Molecular Spiders in One Dimension

Molecular Spiders in One Dimension Tibo r An tal, 1 P . L. Kra pivsky, 2 and Kirone Mallick 3 1 Pr o gr am for Evolutionary Dynamics, Harvar d University, Cambridge, MA 02138, USA 2 Dep a rtment of Physics and C enter for Mole cular Cyb ernetics, Boston University, Bost on, MA 02215, USA 3 Servic e de Physique Th´ eorique, Ce a Saclay, 91191 Gif, F r anc e Molecular spiders are sy nthetic bio-molecular systems whic h hav e “legs” made of short single- stranded segmen ts of DNA. Spiders move on a surface co v ered with single-stranded DN A segments complemen tary to leg s. Different mapp ings are e stablished b etw e en v ari ous models of spiders and simple exclusion pro cesses. F or spiders with simple gait and v arying n um b er of legs w e compute the diffusion co efficien t; when th e hopping is biased we also compu t e their velocity . I. INTRO DUCTION In recent years, chemists have constructed a nu mber of synthetic molecular sys tems which ca n mov e on surfac e s and tra cks (see e.g . [1, 2, 3] and a comprehensive review [4]). One such ob ject is a m ulti-p edal mo lecular spider whose leg s are short single-str a nded segments of DNA [5]. These spiders c a n mov e o n a surface cov ered with single-stra nded DNA se gments, called substrates. The substrate DNA is complementary to the leg DNA. The motion pro ce e ds as legs bind to the s urface DNA thro ugh the W atson-Cr ick mec hanism, then disso cia te, then re- bind aga in, etc. More precis ely , a b ond on the s ubstrate with an attached leg is fir st cleav ed [5], and the leg then disso ciates from the affected substra te (which we shall call pro duct). The leg then rebinds again to the new substrate or to the pro duct leading to the motio n o f the spider. The ra te of attachmen t of a leg of the spider to the substrate and the rate of detachmen t from the substrate are different from the corresp onding rates in volving the pro duct instead of the substrate. Hence for the prop er description o f the motion of a single spider one must keep track of its entire tra jectory . This memor y requirement makes the pr o blem non-Ma rko vian [6, 7] a nd generally int ractable ana lytically even in the cas e o f a single spider . (W e shall a ddress this pr oblem in a se parate pap er [8 ].) Many interacting s pider s add ano ther level of complexity . Even if the rates were the same for the substrate and the pro duct, the prop er ties o f the spider (e.g., the na tur e of its gait) repre s ent a nother ma jor c hallenge. Here we separate this latter issue from the res t: we inv estigate how the s tr ucture of the s pider affects its characteristics (velocity and diffusion co efficient). F urther, w e consider spiders with idealized gait — the g oal is not to mimic complicated (a nd p o orly known) gait of molecular s piders but to show that spiders’ macro s copic characteristics ar e very sensitive to their str ucture and g ait. W e shall mostly focus on a single spider moving on a lattice. W e shall als o a ssume that the r ate of attachmen t greatly e xceeds the r ate o f deta c hment . In this situa tion the relative time when o ne leg is detached is negligible and hence the po ssibility that tw o or more legs are de- tached simult aneously can b e disreg arded. Ther efore the present mo del p osits that s piders remain fully attached and never leav e the sur face [9]. (This differs from the actual situation when a few legs may b e simultaneously detached.) The spiders are defined as fo llows. Legs can jump in- depe ndently a t constant r ates if they do not violate the restrictions b elow. W e mainly consider symmetric spi- ders wher e we set a ll these rates to o ne , or biased spiders whose leg s can o nly move to the right at ra te one, but some sp ecial gaits ar e also inv estigated. The fundamental restriction on the s pider ’s motion is the exclusion pr inc i- ple: Two legs cannot bind to the same site. Additional constraints keep the legs clo se to each other. W e mainly consider tw o types of spiders with the simples t feasible constraints: Cen tip edes (or lo cal spiders). A leg of a centipede can step to nea rest neighbor sites pr ovided that it r e - mains within distance s from the adjac ent legs. (This threshold is a ssumed to b e the sa me for each pa ir of a d- jacent legs.) Spiders (or globa l spiders). Legs o f these spiders can step to nearest neigh bor sites as long a s al l legs remain within distance S . The ab ov e prop erties of the ga it gua rantee that in one dimension the order o f the legs never changes. The ab ov e constraints seem e q ually natural in one dimension, while in tw o dimensions the globa l constraint appears more r ea- sonable. W e shall also briefly discuss a third type of spider where the ne a rest neighbor restrictio n on the hopping is relaxed. F or these quick s piders , legs can s tep anywher e within dis tance S from al l legs. Q uick s piders have be en prop osed and s tudied numerically in Ref. [10]. The ab ov e as sumptions a bo ut the ga it and the dis- regar d of memory effects leav e little hop e for quantit a- tive mo deling, but simplicity ca n help to s hed light on qualitative b ehaviors. Therefore w e study in depth a single spider with a forementioned ga it moving on a one- dimensional lattice, a nd more briefly pro be the influence of the ga it a nd many-spider effects. The rest of this pap er is or ganized as follows. In Sec. I I, we analyze bipeda l spider s (i.e., s piders with tw o legs). This fra mework provides a useful lab or atory to probe v a rious techniques. Bip edal spiders als o closely r esem- ble molec ular motor s [11] and the metho ds develop ed for studying mo lecular motors are fruitful for studying indi- 2 vidual spider s [12, 13]. In Sec. I I I we examine m ulti-p edal spiders (i.e., spiders with L ≥ 3 legs). W e s how that the spider with lo cal constra int and s = 2 is iso morphic to a simple exclusion pr o cess (SEP) on a line with L − 1 s ites and op en bo undary c onditions; an even simpler isomor- phism exists b etw een s piders with global co nstraint and the SE P on the ring. Thes e connections allo w us to ex- tract some spider c haracteris tics from results ab out the SEP . Q uick spiders are briefly inv estigated in Sec. IV. In Sec. V we show that the b ehavior of many interacting spiders ca n also b e understo o d, at least in the pra cti- cally imp ortant s ituation of low s pider density , via the connection with the SE P . Finally we stress limitations of our analy sis a nd discuss p os s ible extensions in Sec. VI. II. BIPED AL SPIDER F or bipe da l spider s, the lo ca l and glo bal constraints are equiv alen t, s ≡ S . F or the simplest mobile bip edal spider, the allowed dista nce b etw een the leg s is one or tw o lattice spacing, i.e ., s = 2. Two p oss ible co nfigurations are (up to translatio n) . . . ◦ • • ◦ . . . and . . . ◦ • ◦ • ◦ . . . (1) where we denote empty sites by ‘ ◦ ’ and filled sites (to which the legs are attached) by ‘ • ’. There are obvious back and for th transitio ns b etw een these configur ations: • • ◦ ⇐ ⇒ • ◦ • and ◦ • • ⇐ ⇒ • ◦ • F or symmetric s piders each leg jumps at rate o ne when po ssible, hence all the a b ove four elementary moves ha p- pen at rate unity . The diffusion co efficient of this bip edal spider is D 2 = 1 4 (2) T o put this in persp ective, w e note that the diffusio n co efficient o f a random walker which hops to the rig ht and left with unit ra tes is D = 1 . Thus adding a leg a nd requiring the legs to stay within distance tw o to each other reduces the diffusion co efficient by a factor 4. Generally for symmetr ic bip edal spiders with arbitrar y s , there are s p ossible configura tions C ℓ lab eled by the int er-leg distance, ℓ = 1 , . . . , s . The trans itions ar e C 1 ⇐ ⇒ C 2 ⇐ ⇒ . . . ⇐ ⇒ C s The diffusion co efficient D s of this bip edal spider is D s = 1 2  1 − 1 s  (3) The ab ov e results apply to symmetric bipedal spiders which hop to the left and right with equal r ates. Molec- ular moto rs usually undergo directed motion [11], and one of the g oals o f future resear ch is to control spiders to mov e prefer en tially in a certain dir ection. Here w e a na- lyze such dire cted motio n theoretically . F or concr eteness, we fo cus on the extreme bia s when each leg can only hop to rig ht at r a te one. F or insta nce, for the bip edal s pider the most compact configura tion evolves via • • ◦ = ⇒ • ◦ • ; the pro c e ss ◦ • • = ⇒ • ◦ • inv olv es hopping to the left and therefore it is for bidden in the biased ca se. F or biased bip edal spiders the velocity and the diffu- sion co efficient ar e given by V s = 1 − 1 s , D s = 1 3  1 − 1 s   1 − 1 2 s  (4) In this section we give a p edestrian deriv ation of (3). The expre s sions (4) for velocity a nd diffusion co efficient can b e derived by utilizing the same technique; instead, we shall extrac t them from more general results for la me spiders (Sec. I I C). T o s et the notation and to explain how we co mpute the diffusion co efficient we beg in with a random walk (which is a one-leg spider). Let P n ( t ) b e the proba bility that the r andom walker is at site n at time t . This qua n tit y evolv es acco rding to dP n dt = P n − 1 + P n +1 − 2 P n (5) One can solve this equation and then use that s olution to extract the diffusio n co efficient. In the case of the s piders, how ev er, master equatio ns ge neralizing (5) are muc h less tractable, and ther efore a mo r e direct wa y of computing the diffusion co efficient is preferable. Here we desc rib e one suc h a pproach [7]. It in volv es tw o steps. First, one ought to determine the mea n-square displacement h x 2 i = ∞ X n = −∞ n 2 P n (6) Then the basic formula [7] D = lim t →∞ h x 2 i 2 t (7) allows to extra ct the diffusion co efficient. F or the ra ndom walk, the mean-sq uare displacement evolv es acco rding to d dt h x 2 i = ∞ X n = −∞ n 2 ( P n − 1 + P n +1 − 2 P n ) (8) T ransforming the fir st tw o sums we obtain ∞ X n = −∞ n 2 P n ∓ 1 = ∞ X n = −∞ ( n ± 1) 2 P n (9) These identities allow us to recast (8) into d dt h x 2 i = ∞ X n = −∞  ( n + 1) 2 + ( n − 1) 2 − 2 n 2  P n = 2 ∞ X n = −∞ P n = 2 3 where the las t equality follows fro m nor malization. Thus h x 2 i = 2 t . Plugg ing this int o (7) we recover the diffusion co efficient of the rando m walker D = 1 . W e now turn to the bip edal spider . W e shall examine in detail o nly symmetric hopping. A. Bip edal Spi der wi th s = 2 F or the bip edal spider with s = 2 there ar e tw o p ossible spider co nfigurations. Denote by P n ( t ) and Q n ( t ) the probabilities that at tim e t the spider is in r esp e c tiv e configuratio ns (1), namely P n = Pr o b[ • • ] , Q n = Pr o b[ • ◦ • ] , (10) with the left leg b eing at site n . The gov erning equations for these probabilities are dP n dt = Q n + Q n − 1 − 2 P n (11a) dQ n dt = P n +1 + P n − 2 Q n (11b) The mean p o sition o f the legs or the ‘cen ter of mass’ of the s pider in a co nfiguration cor resp onding to P n (resp. Q n ) is loca ted a t n + 1 2 (resp. n + 1). Thus the mea n- square displacement is h x 2 i = ∞ X n = −∞ "  n + 1 2  2 P n + ( n + 1) 2 Q n # (12) and it evolves accor ding to d dt h x 2 i = ∞ X n = −∞  n + 1 2  2 ( Q n + Q n − 1 − 2 P n ) + ∞ X n = −∞ ( n + 1) 2 ( P n +1 + P n − 2 Q n ) Utilizing the same tricks as in (9) w e recast the ab ov e equation into d dt h x 2 i = 1 2 ∞ X n = −∞ ( P n + Q n ) = 1 2 (13) The las t identit y is implied by norma lization a nd its v a - lidit y also follows fro m Eqs. (1 1 a)–(11b). Integrating (13) yields h x 2 i = 1 2 t w hich in conjunction with (7) leads to the previo usly a nnounced result, Eq. (2). B. General Case In the general case ( s ≥ 2) we denote P ℓ n = Prob[ • ◦ · · · ◦ | {z } ℓ − 1 • ] (14) the probability to o ccupy sites n and n + ℓ . Thes e pro b- abilities ob ey dP 1 n dt = P 2 n − 1 + P 2 n − 2 P 1 n (15a) dP ℓ n dt = P ℓ +1 n − 1 + P ℓ − 1 n +1 + P ℓ +1 n + P ℓ − 1 n − 4 P ℓ n (15b) dP s n dt = P s − 1 n +1 + P s − 1 n − 2 P s n (15c) where equa tions (15b) apply for 2 ≤ ℓ ≤ s − 1. The mean-squar e displacement is given by h x 2 i = ∞ X n = −∞ s X ℓ =1  n + ℓ 2  2 P ℓ n (16) Using Eqs. (15a) –(15c) a nd a pplying the s ame tricks as ab ov e to simplify the s ums, we obtain d dt h x 2 i = s X ℓ =1 w ℓ − 1 2 ( w 1 + w s ) (17) where w ℓ = P n P ℓ n is the weight of configura tions o f the t yp e C ℓ . The sum on the right-hand side of Eq . (17) is equal to one due to normaliza tio n. T o determine w 1 and w s one do es not need to solve an infinite set of the mas ter equations (15a)–(15 c). Instead, we take Eqs . (15a)–(15c) and sum each of them over all n to yield a clo sed system of equations for the weights dw 1 dt = 2( w 2 − w 1 ) (18a) dw ℓ dt = 2( w ℓ − 1 + w ℓ +1 − 2 w ℓ ) (18b) dw s dt = 2( w s − 1 − w s ) (18c) If initially w 1 = . . . = w s = 1 / s , then Eq s . (1 8 a)–(18c) show that this remains v alid forever. Ev en if we s tart with an arbitrary initial co ndition, all the w eigh ts w s relax exp onentially fast tow ard the ‘eq uilibrium’ v alue 1 /s . Th us the rig ht-hand side of (17) b ecomes 1 − 1 /s yielding h x 2 i = (1 − 1 /s ) t which in conjunction with (7) leads to Eq . (3). C. Heterogeneous Spiders V arious spiders can b e ass e mbled exp erimentally [5 ], including those with distinguishable legs. Here we ana- lyze the coars e-grained pro pe r ties of these ‘lame’ spiders . The bip edal lame spider is character ized by the max- imal separ ation s betw een the legs and b y the hopping rates α and β of the leg s, e.g ., the α -le g hops to the r ight and left with the sa me rate α (whenever hopping is p os- sible) in the symmetric case. F or the bip edal spider with s = 2, the diffusion co efficient is given by D 2 = 1 2 αβ α + β (19) 4 When α = β w e recover the a lready known result telling us that the diffusion co efficient is 4 times smaller than the hopping rate. F or a very lame spider ( α ≪ β ), E q. (19) gives D 2 = α/ 2, so the diffusion co efficient is half the hopping rate o f the very slow leg . T o der ive (19), we first note that the pr obabilities (10) satisfy dP n dt = β Q n + αQ n − 1 − ( α + β ) P n (20a) dQ n dt = αP n +1 + β P n − ( α + β ) Q n (20b) Here we hav e assumed tha t the left leg hops with ra te α and the right leg hops with ra te β . (Recall that in one dimension, the order of the legs never changes.) Using Eqs. (20a)–(20 b) we find that the mean-sq uare displacement (13) evolv es accor ding to d dt h x 2 i = α + β 4 − ( α − β )( u − v ) (21) where u = ∞ X n = −∞  n + 1 2  P n , v = ∞ X n = −∞ ( n + 1) Q n F rom E qs. (20a)–(20b) we deduce that the quantit y u − v ob eys d dt ( u − v ) = α − β 2 − 2( α + β )( u − v ) (22) Equation (22) shows that u − v quic kly appr oaches to ( α − β ) / [4( α + β )]. Plugg ing this into (21) y ields h x 2 i t → 1 4  α + β − ( α − β ) 2 α + β  = αβ α + β which leads to the a nnounced result, Eq. (19). F or the bias ed bipedal lame spider, the drift velocity is given by a nea t fo r mu la V 2 = αβ α + β (23) which r esembles to (19). T o esta blish (23) o ne can use the analo g o f E qs. (20a)–(20a), namely dP n dt = αQ n − 1 − β P n (24a) dQ n dt = β P n − αQ n (24b) Equations (24a)–(24 b) g ive the weights w 1 ≡ ∞ X n = −∞ P n = α α + β , w 2 ≡ ∞ X n = −∞ Q n = β α + β and relation V = ( β w 1 + αw 2 ) / 2 leads to (23). F urther ana lysis o f Eqs. (24a)–(24 b) a llows one to de- termine the diffusion co efficient D 2 = 1 2 αβ α 2 + β 2 ( α + β ) 3 (25) W e do not give a deriv ation of this for m ula since it c an b e extracted from earlie r r e sults by Fisher and Ko lomeisky [12] who in turn used previous findings by Derr ida [13]. It is mor e difficult to co mpute the diffusion co efficient for the bip edal lame s pider with maximal span s > 2. The re s ults of Ref. [1 2] do not cov er the g eneral case, al- though a prop er extension of metho ds [12, 13] may solve the problem. F or the sy mmetr ic bip edal lame s pider with maximal span s ≥ 2, we used an a pproach outlined in Appendix A a nd obtained D s = αβ α + β  1 − 1 s  . (26) F or s = 2, we rec ov er e q uation (19). F or the biased bip edal lame s pider with ma x imal span s ≥ 2 it is again simple to determine the drift velo city . Using an ana log of (24a)–(24b) one gets the weight s a nd then the dr ift velocity is found fro m the relatio n 2 V s = β w 1 + ( α + β ) s − 1 X ℓ =2 w ℓ + αw s The outcome of this co mputation is V s = αβ α s − 1 − β s − 1 α s − β s (27) Spec ia lizing to α = β = 1 (the l’Hospital rule allo ws to resolve a n appare n t singular it y) o ne a rrives at the ex- pression (4) fo r the velo cit y . Finally , the diffusion co efficient for the biased bipeda l spider with a rbitrary s is D s = 1 2 αβ α s − 1 − β s − 1 α s − β s + 1 + s α s − β s α s − 1 − β s − 1 α s − β s α s +1 β s +1 α s − β s + 1 − s α s − β s α s +1 − β s +1 α s − β s α s β s α s − β s . (28) This equation is derived in App endix A. Equation (28) r educes to (25) when s = 2; for s = 3, the diffusion co efficient can b e re-written as D 3 = 1 2 αβ ( α + β )( α 2 − αβ + β 2 )( α 2 + 3 αβ + β 2 ) ( α 2 + αβ + β 2 ) 3 Also whe n α = β = 1, equation (28) reduces to the expression (4) for the diffusion co efficient. II I. MUL TI-PED AL SPIDERS F or the m ulti-p e dal spider, L ≥ 3, we m ust sp ecify the constraint governing the separa tions b etw een the leg s. 5 FIG. 1: Illustration of a centipede, i.e. a spider with local constrain t. All legs step to empty n earest n eigh b or sites at the same rate with adjacent legs staying within distance s from each oth er. A. Centipede s Here we consider centipedes or lo cal spiders where the distance b etw een the j th and ( j + 1) st legs is at most s . See Fig. 1 for an illus tration of such a centipede. In this case the to ta l num ber of c onfigurations is C = s L − 1 since each of the ( L − 1) spacings b etw een adjac en t leg s can hav e s p oss ible v a lues. 1. Main r esults Consider first spiders with s = 2. The co nfigurations for the bipeda l spider are shown in (1); the fo ur p oss ible configuratio ns for the trip o d a re • • • • ◦ • • • • ◦ • • ◦ • ◦ • (29) and genera lly there are 2 L − 1 po ssible co nfigurations. Let D ( L ) b e the diffusion co efficient of an L - leg spider . In the case of sy mmetric hopping (a ll rates ar e one ) D ( L ) = 1 4( L − 1) (30) when s = 2. F or L = 2, this of course a grees with our previous result: D (2) = D 2 = 1 / 4 . F or the bias ed mu lti-p edal spider , the veloc ity is V ( L ) = 1 2 L + 1 2 L − 1 (31) The biase d infinite-leg spider ha s a finite limiting sp eed! More precisely , V ( ∞ ) = 1 / 4 , i.e., the infinite-leg spider drifts 4 times slow er than the single- le g spider . The dif- fusion co efficient o f the bia sed spider is D ( L ) = 3 4 (4 L − 3)! [( L − 1 )!( L + 1)!] 2 [(2 L − 1)!] 3 (2 L + 1)! (32) Note that the diffusion co efficient of the infinite-leg spider v a nishes. Asymptotica lly , D ( L ) ∼ 3 √ 2 π 128 L − 1 / 2 as L → ∞ (33) The ab ove r esults (30)–(32) are v alid when s = 2. W e hav e not succeeded in computing V ( L ) and D ( L ) for ar bi- trary L when the maximal separatio n exceeds t wo, s > 2. The veloc ity and the diffusion co efficient can b e co m- puted for centipedes with s > 2 when the num ber of legs is sufficient ly small. The simplest quantit y is the velocity of biased spiders. When s = 3, we computed the velocity V ( L ) of centipedes with up to s even legs: V (2) = 2 / 3 V (3) = 26 / 45 ≈ 0 . 57 78 V (4) = 2306 / 4301 ≈ 0 . 53 62 V (5) = 22579 32864 491452 44106 56468 591479 ≈ 0 . 51 19 V (6) ≈ 0 . 4960 47642 9 V (7) ≈ 0 . 4848 25979 5 (w e have no t displayed exa ct expres sions for V (6) and V (7) which are the ratios of h uge in tegers.) Note that for biased spiders with s = 3 o ne ca n guess the general expression (31) from e x act res ults fo r V ( L ) fo r small L ; in contrast, no s imple expressio n se e ms to exist for the velocity of bia sed spiders with s > 2. F or symmetrically hopping spiders , we computed the diffusion co efficient when the nu mber of legs is small. Here are the res ults for cen tipede s with s = 3 (the metho d used in calcula tio ns is describ ed in App endix A) D (2) = 1 / 3 D (3) = 22 / 117 ≈ 0 . 1880 D (4) = 53 0 / 405 9 ≈ 0 . 1 306 D (5) = 14573 04063 62990 14576 69284 934841 ≈ 0 . 0999 749 D (6) = 13157 42416 9190800305558220463956878370565 16245 43448 89141072641777603974162004103911 ≈ 0 . 08 09915 19 In contrast to the nea t formula (30) characterizing the s = 2 case, the ab ove num bers lo ok intimidating. F ac- torizing the nominator and denomina tor of D (6) reveals the presence of extraordina ry huge factor s and there by excludes that it c a n b e des crib ed by a formula like (32), let alone (30). Note that a t le a st the L − 1 asymptotic behavior pr edicted b y Eq . (30) rema ins v alid for all s ; for s = 3, in particular , we hav e D ( L ) ∼ AL − 1 with A ≈ 0 . 4 23 when L ≫ 1 . 2. Mapping to the exclusion pr o c ess for s = 2 The deriv ations of ab ov e r esults are complicated since the num ber of c o nfigurations grows exp onentially with L . F urther, the transition rates are c o nfiguration dep endent, e.g., for the four-leg spider config urations • • • • • ◦ • • • • • ◦ • • evolv e with rates 2, 3, 4 for symmetric hopping. (In contrast, for bip edal spiders the num be r of config ur ations grows linearly with s and the transition rates ar e simple.) All this makes the computation o f the diffusion co efficient D ( L ) for arbitrar y L very challenging. The p edestrian 6 calculation is feasible for small L , but even for L = 3, the framework bas ed on rate equations like (15a)–(15c) is very c umber some. F ortunately , spiders with lo cal constraint and s = 2 are rela ted to simple exclusio n pro cesses (SEPs). This allows us to ex tract some pr edictions ab out spider s from previously known results a bo ut SEP s , and to employ the metho ds developed in the con text of SEPs to situatio ns natural in applications to spiders. W e now demonstr ate the rema rk able co nnection b e- t ween centipedes with s = 2 and SEPs. As an example we show that the biased spider is isomorphic to the to- tally asy mmetr ic simple exclus ion pr o cess (T ASEP) with op en b oundary conditions. T o understa nd the isomor - phism, co ns ider for concreteness the trip o d. W e can map configuratio ns (29) o nt o configur ations 00 10 01 11 (34) of the exclusio n pro c e ss on tw o sites with op en b oundary conditions. Here 0 on j th site implies that there is no empt y site b etw een j th and ( j + 1) st legs, while 1 implies that there is an empty site. A hop to the right of an int ernal leg in (29) corresp onds to a ho p to the left o f a particle in (34). F urther, the ho p of an extreme rig h t leg corres p onds to the addition of a par ticle to the extreme right p osition, a nd the ho p o f the e x treme left leg corre- sp onds to the remov al of a particle from the extreme left po sition. The same mapping applies to any L . Thus in this T ASEP ea ch s ite i = 1 , . . . , L − 1 ca n be o ccupied by a particle, a nd each particle hops to the left with rate one if this site is empt y; further, a particle is added to site i = L − 1 with rate one if this site is empty , and a particle is removed from site 1 w ith ra te one if this s ite is o ccupied. Thus we hav e shown that the s = 2 bia s ed spi- der that mov es to the right is equiv alent to the T ASEP with open b oundar ies in which pa rticles hop fr o m r ight to left. A similar ma pping holds betw een the symmetric spider and the symmetr ic exclusion pro c e ss. Derrida, Domany , a nd Muk amel [1 4] hav e shown that Eq. (31) gives the flux in the T ASEP; the isomor phism betw een the flux and velo cit y pr ov es that the velo city of the biased s piders is given by (31). T his result was re- derived by other techniques, e.g., by a pure combinatorial approach [15]. The (muc h mo re co mplicated) deriv ation of the diffusion co efficient in Ref. [1 6] gives (32). 3. Derivation of (30) F or the symmetric spider, it s hould b e p ossible to com- pute the diffusion co efficient (3 0) by using the technique of Ref. [16]. This technique (based on an extensio n of a matrix technique) is very adv a nced. The final res ult (30) lo oks muc h s impler tha n its biased counterpart (32). Hence we hav e sought another der iv a tion, and we hav e found an intriguingly simple pro o f of (30 ) bas ed o n the fluctuation-dissipation fo rmula (s e e App endix B). First w e r ecall that the symmetric spider with L legs hopping in b oth directions with rates equal to 1 is equiv- alent to a symmetric exclusion pro cess on L − 1 sites with op en b oundar y conditions. F or this SEP , al l rates (i.e. hopping r ates in the bulk, entrance a nd exit rates at the bo undaries) are equal to 1. This Markov pro ce ss satisfies detailed balance a nd is at equilibrium; in particular , the mean curr en t, i.e., the velo city of the spider, v anis he s ident ically . The v ariance o f the current corr e s po nds to the diffusion constant of the spider . This v ariance can b e calculated as follows. Consider now a s ymmetric exclusion pr o cess of length L − 1 with op en b oundaries and arbitr ary addition and remov al r ates at the b oundarie s . The sys tem is driven out of equilibrium by pa rticles entering a nd le aving at the b oundaries. In the bulk, each particle hops with rates 1 to the right and to the left (if the corresp onding sites are empty); a pa rticle enters at site 1 with rate α and leav es this site with ra te γ ; similarly a particle e nters another b oundary site L − 1 with rate δ a nd leav es this site with ra te β . Gener ically , these unequal rates lead to a current. The mean v alue of this cur r ent is g iven by (see e.g. Ref. [17]) J = β β + δ − γ α + γ L + 1 α + γ + 1 β + δ − 2 . (35) The equilibrium conditio ns cor r esp ond to α = γ = β = δ = 1 and J = 0. W e now choos e the ra tes on site 1 as fo llows α = exp( ǫ 2 ) and γ = exp( − ǫ 2 ) a nd we keep β = δ = 1 at site L − 1. Then, the current is given by J = tanh ǫ 2 2 L − 3 + 1 cosh ǫ 2 . (36) The Markov ma tr ix of this pr o cess satisfies the gener- alized de ta iled balance condition given by equation (B2) of the App endix B, with y = ± 1 if a particle enters at site 1, o r ex its from site 1 ( y = 0 otherwise). W e ca n then use the fluctuation-dissipatio n formula (see (B4) in the Appendix B) wich tells us that the fluctuation of the current at the first site is given by D = ∂ J ∂ ǫ    ǫ =0 = 1 4( L − 1) , (37) in acco rdance with equation (30 ). 4. Me an- field appr oximation f or s ≥ 3 Simple ex clusion pro cesses have been thoroug hly in ves- tigated (see b o oks and re views [18, 1 9, 20, 21]). Hence one ca n extra c t the r esults ab out spiders from a lready known results ab out SEP . Unfortunately , for spider s with lo cal constra int the mapping onto SEP applies only when s = 2. The spider with L le gs and ar bitrary s can b e mapp ed onto an ex clusion-like proc e s s with L − 1 sites 7 and with op en b oundaries. In this pro cess the maximal o ccupancy is limited, namely the num be r of particles in each site cannot exceed s − 1. The dynamics is simple: one chooses sites with rate one and mov es a particle to the site o n its left; nothing happ ens if the chosen site was empty or the site on the left was fully o cc upied. One also a dds pa rticles to site i = L − 1 and remov es from site i = 1, b oth these pro ce sses o ccur with ra te one; the addition is p ossible as long as s ite i = L − 1 is no t fully o ccupied (contains no more than s − 1 particles). Unfor- tunately , this ne a t pro cess has not b een so lved exactly but it ca n be studied by a mean-field analy sis. T o simplify the analys is, we c onsider cen tipe des with infinitely many legs. W e assume tha t the distance b e- t ween adjacent legs cannot ex c eed s . W e further as sume that the spider’s motion is biased, and limit ours elves to a (mean-field) computation of its velo cit y V ( s ) . First we map the spider onto the g eneralized asy m- metric ex clusion pro cess with at mo st s − 1 particle s p er site. W e then write x j for the densit y of sites with j particles; this is just the density o f gaps of length j + 1 betw een adjacent legs of the spider . The p ossible v a lues are j = 0 , . . . , s − 1. W riting the evolution equa tion for ˙ x j and setting ˙ x j = 0 we obtain ( x j − 1 − x j )(1 − x 0 ) − ( x j − x j +1 )(1 − x s − 1 ) = 0 (38 ) when 1 ≤ j ≤ s − 2. Similarly from ˙ x 0 = 0 a nd ˙ x s − 1 = 0 we get x 1 (1 − x s − 1 ) − x 0 (1 − x 0 ) = 0 (39a) x s − 2 (1 − x 0 ) − x s − 1 (1 − x s − 1 ) = 0 (39b) The obvious no rmalization requir emen t is s − 1 X j =0 x j = 1 (40) As a warm up, co ns ider the first non- tr ivial case s = 3. Due to normalization, it is sufficien t to use (39a)–(39 b). W r iting x 0 ≡ x a nd x 2 ≡ z , we have x 1 = 1 − x − z fro m (40), and (39a)–(39b) b ecome (1 − x − z )(1 − z ) = x (1 − x ) (41a) (1 − x − z )(1 − x ) = z (1 − z ) (41b) These equations are actually iden tical; solving any o f them we ar rive at x = 1 − z + p z (4 − 3 z ) 2 (42) T o c ompute v elo city we return to the original form u- lation. A leg of the spider moves with rate 1 if the site ahead is empty and if the leg b ehind is one o r tw o steps behind. The former event happ ens w ith probability 1 − x , while the latter o ccurs with probability 1 − z . Thus the velocity is V (3) = (1 − x )(1 − z ) (43) Using (42 ) we ge t V = 1 2 (1 − z ) h z + p z (4 − 3 z ) i (44) W e should se lect maximal velo city . The maximum of V ( z ) given by (4 4 ) is re ached a t z = 1 / 3, a nd it r eads V (3) = 4 9 (45) A t the state cor resp onding to the actual (maximal) ve- lo city all densities are equal: x 0 = x 1 = x 2 = 1 / 3. The situation for s > 3 is also simple. Analy z ing re- currence (38) one finds that for all 0 ≤ j ≤ s − 1 the solution is a shifted g eometric progr ession x j = A + B λ j , λ = 1 − x 0 1 − x s − 1 (46) Plugging (46 ) into Eqs. (39a)–(39b) one a chieves the co n- sistency if either A = 0 o r λ = 1. In the latter case the densities are the same, and he nc e they are all e qual to s − 1 due to norma lization r e q uirement (40). The straig h t- forward gener alization of (43) is V ( s ) = (1 − x 0 )(1 − x s − 1 ) (47) and therefore V ( s ) =  1 − 1 s  2 (48) In the complimen tary case of A = 0 the analysis is a bit mor e length y . How ever, the final result is the same. Here is the pr o of. Since x j = B λ j , equa tion (46) gives λ = (1 − B ) / (1 − B λ s − 1 ), which can b e r e-written as B = 1 − λ 1 − λ s (49) F ur ther, (47) be comes V ( s ) = (1 − B )(1 − B λ s − 1 ) (50) Using (49 ) we re c ast (50 ) into V ( s ) ( λ ) = λ  1 − λ s − 1 1 − λ s  2 (51) The maximum of V ( s ) ( λ ) is achieved at λ = 1. Thus the velocity is indeed given b y Eq. (48) . The ab ov e elementary ana lysis is mean-field as we have assumed the v alidit y of the factor ization. The answer is trivially exact for s = 1, and it is known to b e exact for s = 2. F or s = 3, we calculated velocities exactly for small centipedes, see Sect. II I A 1. The limiting L → ∞ v a lue obtained from simulations V (3) ≈ 0 . 41 8 9 is close to the pre dicted mean-field v alue V (3) = 4 / 9 ≈ 0 . 4444. Overall, the assumed factoriz ation is not exact when s ≥ 3. Note that a mo del which differs from our mo del only in the hopping rules ha s b een solved exac tly [22], but there the sta tionary state is a pro duct measure. 8 5. L ame spiders Finally we inv estigate la me centipede spiders whose extreme le ft leg ho ps to the right with rate α a nd the extreme right leg hops to the right with rate β . The ab ov e mean-field a nalysis shows that the velo city of the extreme left leg is α (1 − x 0 ) and the velocity of the extreme r ight leg is β (1 − x s − 1 ). As long as these velo cities ex ceed the bulk velocity (48), the actua l g ap densit y x 0 at the left end a nd x s − 1 at the r ight end will b e higher than their bulk v alues, so the spider will move with velocity (48). This o ccur s as long a s α (1 − s − 1 ) and β (1 − s − 1 ) exceed (1 − s − 1 ) 2 , i.e. α, β ≥ 1 − s − 1 . When at lea st one of the rates is smaller than the thres hold v alue, differe n t behaviors emerge. Overall, the s p eed of the infinite-leg spider exhibits an amusing dep endence on the ra tes α and β : V ( s ) =    (1 − s − 1 ) 2 for α, β ≥ 1 − s − 1 W s ( α ) for α ≤ β , α < 1 − s − 1 W s ( β ) for β ≤ α, β < 1 − s − 1 (52) Thu s if a t least one of the tw o extreme legs has the in- trinsic sp eed less than 1 − s − 1 , the spe e d of the entire spider is solely determined by the slowest leg. T o determine W s ( β ) we note that velo cit y on the right bo undary is V = β (1 − x s − 1 ) = β (1 − B λ s − 1 ) = β 1 − λ s − 1 1 − λ s (53) where in the last step we hav e used (49). Eq uating the velocity given by Eq. (53) with the velocity in the bulk given by E q. (51) we find β = λ 1 − λ s − 1 1 − λ s (54) Thu s the velocity is given by (53) or (51), where para m- eters ar e connected via (5 4). Explicit r esults ca n b e obtained for s up to s = 5. F or s = 2 we recover the celebrated r esult W 2 ( β ) = β (1 − β ) (55) F o r s = 3 the fina l expressio n is still co mpact W 3 ( β ) = β (1 − β ) 1 + √ 1 + 4 b 2 (56) with b = β / (1 − β ). F or s = 4 the result is quite cum- ber some W 4 ( β ) = β 2 λ , λ = 1 6 ∆ − 4 3 ∆ − 1 − 1 3 (57) where we have used the shorthand nota tion ∆ = n 28 + 108 b + 1 2 p 9 + 42 b + 8 1 b 2 o 1 / 3 FIG. 2: Illustration of a sp ider with global constrain t. The legs can step indep endently to nearest neighbor empty sites within a distance S from each other. B. Global Constraint F o r spider s with L legs and maxima l span S b etw een any tw o le g s (see Fig . 2), the glo bal c onstraint r ule limits the maximal dista nce b etw een the extreme legs and the exclusion condition implies that S ≥ L − 1. A spider with maximal distance S = L − 1 is immobile, so we shall tacitly assume that S ≥ L . It is also useful to keep in mind that for a spider sa tisfying the lo cal constr aint r ule the maximal span is ( L − 1 ) s if the maximal distance betw een the adjacent legs is s ; for the bipeda l spider S ≡ s . A spider with globa l constr aint is equiv alen t to the exclusion pro cess on a ring , whe r e each leg is interpreted as a par ticle and the total num ber of s ites is equal to S + 1. F o r such a pr o cess with p erio dic b oundary conditions , a key pro per t y of the statio nary state, which ho lds b oth in symmetric a nd biased cases , is that all co nfigurations hav e equal weight [19]. 1. Configur ations T o count the total num ber of configuratio ns, we set, as usual, the o rigin at the po sition of the extreme left leg, s ee e.g., (29); this allows us to avoid multiple count- ing of configura tions which differ merely by translatio n. W e then note that the other L − 1 legs can o ccupy sites 1 , . . . , S . Thus the to ta l nu mber of configuratio ns is C ( L, S ) =  S L − 1  (58) In the stationary state, the weigh t of a configuration is th us given by w = 1 / C . Let us now calculate the to tal num ber N ( L, S ) o f •◦ pairs in a l l co nfigurations. Eac h co nfiguration b egins with a str ing • ◦ · · · ◦ | {z } a • (59) where a = 0 , 1 , . . . , S − L + 1. Disregarding the par t up to the seco nd leg maps configura tions o f the type (59) with fixed a to configuratio ns o f the spider with L − 1 legs and maximal span S − a . The total nu mber o f •◦ pair s in these latter configurations is N ( L − 1 , S − a ). Configur a tions of the type (59) hav e o f co urse a n a dditional •◦ pair at 9 the b eginning (when a > 0). Therefore N ( L, S ) = S − L +1 X a =0 N ( L − 1 , S − a ) + S − L +1 X a =1 C ( L − 1 , S − a ) The latter sum is simplified by using (58) and the identit y r X p = q  p q  =  r + 1 q + 1  Thu s we arrive at the recurr ence N ( L, S ) = S − L +1 X a =0 N ( L − 1 , S − a ) +  S − 1 L − 2  (60) The solution (found b y a generating function tec hnique or verified by mathematical inductio n) reads N ( L, S ) = L  S − 1 L − 1  (61) Since all c o nfigurations hav e equal weigh t in the sta- tionary state, the veloc it y of the biased spider ca n b e expressed by the total num ber N of • ◦ pairs as V = L − 1 N C , (62) which then leads to V = 1 − L − 1 S (63) (Note that the velocity is zero in the unbiased ca se.) It is more involv ed to calculate the diffusion co efficient, which we obtain b elow separa tely for the unbiased and fo r the biased case. 2. Symmetric hopping The diffusion co efficient of the s ymmetric s pider is given by D = L − 2 N C . ( 64) This equatio n is obtained by applying the fluctuation- dissipation relation, which is v alid b ecause the dy nam- ics of the symmetric spider statisfies deta iled balance (in other words, the sy mmetric spider is a system in thermo- dynamic equilibrium). The Einstein rela tion then implies that D ∝ V , where the velocity V of the biased spider is given by the e x pression (62 ). The ex tr a facto r L − 1 betw een (6 4) and (62) comes fr om the fact that D is the diffusion co efficient of the left leg (or o f the spider ’s center of mass) whereas V is the mobilit y of the bia sed spider where all the legs are asymmetric. Using Eqs. (58) and (61) we rec a st (64) to D = 1 L  1 − L − 1 S  (65) F o r the bip edal spider there is no difference b etw een lo cal and global constra ints. Using L = 2 and S = s we find that Eq. (65) indeed turns int o E q. (3). The calculation of D presented her e is self-contained; we notice tha t the expression (65) can also be found as a specia l case of a general fo rmula derived in [23] for the diffusion constant of a pa rtially asymmetric ex c lusion pro cess. 3. Biase d hopping While the velocity (63) is easily computable, the dif- fusion co efficient was calc ula ted in [2 4] using a matrix Ansatz (see [19] and [23] for a mor e genera l fo rmula). The result is D ( L, S ) = 1 2(2 S − 2 L + 1)  2 S − 1 2 L − 1   S L − 1  − 2 (66) F o r a given num ber o f legs, the diffusion co efficie n t of the most clumsy spider is D ( L, L ) = 1 2 L 2 (67) while the diffusion co efficient o f the most agile spider is D ( L, ∞ ) = 2 2 L − 2 L  2 L L  − 1 (68) When L ≫ 1, the diffusio n co efficient (68) scales as D ( L, ∞ ) ∼ 1 4 r π L (69) More gener ally , the diffusion co efficie nt D ( L, S ) also de- creases as ( π / 16 L ) 1 / 2 when 1 ≪ L ≪ S 1 / 2 . C. Heterogeneous Spiders Each leg of a heter ogeneous (lame) spider may have its own hopping ra te. The bipedal lame s pider was studied in Sec. I I C. One can find explicit ex pressions fo r the ve- lo city and the diffusion co efficient of the lame trip o d and per haps for the lame spider with four legs; the genera l solution for a n arbitrar y L is unknown. Lame spiders are tracta ble if only one or tw o legs have different hopping r ates. Below we co nsider lame spiders whose extreme legs a r e affected. F or concreteness , w e fo cus on spider s with lo cal cons tr aint and s = 2 . The analogy with the T ASEP with op en b ounda r y conditions still applies, the o nly mo dification is that the particle is remov ed from site 1 with ra te α and the par ticle is added to site L − 1 with r a te β . The flux in such a system was found in Ref. [25]; this gives us V L ( α, β ) = C L − 2 ( α, β ) C L − 1 ( α, β ) (70) 10 where we used the s horthand notation C N ( α, β ) = N X p =1 p (2 N − 1 − p )! N !( N − p )! α − p − 1 − β − p − 1 α − 1 − β − 1 Plugging C 0 = 1 and C 1 = α − 1 + β − 1 int o (70) we recov er the express io n (2 3) for the velo city of the bipe da l lame spider; the velo city of the lame tr ipo d is V 3 ( α, β ) = α − 1 + β − 1 α − 2 + α − 1 β − 1 + β − 2 + α − 1 + β − 1 The speed of the infinite-leg spider exhibits an amusing depe ndence on the rates α and β : V ∞ =    1 / 4 for α ≥ 1 / 2 , β ≥ 1 / 2 α (1 − α ) for α ≤ β , α < 1 / 2 β (1 − β ) for β ≤ α, β < 1 / 2 (71) Thu s if b oth rates exceed 1/2 , the sp eed attains a univer- sal (indep endent of the ra tes) maximal v alue V ∞ = 1 / 4 . On the other hand, if at least one of the tw o extreme legs has the intrinsic sp eed less than 1/2, the sp eed of the entire s pider is s olely determined by the s lowest leg. A genera l explicit ex pression for the diffusion co effi- cient is unknown. There ar e tw o sp ecial ca ses, howev er, in which the diffusion co efficient was explicitly ca lc ulated [16]. One is the homogeneous s pider ( α = β = 1 ) when D ( L ) is given b y E q. (32); ano ther particular ca se corr e- sp onds to α + β = 1 when the diffusion co efficient is D L = 1 2 V ∞ ( 1 − L − 2 X k =0 2(2 k )! k !( k + 1)! V k +1 ∞ ) (72) with V ∞ given by E q. (71); since (72) is v alid on the line α + β = 1, we hav e V ∞ = α (1 − α ) = β (1 − β ). As a consistency check one ca n verify that eq ua tions (72) and (25) do a gree : setting L = 2 in the former and α + β = 1 in the latter we indeed obtain the same res ult. The b ehavior of D L for the spider with many legs is again amusing. F or the infinite-leg spider, Eq. (72) yields D ∞ = 1 2 αβ | α − β | when α + β = 1 (73) Thu s o n the line α + β = 1 , the diffusion co efficient v an- ishes only when α = β = 1 / 2. The behavior o f the diffusio n co efficie nt for the infinite- leg spider is particularly nea t, and it had actua lly b een understo o d (in the con text of the T ASEP) for arbitrary α and β . Derrida , Ev ans and Ma llick [16] found tha t D ∞ V ∞ =      0 for α ≥ 1 / 2 , β ≥ 1 / 2 (1 − 2 α ) / 2 f or α < β , α < 1 / 2 (1 − 2 β ) / 2 for β < α, β < 1 / 2 (1 − 2 β ) / 3 for α = β < 1 / 2 (74) The discon tin uit y on the symmetry line α = β < 1 / 2 is esp ecially striking . IV. QUICK SPIDERS In the previous sections we hav e consider ed the sim- plest possible gaits when the spider’s legs can step only to the neighbor ing sites. In this section we briefly ex- plore the behavior of quick spiders. These spider s (in- tro duced in Ref. [10]) differ from previously discussed spiders, namely the le g s of a quick spider can jump ov er several la ttice s ites at once. The only r equirement is to stay within dis ta nce S from the other legs . Hence quick spiders can be in the same states as the corr esp onding global spiders, but more transitions ar e p ossible b etw een the states o f the quick ones. The simplest quick spider has tw o le g s always next to each other ( L = 2 , S = 1). Althoug h such a globa l spider cannot mov e, a q uick s pider ca n put one leg ahead of the other and ca n w alk this wa y . Its motion is completely equiv a lent to a simple ra ndom w alk, hence its diffusion co efficient is D = 1. This is gener ally true for quic k spiders with L legs a nd maximal distance S = L − 1 D ( L, S = L − 1 ) = 1 (75) W e also c omputed the diffusion co efficient o f bip edal quick spiders with arbitr ary S . W e found D (2 , S ) = S ( S + 1)(2 S + 1) 6 (76) This ex pression ca n b e derived using the general formula (64). F or the bip edal spider we can lab el v arious co nfig - uration by the distance 1 ≤ ℓ ≤ S b e t w een the legs . . . ◦ • ◦ · · · ◦ | {z } ℓ − 1 • ◦ . . . (77) T ake the left leg. It can jump to the left up to distance S − ℓ ; the co rresp onding displacement s of the center of mass are ∆ x = − i/ 2 with 1 ≤ i ≤ S − ℓ . The left leg ca n also jump to the rig h t. The dis pla cements are ∆ x = i/ 2 with 1 ≤ i ≤ ℓ − 1, a nd onc e it ov ertakes the right leg, ∆ x = ( ℓ + i ) / 2 with 1 ≤ i ≤ S . T aking also into account that all weigh ts ar e equal, w ℓ = 1 /S , and recalling that jumping of the rig h t leg will give the same contribution, we recas t (6 4) into D = 1 4 S S X ℓ =1 " S − ℓ X i =1 i 2 + ℓ − 1 X i =1 i 2 + S X i =1 ( ℓ + i ) 2 # (78) Computing the sum yields the announced result (76). V. INTERACTING SPIDERS In exp eriments [5], tho usands of spiders a re released, yet their density is usually small. Naively , one can an- ticipate that spider s a re essentially non-interacting. This is co r rect in the ea rlier stag e, t < t ∗ , but even tually spi- ders “realiz e ” the presence of other spiders , and their 11 behavior undergo es a dr astic change from diffusiv e to a sub-diffusive o ne. This intermediate stage pro ceeds up to time t ∗ when spiders explore the ent ire system a nd then the diffusive behavior is restor ed, alb eit with a smal ler diffusion co efficie nt D . Here we compute D and estimate the cro ssov er times t ∗ and t ∗ . Let N spiders b e pla ced o n the ring o f size S . W e assume tha t the spider dens it y n = N / S is low, n ≪ 1; equiv a lent ly the t ypical distance ( n − 1 lattice spacings) betw een neig h bo ring spiders is la rge. Imagine that we know the diffusion co efficient D of an individual spider (e.g., for the bip edal spider with s = 3 we found D = 1 / 3 when each leg hops symmetrica lly with rates equal to one). E a ch spider cov ers ar ound √ D t lattice sites, a nd e q uating √ D t ∗ = n − 1 we arrive at the estimate o f the low er cr ossov er time t ∗ = 1 D n 2 (79) The b ehavior is sub-diffusive in the intermediate time range, t ∗ ≪ t ≪ t ∗ . It is characterized by the ( λt ) 1 / 4 growth of the cov ered line [26]; this so-ca lled single-file diffusion has numerous a pplica tions [2 7, 28]. The ampli- tude λ is fo und b y ma tching ( Dt ∗ ) 1 / 2 = ( λt ∗ ) 1 / 4 which in conjunction with (79) yield λ = D /n 2 . The final b ehavior is again diffusive. In the long time regime, t ≫ t ∗ , we may interpret ea ch s pider a s a n ef- fective pa r ticle hopping to the right or left with ra tes D . The int eraction b etw een spiders is ess e ntially eq uiv alent to exclusion interaction b etw een particles, and hence the system reduces to the SE P . W e can therefore use (65) where we sho uld replace L by N , and we must also mul- tiply the res ult by D s inc e spiders effectively hop with rates D ra ther than one. The term in the brack ets in Eq. (65) reduces to 1 − n ; we can replace it by o ne since n ≪ 1. Therefore Eq. (65) b eco mes D = N − 1 D (80) Thu s exclusio n in teraction greatly reduces the diffusion co efficient. This s trong co op erative effect emer ges even when the densit y is arbitra rily small, the only requir e - men t is that there are many spider s, N ≫ 1. The upper cross ov er time t ∗ is found by equating ( D t ∗ ) 1 / 2 = ( λt ∗ ) 1 / 4 . W e arr ive a t t ∗ = S 2 D = N 2 t ∗ (81) Thu s the a nalogy with SEP ess e n tially so lves the prob- lem in the practically impo rtant limit when the spider concentration is low. Neither memory nor the ga it play any role, one must merely use the diffusio n c o efficient D corres p onding to the actua l gait and c omputed under the assumption that the lattice sites ar e in the pro duct sta te. One s hould remember , of cours e , that the SE P regime is achiev ed when t > t ∗ ; at muc h earlie r times t > t ∗ , the spiders mostly hop o n the pro duct, a nd therefor e the a s - sumption of full attachmen t can b ecome problematic. VI. DISCUSSION A single spider is a s elf-in teracting ob ject. There are t wo sources of interaction b etw een the legs: (i) exclusion (no more than one leg p er site), and (ii) leg s canno t b e to o far apart. Is it po ssible to repr e sent a spider as a n effective single par ticle? The answer is yes — at lea st in simple situatio ns , one can trea t a spider as a diffusing particle. It is far from tr iv ial, of course, to compute the diffusion co efficient o f this pa rticle. F ortunately , natural mo dels o f spiders a re rela ted to simple exclusion pro- cesses. In the course of this work we had an adv an tage of utilizing some b eautiful results a nd p ow erful techniques developed in the studies o f simple exclusion pro ce s ses. Our mo dels certa inly do not take into a ccount all the details o f an ex per imen tal situa tion [5]. F or instance, we assumed that the r e-attachmen t of a le g is very q uic k, so the pro cess is controlled b y detachmen t. Hence spider s remain fully attached and nev er leave the surface. This assumption is impo rtant as o ur a nalysis has relied on the per manent presence of spiders on the surface. Relaxing this assumption do es not make the problem intractable — indeed in recent analyzes of molecular motors the com- plete detachmen t (unbinding) from cytoskeletal fila men ts is a llow ed, see e.g. [29, 30, 3 1, 3 2, 3 3]. F urther, our a na l- ysis of the many-spider situation in Sec. V treats the low density ca se; the analo gy with SEP allow ed us to handle the problem but the ass umed p ermanent presence of the spiders is pa rticularly questionable in this case. Perhaps the mos t ser ious limitatio n of our ana lysis is the dis regard of memory — in exp erimental realizatio ns [5] spiders often affect the environment which in turn affect their motion. The non-Markovian na ture of this problem calls for a se t of new techniques even in the case of a sing le spider. In one dimensio n, the influence of memory can be prob ed analytically for a single bip edal spider [8], and the replacement of a self-in teracting spi- der by an effective particle rema ins v alid, though this effective particle bec o mes an excited random walk which distinguishes visited and unvisited sites. Finally we note that the SEP and its gener alizations o ccur in v arious biological pr oblems ranging from mo tio n of molecular motors [29, 30, 31, 32, 33] to protein s y n- thesis [34, 35, 36, 37]. So me mo dels of protein synthesis resemble complicated mo dels o f spider s. Another in trigu- ing co nnection is b etw een spider s a nd co o pe r ative car go transp ort by s everal molecular moto r s [3 8]. Ackno wledgment s W e are thankful to M. Olah, S. Rudchenk o, G. M. Sch¨ utz, D. Stefanovic, and M. Sto janovic for very useful conv ersations. W e also acknowledge financial suppo rt to the Pr ogram for Evolutionary Dynamics at Har v a rd University b y Jeffrey Epstein (T A), NIH grant R01GM07 8986 (T A), and NSF g rant CHE05 32969 (PLK). 12 APPENDIX A: MA STER EQUA TION A ND FLUCTUA TIONS In this Appe ndix, we explain the genera l forma lism, inspired by Ref. [1 9], that allo ws one to calculate veloc- ities and diffusio n contan ts, and we use this metho d to derive equa tio n (28). A spider ca n b e viewed as a homogeneous Markov pr o - cess with a finite num be r o f internal states. The dy nam- ics of the spider is enco de d in a Markov Matrix M , wher e the no n-diagonal ma trix element M ( C, C ′ ) r epresents the rate of e volution from a co nfiguration C ′ to a different configuratio n C . The qua ntit y − M ( C, C ) is the exit-rate from configur a tion C . The master eq uation for P t ( C ), the probability o f b eing in configuratio n C at time t , is then given b y d dt P t ( C ) = X C ′ M ( C, C ′ ) P t ( C ′ ) . (A1) W e now define Y t as the abso lute po sition of the spider ’s left leg, knowing that at time t = 0, Y t = 0. Betw een t a nd t + dt , Y t v a ries by the discr e te amo unt +1 , 0 o r − 1 that dep ends o n the config uration C ′ at t a nd on the configuratio n C at t + dt . The Markov Matrix M ca n then b e decomp osed in three parts corres po nding to the three p ossible evolutions of Y t : M ( C, C ′ ) = M 0 ( C, C ′ ) + M 1 ( C, C ′ ) + M − 1 ( C, C ′ ) . (A2) F o r example, M 1 ( C, C ′ ) represents the transition rate from a c o nfiguration C ′ to C with the left leg moving one s tep forward (this matrix element v anishes other- wise); M − 1 corres p onds to transitions fo r which the left leg mov es one step backw ards; M 0 enco des transitions in which the left leg stays still. W e ca ll P t ( C, Y ) the joint proba bilit y of being a t time t in the configura tion C and having Y t = Y . A master equatio n, analogous to equation (A1), can b e wr itten for P t ( C, Y ) a s follows : d dt P t ( C, Y ) = X C ′  M 0 ( C, C ′ ) P t ( C ′ , Y ) + M 1 ( C, C ′ ) P t ( C ′ , Y − 1 ) + M − 1 ( C, C ′ ) P t ( C ′ , Y + 1 )  . (A3) In terms o f the g enerating function F t ( C ) defined a s F t ( C ) = ∞ X Y = −∞ e λY P t ( C, Y ) , (A4) the master equation (A3) takes the simpler form : d dt F t ( C ) = X C ′ M ( λ ; C , C ′ ) F t ( C ′ ) , (A5) where M ( λ ; C, C ′ ), which governs the evolution o f F t ( C ), is given by M ( λ ) = M 0 + e λ M 1 + e − λ M − 1 . (A6) W e emphasize tha t M ( λ ), is not a Mar ko v matrix fo r λ 6 = 0 (the sum of the elemen ts in a given column do es not v anish). In the long time limit, t → ∞ , the b ehaviour o f F t ( C ) is dominated by the lar gest eige n v a lue µ ( λ ) of the matrix M ( λ ). W e thus hav e, when t → ∞ , h e λY t i = X C F t ( C ) ∼ e µ ( λ ) t . (A7) This result can b e r estated more pr ecisely a s follows : lim t →∞ 1 t log h e λY t i = µ ( λ ) . (A8) The function µ ( λ ) co nt ains the complete information ab out the cumulan ts of Y t in the long time limit. F or example, the velocity V and the diffusion co efficient D of the spider are g iven by V = lim t →∞ h Y t i t = d µ ( λ ) d λ    λ =0 = µ ′ (0) , (A9) D = lim t →∞ h Y 2 t i − h Y t i 2 2 t = µ ′′ (0) 2 . (A10) One there fo re needs to calculate the function µ ( λ ). F or simple problems such a s the bipedal spider with s = 2, µ ( λ ) can b e deter mined explicitely (b ecause M ( λ ) is a 2 b y 2 matrix). In general, the most efficient technique is to p erform a p erturba tive ca lculation of µ ( λ ) in the vicinity of λ = 0 (r ecall that µ ( λ ) v a nis hes at λ = 0). This p ertur ba tive a pproach is v ery similar to the one used in Qua n tum Mechanics, the ma jor difference being that M ( λ ) which plays the r ole of the Hamiltonian is not, in general, a symmetric matrix and its r ight eig env ectors are different from its left eigenvectors. By definition, we hav e M ( λ ) | µ ( λ ) i = µ ( λ ) | µ ( λ ) i , h µ ( λ ) | M ( λ ) = µ ( λ ) h µ ( λ ) | . (A11) Using e q uations (A6), (A9), and (A10), we can write the following perturba tive expa nsions in the vicinity of λ = 0, M ( λ ) = M + λ ( M 1 − M − 1 ) + λ 2 ( M 1 + M − 1 ) 2 . . . µ ( λ ) = V λ + D λ 2 + . . . | µ ( λ ) i = | 0 i + λ | 1 i + λ 2 | 2 i + . . . , h µ ( λ ) | = h 0 | + λ h 1 | + λ 2 h 2 | + . . . where M is the orig inal Mar ko v matr ix o f the sys tem, | 0 i is the stationa r y state and h 0 | = (1 , 1 , . . . , 1 ) is the left ground state of M . W e now substitute these p er turbative expansions in (A11) and identify the terms with the s a me power o f λ . Using the left eigenv ector h µ ( λ ) | , we obtain h 0 | M = 0 , (A12) h 1 | M = V h 0 | − h 0 | ( M 1 − M − 1 ) , (A13) h 2 | M = D h 0 | − 1 2 h 0 | ( M 1 + M − 1 ) + V h 1 | − h 1 | ( M 1 − M − 1 ) . (A14) 13 Multiplying these equations by the rig h t g round state | 0 i of M , and using the fac t that M | 0 i = 0 and h 0 | 0 i = 1, the following formulae for V and D are derived a s solv abilit y co nditio ns for Eqs . (A12)–(A14) : V = h 0 | M 1 − M − 1 | 0 i , (A15) D = h 1 | M 1 − M − 1 | 0 i + 1 2 h 0 | M 1 + M − 1 | 0 i − V h 1 | 0 i . (A16) W e obser ve that in order to calcula te V we only need to know the gro und state of M . How ev er, the calculation of D requires the knowledge of h 1 | , obtained by s o lving the linear equation (A13). W e rema rk that simila r ex - pressions can be obtained starting from the expansio n of right eigenvector | µ ( λ ) i . W e now sp ecialize this framework to the case of the heterogeneo us bip edal la me spider with s internal states. The Markov Matrix is then an s × s matrix M = M 0 + M 1 since M − 1 v a nishes identically . The matrix M 0 is given by M 0 =         − α 0 α − ( α + β ) β − ( α + β ) . . . . . . β − ( α + β ) 0 β − α         and the matr ix M 1 is M 1 = α         0 1 0 0 1 0 1 . . . . . . 0 1 0 0         The stationary state of M is | 0 i = ( p 0 , p 1 , . . . , p s − 1 ) with p k = α − β α s − β s α s − k − 1 β k for k = 0 , . . . , s − 1 . (A17) This expr ession, together with (A15), leads to the for- m ula (27) for the spider veloc it y . In or de r to derive the expr ession of the diffusion co- efficient, we need to solve equation (A13 ). One c a n verify that the solution of this equatio n is g iven by h 1 | = ( q 0 , q 1 , . . . , q s − 1 ) where q k = ( k + 1 ) V − α β − α + αβ − αV ( β − α ) 2  1 −  α β  k  (A18) for k = 0 , . . . , s − 1 . Inserting eq uations (A17) and (A18 ) int o the gener al expre ssion (A16) leads to the for- m ula (28). W e also used the ab ov e metho d to deter mine the velo c- it y and the diffusion co efficient for centipedes w ith s = 3 . The results (Sect. II I A 1) were o btained b y explicitly constructing the matrices M 0 , M 1 , and M − 1 , and p er- forming exact computations using Maple . These compu- tations a r e feasible when the num ber of le g s is sufficiently small. (The to ta l n umber of configuratio ns is 3 L − 1 for centipedes with s = 3, and hence the order of matrices M 0 , M 1 , M − 1 quickly grows with L .) APPENDIX B: GENERALIZED DET A ILED BALANCE RELA TION F o r the symmetric spider, the three ma trices M 0 , M 1 and M − 1 , introduced in (A2) to take in to ac count the total displacement of the spider , satisfy the following de- tailed balance r elation M y ( C, C ′ ) P eq ( C ′ ) = M − y ( C ′ , C ) P eq ( C ) (B1) where the equilibrium measur e is deno ted by P eq and y = 0 , ± 1. Equation (B1) implies that the velo city o f the spider v a nishes. 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