple of muli-mode ive mer i o be publihed elewhere.

13 FIGURES FIG.. odel yem nd ypil rjeorie. () The ingle-mode model oni of wo inernl e, Γ nd Γ. The ingle-mode ive mer in Γ ± e perform elfpropelled, direed moion wih veloiy ± v under rndom fluuing fore exered from medium. The ohi rniion beween inernl e i hrerized by he re onn, k. (b) The muli-mode model oniing of hree inernl e: pive rnpor e, Γ, in ddiion o ive rnpor e, Γ nd Γ. The muli-mode mer perform undireed, rndom moion in e Γ, bu perform direed, elf-propelled moion wih veloiy in e Γ ±. k nd k repreen he ohi rniion re from he pive Γ o he ive ± Γ e nd from he ive ± Γ o he pive Γ e, repeively. For eh model, ypil ime re of he poiion i hown. Color in he ive mer digrm nd rjeory repreen he ell inernl e. 3 ± v

14 FIG.. PDF for men veloiy diribuion. The ime dependen men veloiy diribuion, () fs ( v,) for he ingle-mode model nd (b) f ( v,) for he muli-mode model wih R =.5. In boh () nd (b), he men veloiy diribuion i diplyed ring from rbirry uni ime, T, nd he relxion ime of he veloiy-veloiy uo-orrelion funion i u e o be T u. The men veloiy diribuion for he muli-mode model, wih hree differen vlue of 4 = T () for he ingle-mode model nd (d) u. (line) nlyi reul (irle) ohi imulion reul. In (d), he vlue of R i e o be.5, in whih e he hree e re eqully probble equilibrium. (e) Dependene of he roo-men-qure veloiie, or he ndrd deviion of he men veloiy diribuion on he relxion peed meured by, nd (f) he men veloiy diribuion = Tu for he muli-mode model wih hree differen vlue of R : (blue doed line) R = ; (blue olid line) R =.5 ; (blk line) R =.5 ; nd (red line) R = 5. The vlue of i T u. In he mll R limi, f ( v,) pprohe fs ( v,). The Guin diribuion wih he me men nd vrine f ( v,) for R = 5 i ploed red doed line. The vlue of he oher prmeer re e o be D = nd v = 5 for ll e.

15 FIG. 3. en qure diplemen nd non-guin prmeer. () Time-dependen men qure diplemen (SD). The ingle-mode nd muli-mode model hre he me SD, given in Eq. (6). The vlue of i e o. The vlue of he oher prmeer, D nd D re e o be D = nd D =. (line) nlyi reul (irle) ohi imulion reul. (b) Dependene of x () on ime for he hree e wih =. (red), = (blk), nd = (blue). The effeive diffuion oeffiien inree from D o D D, whoe rniion ime le i deermined by. The green line repreen he bllii moion ( x ( ) D D ( ) ) orreponding o eh e. () The non- Guin prmeer, α ( ), for he ingle-mode model (blue line) nd for he muli-mode model wih vriou vlue of R (blk line). The wo red line repreen α ( ) for he wo riil vlue of R,.6 nd. (d) () α i lwy negive when 5 R < 5.6, bu poiive ll ime when R >. When.6 < R <, α () wihe from hor-ime negive regime o long-ime poiive regime. (e) A repreenive e for he ime- α wih R =.75 i ploed blk line. The wo red line dependen wihing, repreen ( ) α for he wo riil vlue of R, hown in (). The non-guin prmeer vnihe boh in he hor ime nd he long ime limi, mening h he iniil diribuion i del funion, Guin wih zero vrine, nd he finl diribuion obey he Guin enrl limi heorem. A he four ime poin mrked by he olid irle, he men veloiy

16 diribuion (blue line) nd heir orreponding Guin diribuion (blk line) re ploed ording o he qure men veloiy in he ine. The red irle mrk he defiieny in populion of he ive mer in he high men veloiy region, ompred o heir orreponding Guin diribuion. (f) Deviion of he men veloiy diribuion from Guin he wo ime poin mrked by he filled blk irle. The red irle here nd in he ine in (e) boh repreen he me pil regime. 6

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18 [7]. Theve, J. Tkiko, V. Zburdev, H. Srk, nd C. Be, Biophy. J. 5, 95 (3). [8] D. Cmpo nd V. endez, J. Chem. Phy. 3, 347 (9). [9] N. ki, H. iyohi, nd Y. Tuhiy, Prooplm 3, 69 (7). [3] H. iyohi, N. ki, nd Y. Tuhiy, Prooplm, 75 (3). [3] F. Peruni nd L. G. orelli, Phy. Rev. Le. 99, 6 (7). [3]. Shienbein nd H. Gruler, Bull. h. Biol. 55, 585 (993). [33] H. Tkgi,. J. So, T. Yngid, nd. Ued, PLoS One 3, e648 (8). [34] D. Selmezi, S. oler, P. H. Hgedorn, N. B. Lren, nd H. Flyvbjerg, Biophy. J. 89, 9 (5). [35] S. I. Deniov, W. Horhemke, nd P. Hnggi, Eur. Phy. J. B 68, 567 (9). [36] H. Riken, The Fokker-Plnk equion : mehod of oluion nd ppliion (Springer- Verlg, New York, 996), nd edn., Springer erie in ynergei,, 8. [37] See Supplemenl eril for he derivion of he eond nd fourh momen of diplemen, for PDF of diplemen limiing ime le, for men veloiy diriubion nd ionry diribuion, for more deil on he exended model, for he relxion ime, for non- Guin prmeer in ll rnge, nd for he ohi imulion mehod. [38] See Supplemenl eril for rjeorie of eh model. [39] H. U. Bodeker, C. Be, T. D. Frnk, nd E. Bodenhz, Europhy. Le. 9, 85 (). [4] B. Wng, J. Kuo, S. C. Be, nd S. Grnik, N. er., 48 (). 8

19 Supplemenl eril for Super-Guin, uper-diffuive rnpor of muli-mode ive mer Seungoo Hhn, Snggeun Song,,,3 De Hyun Kim,,,3 Gil-Suk Yng,,,3 Kng Tek Lee, 4 Jeyoung Sung,,3, Creive Reerh Iniiive Cener for Chemil Dynmi in Living Cell, Chung-Ang Univeriy, Seoul 6974, Kore. Deprmen of Chemiry, Chung-Ang Univeriy, Seoul 6974, Kore. 3 Nionl Iniue of Innovive Funionl Imging, Chung-Ang Univeriy, Seoul 6974, Kore. 4 Deprmen of Chemiry, Gwngju Iniue of Siene nd Tehnology, Gwngju 65, Kore Conen A. Derivion of he eond nd fourh momen of diplemen for he muli-e model B. Derivion of he eond nd fourh momen of diplemen for he ingle-e model 4 C. Probbiliy deniy funion of diplemen wo limiing ime le 5 D. en veloiy diribuion nd ionry diribuion 7 E. Dynmi of he muli-e model 8 F. Convergene of P (,) x o PS (,) x he mll R limi G. Relxion ime of he wo olvble model 9 8 H. Generl model I. Time orrelion funion for wo olvble model J. Derivion of hor ime men veloiy diribuion from Eq. () K. Sohi imulion mehod 3 L. Referene 4 Fig. S. Non-Guin prmeer 5

20 A. Derivion of he eond nd fourh momen of diplemen for he muli-mode model The muli-mode ive mer model h hree inernl e, Γ, Γ, nd Γ. Eh inernl e regule he direion nd peed of given ive mer explined in Fig. (b). Bed on he iniil ondiion h he inernl e re iniilly in equilibrium nd he iniil poiion of ive mer i zero, hree imulneou equion re obined from Eq. (4) by pplying he Fourier rnform nd he Lple rnform o Pi ( x, ) wih i,,. The oluion of he imulneou equion provide hree probbiliy deniy funion (PDF) for he individul inernl e in he Fourier-Lple domin, wrien ( χ ( w, ) i ( D ) ) eq p ( χ vw ) χ ( w, ) i ( D ) p vw w k k eq P ( w, ) P ( w, ) w, = ( Dw )( χ ( w, ) vw ) kv w P w, eq p vw w k k χ w, Dw k Dw k k. (S) wih In Eq. (S), w nd repeively denoe he Fourier rnform of poiion x nd he Lple rnform of ime. The ilde indie h he funion re repreened in he Fourier-Lple eq eq eq domin. p, p, nd p denoe he equilibrium probbiliie of he e Γ, Γ, nd Γ, repeively. A ummion of he PDF given in Eq. (S) provide he PDF of muli-mode ive mer, whih i given by (, ) (, ) (, ) (, ) eq χ ( w, ) p vw P w P w P w P w = ( D )( χ ( w) ) vw w, vw k The denominor in Eq. (S) i ubi funion of ( ) ( ) funion C ( w) i 3. (S) D z w z k k z k k k vw z kvw. If we ume he roo of he ubi P ( w, ) wih (,, 3), he PDF n be rewrien i 3 = kvw Dw Dw k k i= Dw Ci ( w). (S3) 3 3 = kvw Dw Dw k k i= Dw Ci ( w) j i Ci( w) Cj( w) Invering in P( w, ) genere he Fourier-domin PDF, wrien 3 3 P ˆ ( w D w k k C i w, ) vw e = i Ci( w) k k e = j i Ci w Cj w. (S4)

21 Eq. (S) nd (S4) n boh be ued o derive he nlyi oluion for he men qure diplemen (SD) of he muli-mode model. One wy i o ue he eond pril derivive P ˆ w,, while he oher wy, whih i n eier wy o obin he ime-domin SD, i of o pply he invere Lple rnform o he eond pril derivive of P ( w, ) x = lim = L lim w w, wrien Pˆ w, P w,, (S5) w w where L denoe he invere Lple rnform of ime. The ime-dependen SD of hi model i given in Eq. (6). The nlyi oluion for he fourh momen of diplemen i obined uing he following equion: x 4 4 P = lim = w 4 w vk D. (S6) k ( w, ) 4 ( k)( k k ) 3 vk 3D kd v k k Appliion of he invere Lple rnform o Eq. (S6) provide he fourh momen of diplemen in he ime domin, wrien x 4 ( ) = ( ) D D 5 / / 3 / R e R 3R 3 R e 3 4D ( 3 6 3) R R R / ( R ) ( R ) ( R D D) e ( R ) ( R R ( R ) D D) ( R ) ( )( ) (S7) The eond nd fourh momen expreed in he ime domin re ombined o produe he α, uh non-guin prmeer, 3

22 α R ( ) wih ( ) 4 x = κ ( ) 3 x 3 ( ) e 4e 5 4e ( e 8e 8e 4 ) R () D = 4 ( e e ) R ( R ) x ( ) 3 ( 6e 6 4e ) R R 5 e e R R x D ( ) =, (S8) D () () E D ( R ) where κ ( ) denoe kuroi nd hown in Fig. 3 (b), he relxion ime, In Fig. S, ( ) ondiion of α in ll rnge of R nd R (). D R D v. On he log-le ime xi () =, hif he ( ) α urve well he SD urve. R i nlyilly evlued nd ploed under he D D =, where he red line re he wo line hown in Fig. 3() nd 3(e) nd he blk line mrk border line wihing from ub-guin o uper-guin D D, he border line i invrin on he hnge of given R. Alhough α ( ) depend on he D D rio in Eq. (S8). Thu, ( ) ll ime, nd ( ) 5 R R α R wih R le hn 5 i ub-guin α wih R lrger hn i lwy uper-guin ll ime. When < <, P (,) hown in Fig. 3(e) nd S. x n wih from ub-guin o uper-guin over ime, B. Derivion of he eond nd fourh momen of diplemen for he ingle-mode model The ingle-mode model h wo inernl e, Γ nd Γ. Eh inernl e regule he direion nd peed of ive mer, explined in Fig. (). Bed on he iniil ondiion h he inernl e re iniilly in equilibrium nd he iniil poiion of ive mer i zero, wo imulneou equion re obined from Eq. (3) by pplying he Fourier rnform nd he Lple rnform o Pi ( x, ) wih i,. The nlyi oluion of he imulneou equion provide wo PDF in he Fourier-Lple domin, wrien 4

23 ( w, ) P Dw k ivw = P ( w, ). (S9) vw ( )( ) Dw k ivw Dw Dw k The PDF of ive mer for he ingle-mode model in Fourier-Lple domin i wrien S (, ) (, ) P ( w, ) P w P w = vw Dw Dw k Dw k ( )( ). (S) Appliion of he invere Lple rnform o Eq. (S) genere he PDF of ive mer repreened in he Fourier domin ( ˆ ) (, ) e k Dw PS w = k oh ( Λ ) inh( Λ) Λ wih Λ k vw. (S) From hi funion, he ime-dependen eond nd fourh momen of diplemen re imple o obin. The ime-dependen SD of hi model i given in Eq. (6). The fourh momen of diplemen i lo evlued from he PDF x 4 where = D D D D D e 3D e, (S) ( ) ( ) ( ) D = v. The eond nd fourh momen expreed in he ime domin re ombined o produe he non-guin prmeer, α ( ), uh α ( ) D D = ( 5 e 4e 4e ) = β x ( ) x ( ) wih ( ) x D E D = D nd β ( ). (S3) 5 e 4e 4e In Eq. (S3), α ( ) i le hn or equl o zero beue ( ) equion i equl o ( ) ( ) β in ll ime rnge, nd he α R of he muli-mode model he mll R limi α ( ) limα R ( ) R =. C. Probbiliy deniy funion of diplemen wo limiing ime le The diffuion dynmi of he model i highly dependen on he relxion ime,, of he veloiy, v ( Γ ). A hor ime ( ), given ive mer minin i direion nd 5

24 mgniude of veloiy, nd eh unrelxed veloiy produe hree individul pek in he PDF of diplemen. The PDF of diplemen hor ime i derived from Eq. (S), whih i wrien P ( D i ) eq p ( D w ) ( Dw i vw) eq p w vw P hor, ( w, ) P hor, ( w, ) = P, (, ) e hor w q p (, ) x hor ime i wrien P, hor ( x, ) = p e 4π D. (S4) ( x v ) x ( xv ) eq 4D eq 4D eq 4D p e p e, (S5) D. The hree pek in where he diribuion i Guin wih vrine of P, hor x, re pproximed ingle Guin funion wih mll vrine very hor ime ( D v ) nd grdully epre ime inree. ), he pek for individul v ( Γ) re gin inermingled ino ingle A long ime ( Guin nd follow he diribuion, wrien ( w, ) ( w) ( w ) ( i vwk ) ( k) ( i ) eq P p long, eq P long,, = p ( vw k k ), (S6) Deff w P,, eq long p vw k where D D eff i equl o D. The PDF P (, ) x long ime i wrien P, long ( x, ) x eq 4Deff p D R x = e. (S7) 4π D Deff D ef f eff In Eq. (S7), deviion from he Guin diribuion i proporionl o p eq D R D, eff long ime. eq where p D D eff i lwy le hn, nd i lrger hn Therefore, p eq D R Deff i muh mller hn if R i finie uffiienly long ime. Thu, he PDF pprohe he Guin diribuion uffiienly long ime, whih i in ordne wih he Guin enrl limi heorem. In ummry, he PDF of diplemen pprohe he del funion very hor ime, beue elf-propelled veloiy i muh weker hn he veloiy ued by he rndom fluuing fore. A reul of he onribuion of rndom fluuion being diiped, he del funion i pli ino individul pek reled o he veloiy of eh inernl e, where 6

25 he elf-propelled veloiy how no vriion. Eq. (S5) explin he wo differen funionl form of he PDF. A long ime, he PDF pprohe he dipered Guin diribuion beue he vrine for eh diribuion i proporionl o ime. f D. en veloiy diribuion nd ionry diribuion The men veloiy, v (), i defined by v () x (). The men veloiy diribuion, (, ) v, i direly obined from he PDF of diplemen wih proper normlizion onn. A hor ime ( f ), he men veloiy diribuion, f (, ) ( v v ) v ( vv ) eq 4D eq 4D eq 4D, hor ( v, ) = p e p e p e 4π D v, i wrien. (S8) where he diribuion i Guin wih vrine of D. Beue he vrine i inverely proporionl o, he brodne of he individul pek in Eq. (S8) how he oppoie pern P x,, whih pper in Eq. (S5). A long ime ompred o he individul pek in ), f (, ) ( (S9) f f, long,, long v i wrien ( v, ) v eq 4Deff p D R v = e 4π D Deff D eff eff v pprohe he del funion ime inree. The vrine of he individul pek in P (, ) x i proporionl o boh hor nd long ime. If we define new vrible, q( x ), hen he ionry diribuion n be obined wo he ime-limiing e. A hor ime ( wrien ( q v ) q ( qv ) 4D 4D 4D g, hor ( q, ) = pe pe pe 4π D The vrine of he diribuion, (, ) ), he ionry diribuion, g (, ).. q, i (S) g q, hor ime i equl o D nd doe no vry wih ime, however, he inervl beween he pek doe grdully inree ime inree. A long ime ( g q, i wrien ), 7

26 g g, long, long, ( q, ) = 4π D eff e q eq 4D p eff D R q Deff D eff. q onverge o he Guin diribuion wih vrine of D eff. (S) E. Dynmi of he muli-mode model For he muli-mode model, given ive mer i opered by vrible ompoed of hree diree e: Γ, Γ, nd Γ. If he ive mer wih Γ i e i loed in he infinieiml re dx, hen he probbiliy of finding he ive mer n be wrien ρ Γ, x, dx. The PDF ifie he onervion lw, wrien i 3 ( x ) dx ρ Γ i,, =. (S) i From he onervion lw, he oninuiy equion for he PDF i wrien 3 ρ( Γ i, x, ) = ( x ρ( Γ i, x, )) Κi jρ( Γ i, x, ) Κ j iρ( Γ j, x, ) x, j i where x nd Κ i j denoe he ime derivive of x nd he re onn from e Γ i o Γ j. Here, we onider he moion of ive mer in n overdmped environmen where elerion i zero. The ime derivive of n ive mer poiion i wrien dx = v( Γ ) γ ξ( ), (S3) d where ξ ( ) repreen he rndom fluuing fore modeled Guin whie noie. The enemble verge of ρ ( Γ i, x, ) over he Guin whie noie give he oberved PDF P( Γ i, x, ) []. The ppliion of he umuln expnion give Eq. (3) nd (4), where we e P( Γ i, x, ) o Pi ( x, ) for he ke of impliiy. Inernl-e-dependen veloiy, v( Γ ), depend on he e v( Γ ) = v, v( Γ ) =, nd v( Γ ) = v. F. Convergene of P (,) x o PS (,) x he mll R limi The populion rio, R( p eq ( p eq eq p ) k k) = =, of he pive e o he ive e module he hpe of he probbiliy deniy of he ive mer, P (,) x, in he muli-mode model. Applying wrien k = nd 8 k R = o Eq. (S) produe P ( w, ),

27 P ( w, ) = = In he limi of ( Dw )( Dw R ) v w R ( R ) ( D )(( D )( D R ) ) R ( Dw )( R RDw R ) v w R ( R ) ( ) ( )( ) R w w w v w v w D w D w R RD w R v w v w R, P ( w, ) i wrien. (S4) D w lim P ( w, ) =. (S5) R w w v w ( D )( D ) The relxion ime of he ingle-mode model i produe P( w, ) S = k. Applying k = o Eq. (S) Dw P. (S6) S, = vw ( w) ( Dw )( Dw ) ( w, ) i he me P ( w, ) P S in he mll R limi. G. Relxion ime of he wo olvble model In he high friion regime, where we n fely negle he ineril erm in he Lngevin equion, he veloiy, x (), of ive mer wih friion onn, γ, n be wrien he um of wo omponen: x v, (S7) () = ( Γ ()) γ ξ() where v ( Γ ) nd γ ξ() repreen he veloiy omponen of elf-propelled, bllii moion, whih i dependen on he inernl e, Γ, nd he veloiy omponen ued by he rndom fluuing fore. If we ume h he iniil poiion of ive mer i zero, hen he ime inegrion of Eq. (S7) produe he ime-dependen poiion, wrien ( ) x = v( Γ ( )) γ ξ d. (S8) From Eq. (S8), he SD of he ive mer n be evlued from he veloiy orrelion funion, wrien ( ) x = d d v Γ v Γ ξ ξ = D d v v γ, (S9) 9

28 where we denoe v Γ ( ) in hor v obin he following equion: Beue ( ) ( ) = D ( e ). By ompring Eq. (S9) wih Eq. (6), we d v v. (S3) D i equl o v normlized ime orrelion funion of veloiy, φ (), φ v () () (), he eond derivive of eh ide of Eq. (S3) provide he v = v v v = e, (S3) where i given by k for he ingle-mode model nd k for he muli-mode model. H. Generl model In generl, given ive mer move in mulidimenionl pe, d, nd i rndom fluuing fore h finie relxion ime,. To obin nlyi oluion for hi generl model, he veloiy of ive mer orreponding o Eq. () i generlized o () = ( Γ ()) γ () p r v ξ, (S3) where eh bold ymbol denoe he d-dimenionl veor orreponding o eh lr quniy in Eq.(). The inegrion of eh ide of Eq. (S3) from o produe he ime- r, wrien dependen poiion, ( ) = ( ( ) Γ ) r v γ ξ d, (S33) where we ume he iniil poiion i zero. From Eq. (S33), he SD i wrien ( ) d ( ) r = γ ξ( ) ξ() v ( ) v () = = d d D p φξ ( ) Dφv ( ) r ξ ( ) r ( ) v, (S34) where φ ξ () denoe he normlized ime orrelion funion, rndom fluuing fore, ξ (), nd he relxion ime, p, i defined ξ( ) ξ() ξ (), of he p dφξ (). D d γ d Here, he diffuion oeffiien for pive moion i defined by = ξ ξ (), nd he diffuion oeffiien for elf-propelled moion i defined by D = d d v () v. The SD oni of wo independen movemen from he diffuive mode nd he elf-ive mode. The diffuive mode onribuion o he SD i

29 defined r dd d ( ) ξ φ ( ), while he elf-propelled mode onribuion i defined dd d ( ) p ξ r φ ( ). If we ume h he diribuion of ξ () v v i Guin, hen he nlyi oluion for he fourh momen of diplemen n be wrien 4 ξ ξ v v 4 ( ) = ( d ) ( ) ( d ) ( ) ( ) ( ) r r r r r, (S35) where r 4! d d d d v ( ) v ( ) v ( ) v ( ). The non-guin v prmeer for he generl model i wrien α R ( ) ( ) d r = d r ( ) wih ( ) A hor ime, ( ) 4 r v ( ) r ( ) ( ) ( ) 4 α 4 3 v ( ) r α d v d. (S36) v r v α R pprohe zero beue ( ) ( ) α ( ) lo pprohe zero beue ( ) R α v pprohe zero. r r v. A long ime, I. Time orrelion funion for wo olvble model Time orrelion funion of he veloiy omponen, D nd ( ) v, of elf-propelled moion re ued o lule α well he eond nd fourh momen of diplemen. In our model, beue v i only dependen on he inernl e, we nlyilly obin he ime evoluion of inernl e probbiliie P P = P G P wih for he ingle-mode model nd P P P = G P wih P P ( k) inh ( k) ( k) oh ( k) k oh G = E (S37) inh

30 G = k k e k k e k e k k e k k k k k k e k k e k k e ( k k) k ( k k ) k ( k k ) e k k ( k k) kk k ( k k) k ( k k) k ke ( k k) e ( k k ) e e k k k k k ke k (S38) for he muli-mode model. We obin he veloiy uoorrelion funion, v ( ) v (), hrough he following equion: v( v ) () = v Γ v Γ G Γ, Γ, PΓ,, (S39) j i j i i i, j, G denoe rniion mrix from Γ i o where ( Γ j, Γi,) Γ j fer ime ping. The rniion mrie re hown in Eq. (S37) for he ingle-mode model nd in Eq. (S38) for k he muli-mode model. The lulion reul of v( v ) () re ve for he inglemode model nd pve k for he muli-mode model, whih oinide wih Eq. (S3). By luling he SD hrough he ppliion of he luled veloiy uoorrelion funion o Eq. (S9), we obin Eq. (6). The four-ime veloiy uoorrelion funion, v ( ) v ( ) v ( ) v ( ), i obined by he following equion, wrien 4 3 v ( ) v ( ) v ( ) v ( ) 4 3 i, j, k, l, = v Γ v Γ v Γ v Γ l k j i (,, ) (,, ) (,, ) P(, ) G Γ Γ G Γ Γ G Γ Γ Γ l 4 k 3 k 3 j j i i. 4 k 4 3 The lulion reul of v( 4) v( 3) v( ) v( ) re ve for he ingle-mode 4 k model nd 3 k e e 3 k e 4 pv R for he muli-mode model. The four-ime veloiy uoorrelion funion n be ued o genere 4 r in Eq. (S35), nd heir reul re equl o he fourh momen of diplemen whih re wrien in Eq. (S7) nd (S). v J. Derivion of hor ime men veloiy diribuion from Eq. () In our model, he veloiy of ive mer oni of wo omponen in Eq. (). If we ume h he wo omponen re independen, hen he men veloiy diribuion i wrien (, ) = δ ξ ( ( ξ) ) (, ) ξ ( ξ, ) f v dv dv v v v f v f v, (S4)

31 where v nd f (, ) v repeively denoe he veloiy omponen ued by elfpropelled moion nd i diribuion funion; v ξ nd f ( v, ) ξ ξ denoe he veloiy omponen due o he rndom fluuing fore nd i diribuion funion. A hor ime, he veloiy omponen, γ ξ(), ued by he rndom fluuing fore i lredy relxed nd follow Guin diribuion wih vrine of D, where he elf-propelled moion pproximely minin i direion. The wo diribuion funion hor ime n be wrien vξ 4D, 4 eq = nd f ( v, ) = p δ ( v v ) f v e D ξ ξ π. (S4) i i i Γ By pplying Eq. (S4) o Eq. (S4), he men veloiy diribuion funion hor ime n be rewrien iu( v ( v vξ )) eq fhor ( v, ) = du dvdvξ e pi δ v vi fξ vξ, π π i Γ = eq iu( v v i ) iuv p i due ξ dvξ e fξ ( vξ, ) i Γ i Γ eq (, ) δ (, ) = p dv f v v v v = p f v v eq i ξ ξ ξ i ξ i ξ i i Γ ( ) v v i 4D eq = pi e. (S4) 4π D i Γ Eq. (S4) i equivlen o he men veloiy diribuion for he muli-mode model hor ime, whih i hown in Eq. (S8). K. Sohi imulion mehod Our ohi imulion mehod oni of boh he Brownin dynmi for he ime evoluion of n ive mer poiion nd he Gillepie mehod for he ohi rniion beween inernl e [,3]. For he Brownin dynmi, we numerilly inegre Eq. (S3) ξ ' x = x v Γ D, (S43) where x( ),, nd ξ '( ) denoe he ive mer poiion ime, he ize of he ime ep, nd he Guin rndom number wih G (,), repeively [3]. For he Gillepie mehod, we ume h he rniion beween inernl e for he muli-mode model re forbidden, exep hoe rniion deribed by he following four unimoleulr reion: 3

32 Κ = k Γ Γ, Κ = k Γ Γ, Κ = k Γ Γ, nd Κ = k Γ Γ []. The reion onn for he forbidden rniion re e equl o zero. Our ohi imulion proeed follow:. Rndomly hooe n inernl e of ive mer bed on he equilibrium populion beween e nd e he iniil poiion equl o zero. Se he eleed e o he urren e, Γ.. Bed on he urren e, lule he wiing ime for reion uing he equion, =ln ( RN ) Κ j, where RN denoe n evenly diribued rndom number j beween nd, beue onenrion of he eleed e i nd he onenrion for he oher e i zero. Only he Γ e h wo reion ph wih equl probbiliy, nd he oher e hve only one ph for e rniion. 3. Unil he wiing ime i over, evolve he ime-dependen poiion uing Eq. (S43) wih he e-dependen veloiy v( Γ ) nd given ime inervl. 4. Afer finihing he ime evoluion in ep 3, hnge he urren e o he e deermined by he rniion in ep. Reurn o ep when he elped ime of he rjeory i le hn he ime limi of he rjeory. 5. Reurn o ep unil uffiien rjeorie re olleed. We ue 5, rjeorie o obin he veloiy diribuion nd he eond nd fourh momen of he diplemen diribuion. L. Referene [] H. Riken, The Fokker-Plnk equion : mehod of oluion nd ppliion (Springer-Verlg, New York, 996), nd edn., Springer erie in ynergei,, 8. [] D. T. Gillepie, J. Phy. Chem. 8, 34 (977). [3] D. L. Ermk, J. Chem. Phy. 6, 489 (975). 4

33 Figure S. Non-Guin prmeer. ( ) α in ll rnge of R nd re ploed when D D i equl o.. The diribuion of diplemen i ub-guin (uper-guin) in ll ime rnge if R<.6 (R>.), hown in Fig. 3(d). The wo horizonl red line repreen α for he wo riil vlue of R:.6 nd. In he rnge.6 < R <., he diplemen diribuion wihe from ub-guin o uper-guin long he ime xi. The boundry beween he ub-guin nd he uper-guin diribuion i repreened by he blk line. 5