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Duty-ratio of cooperative molecular motors Nadiv Dharan1 and Oded Farago1,2 1Department of Biomedical Engineering, 2Ilse Katz Institute for Nanoscale Science and Technology, Ben Gurion University, Be’er Sheva 84105, Israel Molecularmotorsarefoundthroughoutthecellsofthehumanbody,andhavemanydifferentand important roles. These micro-machines movealong filament tracks, and havetheability to convert chemical energy into mechanical work that powers cellular motility. Different types of motors are characterized by different duty-ratios, which is the fraction of time that a motor is attached to its 2 filament. In the case of myosin II - a non-processive molecular machine with a low duty ratio - 1 cooperativitybetweenseveralmotorsisessentialtoinducemotionalongitsactinfilamenttrack. In 0 this work we use statistical mechanical tools to calculate the duty ratio of cooperative molecular 2 motors. The model suggests that the effective duty ratio of non-processive motors that work in cooperation is lower than the duty ratio of the individual motors. The origin of this effect is the n a elastic tension that develops in the filament which is relieved when motors detach from thetrack. J PACSnumbers: 2 2 I. INTRODUCTION motors. A recentmotility assayof myosin II motors and ] h actinfilamentswithalternatingpolaritieschallengedthis p prediction. It was found that the characteristic reversal Motor proteins are molecular machines that convert - times of the bidirectional motion in this motility assay o chemical energy into mechanical work by ATP hydrol- i ysis. They “walk” on the microtubule and actin cy- were macroscopically large, but practically independent b of the number of motors [10]. This observationhas been . toskeletonandpull vesiclesandorganellesacrossthe cell s explained by a model that accounts for the elasticity of [1]. Motor proteins can be classified into processive and c the actin filament [10, 11]. It has been shown that the i non-processivemotors. Theformerclassincludes motors s motors indirectly interact with each other via the ten- y like kinesins, which travela long distance alongtheir cy- sile stress thatthey generatein the elastic filament. The h toskeleton track (microtubules) without detachment [2]. elasticity-mediated crosstalk between the motors leads p Inthelatterclass,wefindmotorslikemyosinsthatmake [ only a single step along their tracks (actin filaments) to a substantial increase in their unbinding rates, mak- ingeachmotoreffectivelylessprocessiveandeliminating before disconnection [3]. While processive motors can 1 the exponential growth of τ with N [12]. v move cargoes by operating individually, non-processive rev The reduction in the duty ratio (the fraction of time 4 motorsneedtoworkincooperationtogeneratesubstan- that each motor spends attached to the filament track) 3 tial movement. Cooperative action of myosin motors is 5 implicatedinavarietyofcellularprocesses,includingthe of the cooperative motors can be explained as follows: 4 During bidirectional motion, the elastic filament is sub- contractionof the contractile ring during cytokinesis [4], . jectedtoatug-of-warbetweenmotorsthatexertopposite 1 adaptation of mechanically activated transduction chan- 0 nels inhair cells inthe inner ear[5],andmuscle contrac- forces, which leads to large stress fluctuations along the 2 tion [6]. Another important example of cooperative mo- elastic filament. These stress fluctuations are the origin 1 ofthe elasticity-mediatedcrosstalkeffect. Detailedanal- tordynamicsisfoundinmotility assays,wherefilaments : ysis shows that the typical elastic energy stored in the v glide over a surface densely covered by motor proteins i [7]. actin filament scales as [10, 11] X One of the more interesting outcomes of cooperative E r a actionofmolecularmotorsistheir abilitytoinduce bidi- =αNn (1) k T B rectional motion. Bidirectional motion is observed when a filament is subjected to the action of two groups of where N is the number of motors, n is the number of motors that engage in a “tug-of-war” contest and exert connected motors, and α is a dimensionless parameter ∗ forces in opposite directions [8]. The motor party that (which is closely related to the parameter β , to be de- exerts the larger force determines the instantaneous di- fined below in eq. 3). For actin-myosin II systems, α is rection of motion, which is reversed when the balance of rathersmall(forthe motilityassaydescribedinref. [10], forcesshiftsfromonegrouptotheother. Earliertheoret- α 2 10−3); but the energy releasedupon the detach- ∼ × icalmodelssuggestedthatthatthecharacteristicreversal ment of a single motor (n n 1): ∆E/k T = αN, B → − − timeofthebidirectionaldynamics(i.e.,thetypicaldura- can be quite large if N is large. This last result implies tionoftheunidirectionalintervalsofmotion),τ ,grows thatforlargeN,thetransitionsofmotorsbetweentheat- rev exponentially with the number ofmotors N [9]. Inthese tached and detached states are influenced by both ATP- earliermodels,themovingfilamentwastreatedasarigid driven (out-of-equilibrium) processes as well as by ther- rod which does not induce elastic coupling between the mal (equilibrium) excitations. The latter will be domi- 2 f f 0 k 0 Thus, h(x) gives the total force applied on the chain up s to the point x, with h(x = 0) = 0 and h(x = N) = n, where n is the number of monomers connected to mo- motors tors in a given configuration. The solid line in fig. 2 shows the function h(x) corresponding to the configura- FIG.1: Theactinelasticfilamentisrepresentedasachainof tion of 5 nodes depicted in fig. 1. To calculate the elas- nodes,connectedbyidenticalspringswithspringconstantks. tic energy of a configuration,we introduce the function EachnodeiseitherconnectedtoasinglemyosinIImotor-in g(x) = h(x) (n/N)x, which is depicted by the dashed whichitexperiencesaforceofmagnitudef0,ordisconnected lineinfig.2a−ndgivesthe totalexcess forceaccumulated - in which it experiences no force. up to x. The elastic energy can then be expressed as: nated by the changes in the elastic energy of the actin (seeeq.1)andnotbytheenergyoftheindividualmotors. Ejel =β∗N−1g2(x ) β∗ Ng2(x)dx, (2) For the actin filament with alternating polarities, the k T i ≃ b i=1 Z0 elasticity-mediated crosstalk is a cooperative effect that X reduces the degree of cooperativity between motors (de- where creases τ ) by decreasing their duty ratio (i.e., their rev f2 attachment probability). But as discussed above, much β∗ = 0 (3) 2k k T of the strength of this effect is related to the large stress s B fluctuations that develop in the elastic filament due to is the ratio betweenthe typicalelastic energyofa spring the opposite forces applied by the antagonistic motors. f2/2k and the thermal energy k T. 0 s B In view of this fact, it is fair to question the significance To determine the mean number of connected motors, of this effect in the more common situation where polar one needs to calculate the partition function filaments move directionally under the action of a single family of motor proteins. In this work we analyze this N problem and show that the elasticity-mediated crosstalk Z = pn(1 p)N−nzn, (4) − effect in this system is indeed much smaller, but not en- n=0 X tirely negligible. Our analysis of this effect is presented where p is the attachment probability, i.e. duty ratio, in the following section II. The magnitude of this effect of a single motor, and z is the partition function of n in acto-myosinsystems is estimated in section III. all the configurations with exactly n connected motors. The function z can be calculated by tracing over all n the functions g(x) corresponding to configurations with II. STATISTICAL-MECHANICAL MODEL n connected motors. Mathematically, the condition that exactlynmotorsareconnectedcanbeexpressedthrough In what follows we model the elastic actin filament as the following constraint on the function h(x). a chain of N equally spaced nodes connected by N 1 − identical springs with spring constant ks (see fig. 1). In N dh α the chain reference frame, the i-th node is located at lim dx=n. (5) α→∞ dx x =(i 1)∆l, where i=1,...,N and ∆l is the spacing Z0 (cid:12) (cid:12) i − (cid:12) (cid:12) betweenthenodes. Forbrevityweset∆l=1. Thechain (cid:12) (cid:12) (cid:12) (cid:12) liesona“bed”ofmotors,whereeachnodemaybeeither freeandexperiencenopullingforce(f =0),orattached i to one motor in which case it is subjected to a force of 2 h(x) magnitude f = f . Different configurations of the sys- i 0 g(x) tem are defined according to which nodes are connected 1.5 to motors and which are not. For a given configuration j, the elastic energy of the chain is given by the sum of 1 energies of the springs: Eel = N−1F2/2k , where F j i=1 i s i is the force applied on the i-th spring. The forces F are 0.5 i P calculated as follows: We first calculate the mean force f¯ = N f /N, and define the excess forces acting 0 i=1 i on th(cid:16)e nodes: (cid:17)f∗ = f f¯. The force on the i-th spring P i i− -0.5 is then obtained by summing the excess forces applied on all the monomers located on one side of the spring: -1 Fi =− il=1fl∗ = Nl=i+1fl∗. 0 1 2 x 3 4 5 Itismoreconvenienttoanalyzetheproblemusingcon- P P FIG.2: Thefunctionsh(x)andg(x)whichcorrespondtothe tinuous functions. Let us introduce the function h(x) configuration shown in fig. 1. which, for x < x < x , has slope +1 if the monomer i i+1 at x is connected to a motor, and a slope 0, otherwise. i 3 To allow an analytical solution, we approximate this constraint by setting α = 2, in which case eq. 5 can be expressed in terms of g(x)=h(x) (n/N)x as − N dg 2 n n dx=N 1 (6) dx N − N Z0 (cid:18) (cid:19) (cid:16) (cid:17) With eq. 6, the partition function z is given by n N N dg 2 n n z =B(n,N) [g(x)]exp β∗ g2(x)dx δ dx N 1 (7) n Z D − Z0 ! "Z0 (cid:18)dx(cid:19) − N (cid:16) − N(cid:17)# where δ is Dirac’s delta-function. The function B(n,N) is introduced in eq. 7 in order to compensate for the error introduced by the approximated constraint eq. 6. We will determine this function through the requirement that for ∗ β =0, i.e. in the absence of elastic crosstalk between the motors, N N! zn β∗=0 = = , (8) | n n!(N n)! (cid:18) (cid:19) − which is simply the number of ways to choose n out of N monomers. In order to calculate the partition function z , we use the Fourier space representation of δ(x), n 1 i∞ δ(x a)= ew(x−a)dw, (9) − 2πiZ−i∞ and the Fourier series of g(x), N/2−1 g(x)= gkei2Nπkx. (10) k=X−N/2 Substituting eqs. 9 and 10 into eq. 7 yields: 1 i∞ n n 8π2 z =B(n,N) dw [g ]exp wN 1 exp g2 k2w+2Nβ∗ . (11) n 2πiZ−i∞ Z D k h N (cid:16) − N(cid:17)i× "−Xk k(cid:18) N (cid:19)# Tracing over g can be readily performed, giving k 1 i∞ n n N/2 π z =B(n,N) dwexp wN 1 . (12) n 2πiZ−i∞ h N (cid:16) − N(cid:17)ikY=02Nβ∗+8π2k2w/N The integral over w can be evaluated using the method of steepest descent, which yields:  z B(n,N)eG(w0), (13) n ≃ where, n N/2 1 8π2 G(w) = n 1 w ln k2w+2Nβ∗ (14) − N − π N (cid:16) (cid:17) Xk=0 (cid:20) (cid:18) (cid:19)(cid:21) n n 1 2e−2N β∗ π2w N w 1 ln π2w+β∗ tan−1 , ≃ ( N (cid:16) − N(cid:17)− 2 (cid:20) π (cid:0) (cid:1)(cid:21)−rπ2w s β∗ !) and w satisfying 0 dG n n 1 1 β∗ π2w = 1 + tan−1 0 =0. (15) dw(cid:12) N − N − 2w0 2sπ2w03 s β∗ ! (cid:12)w0 (cid:16) (cid:17) (cid:12) (cid:12) (cid:12) 4 ∗ For β 1, one gets ≪ N N N N w β∗. (16) 0 ≃ 2n N n −s8n N n (cid:18) − (cid:19) (cid:18) − (cid:19) From eqs. 8, 13, 14, and 16, one finds that N N π (N/2) N3 N/2 B(n,N)= e−G(w0) = . (17) n n e3 n(N n) (cid:18) (cid:19) (cid:18) (cid:19)(cid:16) (cid:17) (cid:18) − (cid:19) Inserting eq. 16 into eq. 14, and expanding G(w ) in powers of √β , yields 0 ∗ n(N n) G(w0) G(w0)β∗=0 − β∗. (18) ≃ | − 2 r ∗ Finally, for β 1, the partition function z is obtained by substituting eqs. 17 and 18 into eq. 13, which gives: n ≪ N n(N n) z exp − β∗ . (19) n ≃(cid:18)n(cid:19) −r 2 ! In order to calculate the partition function Z, one needs to substitute eq. 19 into into eq. 4, which gives: N N n(N n) Z = pn(1 p)N−nexp − β∗ . (20) n=0(cid:18)n(cid:19) − −r 2 ! X In the thermodynamic limit (N 1), the sum in eq. 20 is dominated by one term which corresponds to the mean ≫ number of attached motors n . This term is given by h i p(1 p) n =N p (1 2p) − β∗ . (21) h i " − − r 8 # From eq. 21 we identify the effective attachment probability as n p(1 p) p h i =p (1 2p) − β∗ (22) eff ≡ N − − 8 r Notice that the second term on the right hand side of the state (connected/disconnected) of a randomly cho- eq. 22 is anti-symmetric around p = 1/2, and that for sen node changes, and the other in which two randomly p < 1/2 (p > 1/2), the effective attachment probability chosen nodes with opposite states change their states si- p issmaller(larger)thanp. Thisobservationisdirectly multaneously. For each move attempt, the model en- eff relatedtothefacttheelasticity-mediatedcrosstalkeffect ergy of the chain is recalculated, and the move is ac- isdrivenbythetendencytoreducetheforcefluctuations cepted/rejected according to the conventional Metropo- along the elastic filament. For p < 1/2 (p > 1/2) the lis criterion. Our MC results are summarized in fig. 3. ∗ forcefluctuationsarereducedbythedetachment(attach- For β < 0.2, we find an excellent agreement between ment) of motors, which brings the system closer to the our computational results and eq. 22 (which has been ∗ limiting case p = 0 (p = 1) where the force fluctuations derived for β 1). Notice that eq. 22 does not include vanish. any fitting par≪ameters. For larger values of β∗, eq. 22 overestimates the decrease in p . Totestthevalidityandrangeofapplicabilityofeq.22, eff we conducted Monte Carlo (MC) simulations of elastic chains of N = 1000 monomers with p = 0.05, which is the typical duty ratio of myosin II motors [13]. Sys- III. ACTIN-MYOSIN II SYSTEMS ∗ tems corresponding to different values of β were sim- ∗ ulated using the parallel tempering method. The sim- Eq. 3 relates the dimensionless parameter β to three ulations include two types of elementary moves (which physical parameters of the system - the temperature T, are attempted with equal probability) - one in which the typicalforce exertedby the motorsf ,and the effec- 0 5 0.05 to the elasticity cross-talk effect discussed here. It also MC Simulations Eq. 22 stems from the fact that p also depends on the distance and orientation of the motor head (both in space and 0.04 time) with respect to the associated binding site. These may vary between the motors which, hence, should have different attachment probabilities [23]. peff0.03 Using the above mentioned values of system parameters, one finds that for acto-myosinsystems the parameter β∗ lies within the rangeof5 10−4 <β∗ <5 10−3. 0.02 Setting the duty ratio of myosin×II to p∼= 0.∼05 (×as used ∗ in the MC simulations)we find that for this rangeof β , the effective duty ratio is slightly lower than p and lies 0.01 0 0.05 0.1 0.15 0.2 0.25 0.3 within the range of peff =0.97p (for β∗ =5 10−4) and β∗ p =0.90p (for β∗ =5 10−3). Using a low×er estimate eff × fortheduty ratiop=0.02[22],wefindthatthe effective FofIGth.e3:dimTheneseioffnelcetsisvepaartatamcehtmerenβt∗.proTbhaebicliitrycleassdaenfuontecttiohne drauntgyeroaftiβo∗d.rops to 0.83p <∼ peff <∼ 0.95p for the same resultsoftheMCsimulations,whilethesolid linedepictsthe analytical approximation for β∗ ≪1, eq:22. Asstatedearlier,thecooperativeactionofmyosinmo- tors “compensates” for the non-processive character of theindividualmotors. Theforcegeneratedbyagroupof tive spring constant of the actin filaments segment be- N motors is F = Npefff0. Which force f0 maximizes h i tween between two binding sites of the motors, ks. The the effective force per motor feff = hFi/N and, hence, latter parameter can be further expressed as the force production of the collectively working motors? From eq. 3 and 22 we find that the maximum value of YA f is achieved when the force of the individual motors k = , (23) eff s l is where Y is Young’s modulus of the actin, A is the cross- 2p k k T sectional area of an actin filament and l is the distance fmax = s B (24) between binding sites. For myosin II motors, forces in 0 1 2psp(1 p) − − the range of f = 5 10 pN have been measured ex- 0 − Setting the values of the system parameters as above perimentally [13, 14]. The actin-cross sectional area (in- (Y = 2.8 GPa, A = 23 nm2, l = 38 nm) and taking cludingthetropomyosinwrappedaroundtheactinhelix) is A = 23 nm2, and the Young’s modulus of the actin- p=0.02as the duty ratioofa single motor,we find that fmax 25 pN, for which f 0.25 pN. For forces in tropomyosinfilament is Y =2.8GPa [15,16]. The value 0 ≃ eff ≃ the range of f 5 10 pN, which are typically mea- of l is somewhat more difficult to assess. One possibility 0 ≃ − sured for myosin II motors [13], the effective mean force is that l 5.5 nm, which is simply the size of the G- ≃ per motor is f 0.1 0.15 pN, which is about half of actin monomers, each of which includes one binding site eff ≃ − theoptimaleffectiveforcef (fmax). We,thus,conclude for myosinmotors [17]. Another possible value is related eff 0 thatmysoinIImotorsworkquiteclosetoconditionsthat to the double helical structure of F-actin and the fact that it completes half a twist about every 7 monomers, maximize their cooperative force generation. i.e. every 38 nm [18]. Since the binding sites follow ≃ a twisted spatial path along the double helix, many of them remain spatially unavailable to the motors. In the IV. DISCUSSION motility assay,the motorsare locatedunderneath the F- actin, and the distance between the binding sites along There has recently been a considerable interest in the thelineofcontactwiththebedofmotorsisl =38nm. A collective behavior of molecular motors, especially in re- thirdchoiceforl,whichmaybemorerelevanttoskeletal lation to cooperative dynamics of cytoskeletal filaments muscles,isl=14nm. Thisvalueisderivedfromthefact inmotilityassays. Moststudieshavefocusedonthebidi- that within the sarcomere (the basic contractile unit of rectional motion arising when the filament is driven by the muscle) an averagenumber of three thick myosinfil- two groups of antagonistic motors, or in the case when amentssurroundonethinactinfilament. Theseparation one motor party works against an external force. The betweencollinearmotorheadsalongthethickfilamentis present work is motivated by our recent studies of bidi- 43nm[19–21],andbecausethe actinissurroundedby rectional motion in acto-myosin motility assays which ∼ three thick filaments, the distance between the motors demonstrated that the duty ratio of the motors (and, along the actin is l =43/3 14 nm. hence, the level of cooperativity between them) is re- ≃ For the duty ratio p, the range of experimental val- ducedbythe elasticityofthe actinbackbone. Inthis pa- ues is scattered and varies from p = 0.01 0.02 [22] to per we extended our studies to motility assays in which − p = 0.05 [13]. The uncertainty may be partially related similar motors act on a polar actin filament without a 6 counter external force. To single out the filament elas- motorforcesf =fmotor fMF,wecanrewriteeq.A1as i i − i ticitycrosstalkfromotherpossiblecollectiveeffects(e.g., thoseassociatedwithnon-equilibriumATP-assistedpro- fi−fiviscous+Fi−Fi−1 =0. (A2) cesses and with the elasticity of the motors themselves For the chain’s center of mass (CoM), the equation of [24–26]), we neglectmotor-to-motorvariationsanduse a motion reads modelinwhichthemotorsarecharacterizedbytwomean quantities: their attachment probability to a rigid (non- N N elastic) filament p, and the mean applied pulling force FCoM = f fviscous =0; (A3) f . We expect this mean field description of the motors i− i 0 i=1 i=1 to hold when the number of motors N becomes large. X X We calculate the attachment probability to the elastic but the viscous drag force is distributed uniformly along filament p < p from the partition function associated the chain (i.e., it is equal for all masses) and, therefore, eff with the filament elastic energy. The elastic energy of eq. A3 gives filament can be treated as an equilibrium degree of free- dom(of a system whichis inherently out-of-equilibrium) fviscous = Ni=1fi =f¯. (A4) because the mechanical response of the filament to the i N P attachment/detachment of motors occurs on time scales Using this last result, eq. A2 reads whicharefarshorterthanthetypicalattachmenttimeof the motors (see discussion in ref. [12]. The assumption ∗ thattheactinfilamentisinmechanicalequilibriumisalso Fi−Fi−1 =−fi, (A5) made in theoretical studies of intracellular cargo trans- ∗ where f is the excess force. Since F = 0, we find that port [8] and muscle constraction [27]) Our calculation i ∗ ∗ 0 ∗ ∗ F = f . Then, F = F f = f f ; and, in shows that p is only slightly smaller than p. This re- 1 − 1 2 1 − 2 − 1 − 2 eff general, sultisverydifferentfromourpreviousfindingsformotil- ity assays of bidirectional motion, where the elasticity- N mediated crosstalk effect is substantial. Finally, we note F = f∗. (A6) thatalthoughinboththisandpreviousstudieswefound i − i i=1 X thatp <p[11,28], the opposite relationcannotbe ex- eff cluded under certain conditions (e.g., when the filament experiences external forces as in single molecule experi- m k i−1 i i+1 s ments or muscle contraction). We intend to address this opposite scenario in a future publication, in which we investigate the variations in both p (the ATP-hydrolysis motor forces f = f related attachment probability) and p (which also in- i 0 eff cludesthecontributionduetotheelasticcrosstalkeffect) with the muscle contraction velocity. viscous forces WethankAnne BernheimandSefiGivliforusefuldis- cussions. fviscous =−f¯ i spring forces Appendix A: The elastic energy Fi−1 Fi−1 Fi Fi FIG.4: Theactinismodeledasachainofmassesandsprings We model the actin filament as a linear chain of iden- thatmovesinahighlyviscousmedium. Threetypesofforces tical particles of mass m connected by elastic (massless) arepresentinthesystem: themotorforces,auniformviscous springs with spring constant k (see fig. 4). Three types s drag,andthespringforces. 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