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Filament depolymerization by motor molecules Gernot A. Klein,1 Karsten Kruse,1 Gianaurelio Cuniberti,1,2 and Frank Ju¨licher1 1Max Planck Institute for Physics of Complex Systems, D-01187 Dresden, Germany 2Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany (Dated: February 9, 2008) Motorproteinsthatspecificallyinteractwiththeendsofcytoskeletalfilamentscaninducefilament depolymerization. Aphenomenologicaldescriptionofthisprocessispresented. Weshowthatunder certain conditions motors dynamically accumulate at the filament ends. We compare simulations of two microscopic models to the phenomenological description. The depolymerization rate can 5 exhibit maxima and dynamic instabilities as a function of the bulk motor density for processive 0 depolymerization. We discuss our results in relation to experimental studies of Kin-13 familiy 0 motor proteins. 2 PACSnumbers: 87.16.Nn,87.16.-b,02.50.Ey n a J Manyactiveprocessesincellsaredrivenbyhighlyspe- from the depolymerizing filament end is denoted ρ(x). 4 cialized motor proteins which interact with filaments of Here,weuseareferenceframeinwhichthedepolymeriz- 2 the cytoskeleton. Important examples are cell locomo- ing endis locatedatx=0 foralltimes. Motorsoccurin ] tion, cell division and the transport of organelles inside the bulk solution at concentration c. They bind to and C the cell [1]. Cytoskeletal filaments are linear aggregates detach from filaments with rates ω c/ρ and ω , re- a max d S of proteins, for example actin and tubulin. Actin fil- spectively, where ρ is the maximal density of motors max o. aments and microtubules are dynamic and can rapidly for which binding sites on the filament saturate. Bound i changetheirlengthsbyadditionandremovalofsubunits motors diffuse along the filament with a diffusion coeffi- b at the ends [1, 2, 3]. Filaments show a structural asym- cient D and may also exhibit a directed average motion - q metry, which provides a direction for motion and force withvelocityv . Note,thatv ingeneraldependsonthe 0 0 [ generation of bound molecular motors. These proteins density of motors ρ [10, 11, 12, 13]. The density profile 1 areableto transducethe chemicalenergyofafuelwhich along the filament then obeys v is ATP, to mechanicalworkwhile interacting with a fila- 2 ment [4, 5]. ρ ∂ ρ+∂ j = ω c 1− −ω ρ . (1) 3 t x a d (cid:18) ρ (cid:19) 0 In addition to generating forces along filaments, mo- max 1 torscanalsointeractwithfilamentends wherethey may Thecurrentofmotorsisgivenbyj =−D∂ ρ−vρ. Here, 0 influence the polymerization rate and thus the filament x v = v +v is the total velocity of motors with respect 5 length. Examples are provided by the members of the 0 d 0 to the filament end, with vd ≥ 0 denoting the depoly- Kin-13 subfamily of kinesin motor proteins [6, 7, 8]. A / merizationvelocity. Itis relatedto the rate Ω of subunit o particular example is the mitotic centromere-associated removal from the end by v =Ωa/N, where a is the size bi kinesin (MCAK) which regulates the length of micro- of a subunit and N the nudmber of protofilaments in the - tubules during celldivision[9]. Inthe courseofcell divi- q filament. sion, the chromosome pairs are separated by the mitotic : We assume that the rate of filament depolymerization v spindle. In this process, shortening microtubules gener- isregulatedbymotorsboundtothefilamentend. There- i X ate forces pulling the chromosomes towards the oppos- fore, the rate of subunit removal is a function Ω(ρ ) of ing poles of the cell. MCAK is localized at the micro- 0 r the motor density ρ = ρ(x = 0) at the end. It is useful a tubule ends which interact with chromosomes [9] and it 0 to systematically expand Ω in powers of ρ has been shown that it induces depolymerization of mi- 0 crotubules [7]. In vitro assays and single molecule stud- Ω(ρ ) = Ω +Ω ρ +Ω ρ2+O(ρ3) . (2) 0 0 1 0 2 0 0 ies have shown that MCAK accumulates at both ends of stabilized microtubules and induces depolymerization Here, we have introduced the expansion coefficients Ω . i ataratewhichdependsonthebulkmotorconcentration The subunit removal rate in the absence of motors Ω 0 whileatthesametimeMCAKmoleculesdonotgenerate in general depends on buffer conditions. In situations directed motion along microtubules [7]. wherefilamentsarestabilized,Ω =0. Formotorswhich 0 In this paper, we discuss the dynamics of motor inducefilamentdepolymerization,Ω >0. Sincetherate 1 moleculeswhichinducetheshorteningoftheendsoffila- Ω saturates for large densities, typically Ω <0. 2 mentstowhichtheybindusingbothaphenomenological The descriptionis completedby specifying the bound- descriptionandmoremicroscopicmodels. Forsimplicity, ary conditions at x = 0 and for x → ∞. At x = 0 we consider one filament end and use a semi-infinite ge- the current j(x = 0) at the filament end equals the net ometry. Thedensityofboundmotorsatadistancex≥0 rateJ atwhichmotorsattachto the filamentend. Since 2 motorsattachedtotheendinducedepolymerization,the rate J(ρ ) is a function of the motor density ρ at the 0 0 end and also depends on buffer conditions. Again, we express J by an expansion in powers of ρ : 0 J(ρ )=J +J ρ +J ρ2+O(ρ3) , (3) 0 0 1 0 2 0 0 Here, J is the rate of direct motor attachments to the 0 end. ThecoefficientsJ andJ characterizehowinterac- 1 2 tions between motors and the filament end influence the detachment rate of motors. If v > 0, motors typically detach from the end and thus J <0. FIG. 1: Regimes of accumulation and depletion of motors Finally, for large x we require that the density ρ at the shrinking filament end as a function of the coefficient approaches the equilibrium value of the attachment- Ω∗2 = Ω2ρmax/(Dωd)1/2 and the bulk motor concentration detachment dynamics c∗ = ωac/ωdρmax for J0 = Ω0 = J2 = 0 and J1/(Dωd)1/2 = ω cρ −0.1,Ω1/(Dωd)1/2 = 0.5. Accumulation occurs above the a max ρ∞ = (4) solid line, depletion occurs below. The grey area indicates ω c+ω ρ a d max theregionofphysicalinterest. Abovethisarea,peff(ρmax)>0 If a motor bound to the end removes a filament sub- whichisforbiddenbystericexclusionofparticles. Belowthis area, filaments polymerize at high motor concentration. unit, it may fall off the filament with this subunit or it may stay bound to the filament. The tendency of a motor to stay attached while removing subunits can be described by its processivity. In our phenomenological Our phenomenologicaldescription revealsthat motors description, we define the effective processivity can dynamically accumulate at the filament end even if their binding affinity to the end is not largerthan in the J(ρ )−J(0) 0 peff =1−(cid:12)Ω(ρ )−Ω(0)(cid:12) . (5) bulk, Ωa = 0. In this process, motors which bind along (cid:12) 0 (cid:12) the filament are subsequently captured by the retracting (cid:12) (cid:12) It differs from the proce(cid:12)ssivity of an is(cid:12)olated motor due filament end. Dynamical accumulation of motors is a to collective effects resulting from interactions between collective phenomenon and requires a sufficiently large motors at the end and thus depends on ρ . If p ≤ 0, effective processivity. 0 eff more motors detach from the end than subunits and the Inordertoobtainaphysicalpictureofthemicroscopic motor-induceddepolymerizationisnon-processive. How- events which influence the effective processivity as a re- ever,if0<p ≤1,agivenmotorcanremovemorethan sult of crowding, we extend discrete stochastic models eff one subunit[17]. for motor displacements along filaments [10, 11, 12, 13] Forsimplicity,weignorethe densitydependence ofv . to capture subunit removal at the ends. Motors are rep- 0 Inthiscase,Eq.(1)approachesforlargetimest,asteady resentedby particles which occupy discrete binding sites state given by ρ =ρ∞+(ρ0−ρ∞)exp[−x/λ] The char- indexed by i = 1,2,3,... arranged linearly on a fila- acteristic length is mentwhichconsistsofasingleproto-filament(N=1),see Fig. 2. Here, i = 1 denotes the binding site at the fila- 2D λ= . (6) mentend. Eachsite iseitherempty(ni =0)oroccupied v+[v2+4D(ωac/ρmax+ωd)]1/2 (ni = 1). Particles move stochastically to neighboring empty sites with rate ω¯ in both directions. Here, we The steady statevalue ofρ is obtainedby insertingthis h 0 assume again that v can be neglected as compared to expression in Eq. (3). In the following, we consider the 0 v . In addition, particles attach to and detach from the case where v ≫ |v | and the spontaneous velocity v d d 0 0 lattice with rates ω¯ c and ω¯ , respectively. The rates of can be neglected. Indeed, experimental observations of a d particleattachmentanddetachmentattheendsitei=1 MCAK show that vd ≫ |v0| [7]. Depending on param- differ from the bulk rates and are denoted Ω¯ c and Ω¯ . eters, motors either accumulate or deplete at the fila- a d We firstconsider the situation where only one particle mentend,seeFig.1. Accumulationattheendoccursfor ispresent. Ifthisparticleisnotboundattheend,theend Ω >Ω(a), where 2 2 is stable. If the particle is bound at i = 1, this subunit Ω +J ρ Ω +J ρ J ρ is removed from the filament with rate Ω¯. This process Ω(2a) = − 1 ρ∞2 max − 0 ρ2∞1 max − 0ρ3∞max.(7) can occur in two different ways: (i) with probability p¯ the particle stays bound to the new filament end after Motor accumulation can exhibit a reentrant behavior as the first subunit is removed; (ii) with probability 1−p¯ a function of increasing bulk motor concentration, see theparticledetachesfromthe filamenttogetherwiththe Fig. 1. removed subunit. If p¯is close to one, a single particle is 3 w c w pW w h a d (1-p)W FIG.2: Discretemodelofmotor-inducedfilamentdepolymer- ization. Motors attach to emptysites at rate ω¯acand detach with rate ω¯d. The hopping rate to free neighbouring sites is ω¯h. Occupied sites are removed from the end with rate Ω¯. With probability p¯, the particle remains attached to the end when a subunitis removed. FIG.3: Thevelocityvd∗ =vd/a(ω¯hω¯d)1/2ofdepolymerization processive and can repeatedly remove subunits from the asafunctionofthebulkmotorconcentrationc∗ =ω¯ac/ω¯dob- end without falling off. tainedinsimulationsofmodelAfordifferentvaluesofthepro- Processive removal of a subunit requires simultaneous cessivityp¯=0,0.9and1(symbols). Forcomparison thecor- interaction of a motor with the end and the adjacent responding solutions of the phenomenological equations are displayed(lines). Inset: Theaccumulationofmotors,charac- subunit. Therefore, this process is affected by the pres- ence of other particles bound to the filament near the tfaerrizfreodmbtyhetheendraitsioshρo0w/nρ∞asoaffmunoctotirondeonfscit∗yfoarttthheeseanmdeasnitd- end. In particular, if site i = 2 is occupied while sub- uations. Parameter values are ω¯a = Ω¯a,ω¯d = Ω¯d = 0.008ω¯h unit i = 1 is removed, the new end site is already oc- and Ω¯ =4ω¯h. cupied and the particle at i = 1 cannot stay attached. We distinguish two cases with different behaviors in this situation. Model A describes the case where subunit removal by a motor requires an empty adjacent binding for the values of the coefficients Ω and J introduced i i site, while model B corresponds to the case where the above. For both model A and B we find, using this rate ofsubunit removalis independent ofthe occupation approximation, D = a2ω , Ω = aΩ¯, J = Ω¯ c and h 1 0 a oftheadjacentsite. InmodelA,theprobabilityperunit J = −a(Ω¯ c + Ω¯ + (1 − p¯)Ω¯). The nonlinear coef- 1 a d time to remove the end if n1 = 1 is Ω¯A = Ω¯(1−p¯n2). ficients Ω2 and J2 are model dependent. In model A, Here we assume that the processivity characterizedby p¯ Ω = −a2p¯Ω¯ and J = 0 whereas for model B, Ω = 0 2 2 2 is unaffected by the occupation of the neighboring site. and J =−a2p¯Ω¯. In both models all higher order coeffi- 2 Crowding at the end obstructs cutting and reduces the cients Ω and J vanish. n n rate of subunit removal. In model B, an occupied adja- Fig. 3 displays the depolymerization velocity v ob- d centsite will reduce the processivityofa motorbut does tained in the mean field theory corresponding to model not affect the depolymerization rate, i.e., Ω¯B =Ω¯. A as a function of the bulk monomer concentrationc for WecanrepresentthedynamicsofthesystembyaMas- differentvaluesofp¯. For largec the velocitysaturatesat ter equation for the probabilityP{ni,t} to find a config- vd(ρmax), while it increases linearly for small c. For suf- uration of lattice occupation (n1,n2,..) at time t. This ficientlylargep¯,the velocityvd exhibitsamaximumasa leads to expressions for the rate of change of averageoc- function of c. Increasingp¯further, a dynamic instability cupation numbers valid for i≥2: appears where two stable states with different v coexist d withinarangeofcvalues. Athirdunstablestate isindi- dhn i i = ω¯h(hni+1i−2hnii+hni−1i)+ω¯ach1−nii cated by a brokenline. Results of stochastic simulations dt of model A are shown for comparison. Mean field the- − ω¯ hn i+hΩ¯A,Bn (n −n )i. (8) d i 1 i+1 i oryandstochasticsimulationagreequantitativelyexcept in the vicinity of the dynamic instability. Fluctuations At the filament end, i=1, concealthedynamicinstabilitypresentinmeanfieldthe- dhn i ory. The inset to Fig. 3 shows the relative accumulation dt1 = ω¯hhn2−n1i+Ω¯ach1−n1i−Ω¯dhn1i ρ0/ρ∞ of motors. For sufficiently large p¯, motors accu- − (1−p¯)Ω¯hn (1−n )i. (9) mulate as c is increased. We note that in model B, no 1 2 dynamic accumulation of motors occurs and ρ0 <ρ∞. Using a mean-field approximation, replacing two-point In summary, we have shown that a positive effective correlators hn n i by hn ihn i, we obtain from processivityp ofsubunitremovalisessentialtoachieve i i+1 i i+1 eff Eqs.(8)and(9)differentialequationsandboundarycon- dynamic accumulation of motors at the filament end. ditions identical to Eqs. (1)-(3) with ρ(x = a(i−1)) = This effective processivity is a collective effect and re- hn i/a. This procedure leads to explicit expressions sultsfromstericexclusionofmotorsboundneartheend. i 4 ThephenomenologicaldescriptiongivenbyEqs.(1)-(3)is that a collection of MCAK motors processively depoly- generalandvalidirrespectiveofdetailsofthemechanism merizemicrotubules[7]consistentwithmodelA.Present ofmotorinducedsubunitremovalandofthestructureof datacannotruleoutamechanismakintomodelBwhere thedepolymerizingfilamentend. Wehaverestrictedour- motor accumulation is still possible if the affinity of mo- selves to effects corresponding to the lowest order terms tors to the filament end is high. In future experiments, in the expansions of Eqs. (2) and (3). While higher model B could be ruled out if a a maximum of the de- order terms could lead to additional effects, our simula- polymerizationvelocityatintermediatemotorconcentra- tionsofmicroscopicmodelsindicatethatthesetermsare tion would be observed as suggested by our theory. The unimportant(see Fig. 3). We havefocussedonexcluded members XKCM1 and XKIF2 of the Kin-13 family, can volume effects at the filament end and have considered depolymerizemicrotubuleswithorwithoutaccumulation the case v = 0 where some of these effects in the bulk ofmotors at the end, depending on the conditions under 0 disappear. For v 6= 0, the interplay between bulk and which microtubules have been stabilized [6]. Further- 0 end excluded volume effects could lead to new phenom- more, it has been suggested that processivity is reduced ena which will be subject of future work. under conditions where motors do not accumulate [6]. The stochastic models A and B provide a physical Thus, stabilization of microtubules could influence the picture of the cooperativity and processivity of motors microscopicmechanismsofcollectivesubunitremovalby bound at the end of a single protofilament. Interac- a change in the microtubule lattice structure, leading to tions between motors lead to different rates of subunit reduced processivity p¯or a mechanism similar to model removalinthe twomodels. Inourstochasticsimulations B. forN =1,thedynamicinstabilityofsteadystateswhich The theory developed here is not restricted to motors is found in the mean field analysis is concealed by fluc- oftheKin-13familywhichinteractwithmicrotubulesbut tuations. We expect that for larger numbers of protofil- applies in general to associated proteins which regulate aments this effect of fluctuations is reduced. Therefore the dynamics of filament ends. Actin depolymerization a signature of a dynamic instability could reappear for by ADF/cofilin as well as the polymerizationof actin by microtubule depolymerization leading to bistability and formin are further examples of such processes [15, 16]. switchlikechangesofdepolymerizationvelocities. Inthe In additionto the conventionalactionof motor proteins, mitoticspindlethisinstabilitycouldberelevantforchro- filamentpolymerizationanddepolymerizationbyproces- mosome oscillations, which have been observed [14]. sivelyacting end-binding proteins areexpected to play a Accumulation of motors at the filament end described key role in cytoskeletal dynamics and self-organization. by Eq. (9) can occur as a result of three different mech- We thank Stefan Diez andJoe Howardforfruitful dis- anisms. Motors can accumulate by directly binding to cussions and stimulating collaboration. G.C. acknowl- the filament end if they have a higher affinity to the edges support by the Volkswagen Stiftung. end than to subunits along the filament. This effect dominates if the total velocity v = v + v is small, 0 d v2 ≪ 4D(ω c/ρ + ω ). In this case, the localiza- a max d tionlengthisgivenbythediffusionlengthduringtheat- tachment time λ ≃D1/2(ω c/ρ +ω )−1/2. A second [1] B. Alberts, et al., Molecular Biology of the Cell 4th ed. a max d (Garland, New York,2002). mechanism of accumulation is given by transport of mo- [2] M. Dogterom and S. Leibler, Phys. Rev. Lett. 70, 1347 tors to the end with velocity v ≫v . In the third case, 0 d (1993). motors that bind along the filament are captured by the [3] C.Peskin,G.Oster,Biophys.J.65,316(1993);Biophys. shortening end. This dynamic accumulation mechanism J 69, 2268 (1995) . dominates for v0 ≪ vd and vd2 ≫ 4D(ωac/ρmax +ωd). [4] J.Howard,MechanicsofMotorProteins&theCytoskele- The localization length is λ ≃ D/v . The first and last ton 4th ed. (Sinauer Associates, Sunderland,2001). d casescanleadtoaccumulationatbothendsofafilament [5] F. Ju¨licher. A. Ajdari, J. Prost, Rev. Mod. Phys. 69, 1269 (1997). offinite length, while in the secondcasemotorsaccumu- [6] A. Desai, et al., Cell 96, 69 (1999). late at one end only. [7] A.W. Hunter,et al., Mol. Cell 11, 445 (2003). Ourresultscanbe relatedto experimentsonmembers [8] H. Bringmann, et al.,Science 303, 1519 (2004). of the Kin-13 family of kinesins. The depolymerization [9] L. Wordeman and T.J. Mitchison, J. Cell Biol. 128, 95 velocityvd asafunctionofbulkmotorconcentrationhas (1995). been measured for MCAK and it has been shown that [10] R.Lipowsky,S.KlumppandT.M.Nieuwenhuizen,Phys. MCAK accumulates at both ends [7]. The observed ve- Rev. Lett.87, 108101 (2001). [11] K.Kruse,K.Sekimoto,Phys.Rev.E66,031904 (2002). locity v is consistent with both models A and B since d [12] A. Parmeggiani, T. Franosch and E. Frey, Phys. Rev. it does not exclude the possibility of a maximal velocity Lett. 90, 086601 (2003). for intermediate motor concentrations. Accumulation at [13] S. Klumpp and R. Lipowsky, Europhys. Lett. 66, 90 the endsuggeststhata mechanismsimilar to modelA is (2004). more likely to be at work. Indeed, experiments indicate [14] R.V. Skibbens,et al., J. Cell Biol. 122, 859 (1993). 5 [15] D.Pruyne,et al., Science 297, 612 (2002). maximal density, ρ0 ≤ ρmax = N/a. Thus, the effective [16] I.Sagot, et al., Nat.Cell Biol. 4, 626 (2002). processivity vanishesfor ρ0=ρc with ρc ≤ρmax. [17] Notethatthemotordensityattheendcannotexceedthe