Novel Catchbond mediated oscillations in motor-microtubule complexes
Sougata Guha, MIthun K. Mitra, Ignacio Pagonabarraga, Sudipto Muhuri
NNovel catchbond mediated oscillations in motor-filament complexes
Sougata Guha,
1, 2, ∗ Mithun K. Mitra, † Ignacio Pagonabarraga,
3, 4, 5, ‡ and Sudipto Muhuri § Department of Physics, Indian Institute of Technology Bombay, Mumbai, India Department of Physics, Savitribai Phule Pune University, Pune, India CECAM, Centre Europ´een de Calcul Atomique et Mol´eculaire,´Ecole Polytechnique F´ed´erale de Lasuanne (EPFL),Batochime, Avenue Forel 2, 1015 Lausanne, Switzerland Departament de F´ısica de la Mat`eria Condensada,Universitat de Barcelona, Mart´ı i Franqu`es 1, E08028 Barcelona, Spain UBICS University of Barcelona Institute of Complex Systems, Mart´ı i Franqu`es 1, E08028 Barcelona, Spain (Dated: July 14, 2020)Generation of mechanical oscillation is ubiquitous to wide variety of intracellular processes. Weshow that catchbonding behaviour of motor proteins provides a generic mechanism of generatingspontaneous oscillations in motor-cytoskeletal filament complexes. We obtain the phase diagram tocharacterize how this novel catch bond mediated mechanism can give rise to bistability and sustainedlimit cycle oscillations and results in very distinctive stability behaviour, including bistable and non-linearly stabilised in motor-microtubule complexes in biologically relevant regimes. Hitherto, it wasthought that the primary functional role of the biological catchbond was to improve surface adhesionof bacteria and cell when subjected to external forces or flow field. Instead our theoretical studyshows that the imprint of this catch bond mediated physical mechanism would have ramificationsfor whole gamut of intracellular processes ranging from oscillations in mitotic spindle oscillations toactivity in muscle fibres.
Occurrence of spontaneous oscillations due to molecu-lar motor activity is germane to a variety of intracellularprocesses - ranging from mitotic cell division process [1–4] to oscillations in muscle fiber [5, 6]. While for musclefibers, mechanical oscillations arise due to force depen-dent detachment of myosin motors to actin filaments [5],spindle oscillations are generated during cell division dueto the interplay of the microtubule (MT) filament elastic-ity with stochastic (un)binding of cortical dynein motorsto(off) these filaments [2]. Unravelling the underlyingphysical mechanism by which oscillations are generatedin such motor-biofilament complexes is essential to un-derstanding the spatio-temporal organization of the celland their stability characteristics.The unbinding characteristics of individual motorsconstitutes a crucial determinant of the collective proper-ties of motor-filament complexes [7–14]. Dynein motors- which walk along MTs, and myosin-II - which walkalong actin, exhibit catchbonding [15, 16], where the mo-tor unbinding rate decreases when subject to increasingload forces. This non-trivial force dependence in the un-binding rates leads to novel collective behavior in multi-ple motor-filament complexes, in contrast to slip-bondedmotors such as kinesin [17, 18]. A striking consequenceis known as the paradox of codependence - where inhibi-tion of one species of motor in bidirectional intra-cellulartransport can lead to an overall decline in the motilityof the cellular cargo [17, 19]. The implications of thisubiquitous feature of catchbonding on the functional be-havior has not been studied in context of the mechanicalbehaviour of these complexes.In this letter, we study the implications of the catch-bonded behavior of motors on the stability of motor- filament complexes. We find that the catchbonded na-ture of the motor unbinding process leads to a genericmechanism of generation of spontaneous oscillations andresults in very distinctive stability behaviour, includingbistable and non-linearly stabilised motor-filament com-plexes. We argue that this feature has important ramifi-cations for the stability behaviour and oscillations in mi-totic spindles, where such motor-microtubules complexesare the primary constituents of the spindle structure.In order to shed light on the generic mechanism of gen-eration of spontaneous oscillations and analyze the sta-bility behaviour in such motor-microtubule complexes,we adopt a minimalist approach to describe the stabilityof a specific system comprising of a pair of overlappingMTs, dynein motors and confined passive crosslinkers(Fig. 1). For this minimal arrangement, dynein motorsin the overlap region can crosslink the MTs and gen-erate sliding forces which tend to decrease the overlaplength ( l ), while the confined passive proteins (P) gener-ate an entropic force which tends to increase the overlaplength [20]. The force exerted by N p passive proteinsconfined to the overlap region reads F p = N p (cid:15)/l , where (cid:15) is the passive proteins binding energy. We distinguish be-tween two population of motors: i) n c crosslinked motors- which are bound to both the filaments and hence causemutual sliding of the filaments, while experiencing a loadforce F p , and ii) n b bound motors - which are bound toonly one of the MT filaments, and hence exerts no slid-ing force nor feel the effect of the force due to passiveproteins. The dynamic equation for l can be expressedas, dldt = − v Θ ( n c f s − F p ) (cid:18) − F p n c f s (cid:19) + F p Γ (1) a r X i v : . [ q - b i o . S C ] J u l FIG. 1. Schematic diagram of the antiparallel MT-motorcomplex in presence of passive crosslinkers (P). Unbound mo-tors (U) attach to any of the MTs with rate k Db , bound mo-tors (B) crosslink at rate k b whereas they detach from the MTat a rate k u . Crosslinked motors (C) become bound motorswith detachment rate k u under a load force F p . where f s is the stall force for single dynein motor, v thesingle dynein velocity in the absence of load force, and Γis the friction constant. For simplicity, we have assumeda linear force-velocity relation for the crosslinked dyneinmotors [21], zero backward velocity of these crosslinkedmotors in superstall conditions [12], and that the loadforce F p due to passive proteins is shared equally by thecrosslinked dynein motors [12, 21]. The Heaviside, Θfunction ensures that motor sliding contributes to thechange in l only below the stall condition for forces on thecrosslinked motors, whereas above stall the dynamics ofthe overlap length is governed only by the entropic forcesexerted by the confined proteins.Motor (un)binding kinetics in the overlap region isexpressed in terms of rate equations. The crosslinkeddynein motors are subject to F p and they can exhibitcatchbond behaviour beyond the threshold force, f m [15].Incorporating dynein catchbonding in a phenomenologi-cal threshold force bond deformation model [17, 18], thecrosslinked dynein unbinding rate reads, k u = n c k u exp[ − E d ( F p ) + F p / ( n c f d ) ]where f d denotes the characteristic dynein detachmentforce in the slip region ( F p < n c f m ), and the deformationenergy E d is activated when F p > n c f m , reads [18], E d ( F p ) = Θ( F p − n c f m ) α (cid:20) − exp (cid:18) − F p /n c − f m f (cid:19)(cid:21) where, α measures the catchbond strength, while f denotes the force scale associated with the catchbond de- n c t ( s ) n b , l ( n m ) t ( s )150185220 2 4 . l ( n m ) n c . n b n c ( a ) ( b )( c ) ( d ) FIG. 2. Limit cycle oscillations for (a) n c (red curve) and n b (blue curve) and (b) l as a function of time. The figures showthat these quantities oscillate between n c ∼ − n b ∼ − l ∼ −
205 nm. (c) and (d) depict the variation of l and n b with n c , respectively (red curves). The blue and greendashed lines depict two sample trajectories with different ini-tial conditions eventually falling onto the limit cycle. All thecurves are obtained for ˜ f s = 0 .
76, ∆ n = 6, which correspondsto a point denoted by ‘ (cid:7) ’ in the limit cycle region in Fig. 3(d). formation energy. The dynamic equation for n c is, dn c dt = k b n b − k u n c exp( η ) (2)with, η = F p n c f d − Θ (cid:18) F p n c − f m (cid:19) α (cid:20) − exp (cid:18) − F p /n c − f m f (cid:19)(cid:21) Here, k b and k ou are the rates with which the bound mo-tors are converted to crosslinked state and vice-versa, inthe absence of external load force. Specifically, boundmotors are lost due to conversion to crosslinked motors(rate k b ), and due to unbinding from the MT filament(rate k u ), while the gain terms are due to conversionfrom crosslinked motors (rate k u ), binding of free motorsfrom the bulk onto the overlap region of the filament(rate k Db ), and due to the incoming flux ( J ) of boundmotors from the two ends of the overlap region. Thecorresponding dynamic equation for n b is, dn b dt = k ou n c exp( η ) − ( k ou + k b ) n b + k Db ρ d l + 2 J (3)where, ρ d is the bath dynein motor linear density. In thelimit of l being much smaller than the MT length, theincoming flux from a single end is J = k db ρ d k u (cid:2) v + dldt (cid:3) [22, 23]. The equations are analysed using the rescaleddimensionless variables ˜ l = ˜ f s l/l p , l p = v /k u , ˜ f s = bf s k B T , τ = tk u , l e = b(cid:15)k B T , ζ = l e /l p , ˜Γ = bv Γ k B T , ˜ f = bf k B T , ˜ f d = bf d k B T , ˜ f m = bf m k B T , ∆ n = k Db ρ D v k u k u , γ = k b /k u .The corresponding dimensionless equations are providedin the Supplementary Material.The main striking feature, when filaments are cross-linked by catchbonded motors, is the generation of sus-tained limit cycle oscillations, as shown in Fig. 2. Theseoscillations emerge exclusively from catch-bonding, andresult from the non-linear stabilization of previously un-stable morphologies. The corresponding sustained oscil-lations of n c and n b are displayed in Fig. 2(a) and thoseof l in Fig. 2(b), while Fig. 2(c) and (d) display the limitcycle behaviour in the n c − l and the n c − n b planes,respectively. Qualitatively, in the linearly unstable re-gion, the motor force imbalance increases l , while n c de-creases due to the increased propensity of the crosslinkedmotors to detach, corresponding to slip behaviour. As n c further decreases, motor loading forces increase lead-ing to motor catchbonding, prolonging their attachment,favouring reattachment, and eventually counterbalancingthe entropic forces from passive confined proteins. Thisarrests the increase in l and leads to the overall forcedue to crosslinking motors overpowering the force dueto passive proteins, leading to a decrease of the over-lap length. Effectively, dynein catchbonding results in anegative feedback loop which leads to limit cycle oscilla-tions. Catchbond-mediated oscillations offers a hithertounappreciated mechanism through which mechanical os-cillations can be generated in motor-filament complexes.To comprehensively quantify the effect of catchbond-ing, we analyze the stability of motor-filament com-plexes for biologically relevant regimes by using avail-able experimental data for single motors and passiveproteins. Since the single dynein motor velocity, v o =0 . µ m s − [24], and bare unbinding rate k u = 1 s − [24],then l p = 0 . µm . For a characteristic thermal energy, k B T= 4 . b = 1 . l e = 2 . f s = 1 .
25 pN [27], leading to ˜ f s ∼ . f s = 7pN [28], corresponding to˜ f s ∼ .
17. Reported binding rates, k b = 1 s − [29] and k Db ρ D = 60 µ m − s − , lead to ∆ n = 6, which may varyat least over a range 0 . −
10 on varying physiological con-ditions, while estimated passive crosslinker binding ener-gies, (cid:15) = 2 k B T [10]. Quantitative estimates of the frictioncoefficient of passive crosslinkers propose ˜Γ = 2 . f = 11 . α = 68, ˜ f d = 0 . f is varied for a fixed α , in the ∆ n − ˜ f s plane,where ∆ n is a tunable biological parameter associatedwith the propensity of the motor to bind to the MT fil-ament (Fig. 3), when the catchbonding force threshold differs from the motor stall force, i.e. f m (cid:54) = f s . In theabsence of catchbonding, ( f → ∞ ) the phase diagramhas just two morphologies corresponding to a linearly sta-ble and unstable overlapping MTs. This is shown in Fig.3(a). The corresponding bifurcation diagram is shown inFig. 3(i) for ∆ n = 10, and the force-dependent unbindingrate in this regime is shown in Fig. 3(e).The effect of the catchbond strength can be assessedvarying f . For weak catchbonding ( ˜ f = 24 . f = f m , for which the rateof increase of unbinding rate is slower owing to catch-bonding (see Fig. 3(f)), as shown in Fig. 3(b). For small∆ n (∆ n (cid:46) f s further,the system shows a re-entrant transition where the com-plex is stable, before finally becoming unstable at higher˜ f s . As Fig. 3(j) displays the corresponding bifurcation di-agram for ∆ n = 6, the system always has one single fixedpoint. As ˜ f s is increased, this fixed point changes stabil-ity from stable to unstable to stable to unstable througha series of bifurcations (Hopf, saddle and Hopf). Thisre-entrant behaviour can be qualitatively understood be-cause at intermediate ˜ f s , the load force on individual mo-tors is high enough for them to be catchbonded and thusthe unbinding rate is relatively small compared to pureslip. This leads to a higher number of motors attachingto the filament than in absence of catchbonding, whichin turn implies that the the sliding forces exerted by themotors counterbalance the passive crosslinker force, re-sulting in complex stabilization. At higher values of ˜ f s ,the unbinding rate is sufficiently high, and the remainingmotors can no longer stabilize the MT-motor complex.For larger ∆ n ( ∆ n (cid:38) f s . A representative bifurcation diagram isshown in Fig. 3(k) at ∆ n = 10. For small ˜ f s , the complexdisplays a single stable fixed point. Beyond a critical ˜ f s ,the system undergoes a saddle-node bifurcation leadingto the emergence of a new stable fixed point, in addi-tion to an unstable fixed point. This corresponds to theregion of bistable behaviour. On increasing ˜ f s further,the new fixed point destabilizes via a Hopf bifurcation,leading to a single stable steady state. At even larger ˜ f s ,this stable fixed point disappears via a reverse saddle-node bifurcation and the system become unstable. Thisbistable behaviour arises due to the catchbonded natureof the unbinding characteristics of dynein motors.On decreasing ˜ f further, not only does the linearlystable region become larger, but the region of linearly un-stable configurations ( for intermediate values of ˜ f s ) arestabilized by a non-linear mechanism leading to limit-cycle oscillations. This non-linear stabilization arisespurely due to dynein catchbonding. The bistable re-gions also grow with decreasing ˜ f , as shown in Fig. 3(c)for ˜ f = 15 .
47. For even stronger catchbonding, wherethe unbinding rate decreases sharply beyond f m (see FIG. 3. Stability diagram of a MT-motor complex as a function of ˜ f . (a) ˜ f = 309 .
52 ( f o ≈ f = 24 .
76 ( f o ≈ f = 15 .
47 ( f o ≈
50 pN) and (d) ˜ f = 11 .
98 ( f o ≈ . N p = 100 , k b = k u = 1 /s ( ∴ γ = 1) , v = 100 nm/s, ˜Γ = 2 . , ˜ f d = 0 . f d ≈ .
67 pN), α = 68 , ˜ f m = 0 .
43 ( f m ≈ . b = 1 . (cid:15) = 2 k B T .The redsolid line depicts the boundary between linearly stable (white) and unstable (cyan) regions. Green areas indicate regions wherelimit cycles can be sustained and yellow shaded areas signal regions where the complex displays bistable behaviour. Panels(e-h) depict the unbinding rate of a single dynein motor under load force, f , for the f values of panels a-d, while blue dashedline gives the reference curve when ˜ f s = ˜ f m Panels (i-l) show the bifurcations diagrams as a function of ˜ f s in different regionsof the phase plane, as indicated by the dashed lines in panels (a-c). The solid blue lines indicate a stable branch, while thedashed red lines indicate unstable solutions.FIG. 4. Stability diagram when ˜ f s = ˜ f m , (a) in the absenceof catchbond ( ˜ f = 309 . f = 11 . Fig. 3(d)), the nature of the phase diagram remains sim-ilar, except that the region of stable overlaps grows evenbigger, as shown in Fig. 3(d) for ˜ f = 11 .
98 [18]. Theselimit cycle oscillations are robust even under fluctuationsof the order of the underlying energy scales in the system(see Supplementary Material).Experimental studies suggest that for dynein, catch-bond sets in around the motor stall force, f m (cid:39) f s = 1 . n − ˜ f s plane, when f s = f m . In the absence of catch-bonding, Fig. 4(a), a stable-to-unstable transition is re-covered, cf. Fig. 3(a). In the presence of catchbonding, e.g; decreasing f o , limit-cycle oscillations appear in thephase diagram, see Fig. 3(b), where unstable overlapsare non-linearly stabilized. The region with oscillatorybehaviour increases with enhanced catchbonding, i.e. de-creasing f .For experimentally relevant parameters, the limit cycleoscillations have a typical time period ∼ − s , ampli-tude ∼ . − . µm , and characteristic overlap lengthin the range of 0 . − µm (See Supplementary Mate-rial). These estimates lie within the observable tempo-ral and spatial scale of cellular processes and points totheir biological relevance in context of motor-microtubulecomplexes in particular and more generally in context ofmitotic spindles.In summary, in this Letter, we investigated the func-tional consequences of molecular motor’s catchbondingon the stability of motor-biofilament complexes. We findthat for a pair of overlapping antiparallel biofilamentssubject to sliding forces by motors and entropic forcesby confined passive crosslinkers, the catchbond natureof motor unbinding from the biofilaments manifests as ageneric intrinsic mechanism that generates and stabilizesspontaneous oscillations, and additionally also promotesbistability in biologically relevant regimes.Recent experiments report that kinesin motors exhibitcatchbonding under horizontal load forces [30]; hence,it would be interesting to analyze whether catchbond-driven oscillations are present for such kinesin-MT com-plexes. Controlled experiments on immobilized MTs onglass surfaces [20] offer a promising experimental setupto verify the described collective implications of catch-bonding on motor-filament complexes.The mechanism of nonlinear oscillations we have de-scribed is distinct from previously reported oscillationmechanisms for MT-motor complexes that arise from thecoupling of motor proteins in the cell cortex with theoverlapping MTs [11, 31], and shown to be relevant forunderstanding mitotic oscillations in spindles [2, 11]. Therange of oscillation frequency predicted (0.1-1 Hz) lie inthe same range of previously reported mechanisms and inthe experimentally observed oscillation frequency rangein the mitotic spindle during the metaphase of cell divi-sion [32]. Since the mitotic spindle, in the metaphase,is composed of overlapping MTs that interact with cor-tical motor proteins, and are subject to sliding forcesof crosslinking motors e.g; dynein and Eg5 kinesin, aswell as to kinetochores and chromosomes [3, 4], clari-fying whether these distinct oscillation mechanisms canresult in resonances, with their potential implications forspindle stability, remains an open challenge. Acknowledgements.
Financial support is acknowledgedby MKM for Ramanujan Fellowship (13DST052), DSTand IITB (14IRCCSG009); SM and MKM for SERBproject No. EMR /2017/001335]. I.P. acknowledgesthe support from MINECO/FEDER UE (Grant No.PGC2018-098373-B-100), DURSI (Grant No. 2017 SGR884), and SNF Project no. 200021-175719.SG analyzed the model and data, performed the nu-merics and wrote the manuscript. MKM analyzed themodel and data, and wrote the manuscript. IP and SMdesigned the study, proposed the model, analyzed themodel and data, and wrote the manuscript. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected][1] D. G. Albertson, Developmental Biology , 61 (1984).[2] S. W. Grill, K. Kruse, and F. J¨ulicher, Physical ReviewLetters , 108104 (2005).[3] N. P. Ferenz, R. Paul, C. Fagerstrom, A. Mogilner, andP. Wadsworth, Current Biology , 1833 (2009).[4] P. Malgaretti and S. Muhuri, EPL (Europhysics Letters) , 28001 (2016).[5] S. G¨unther and K. Kruse, New Journal of Physics , 417(2007).[6] D. Sasaki, H. Fujita, N. Fukuda, S. Kurihara, and S.Ishiwata, Journal of Muscle Research and Cell Motility , 93 (2005).[7] T. Gu´erin, J. Prost, P. Martin, and J.-F. Joanny, Current Opinion in Cell Biology , 14 (2010).[8] D. Bhat and M. Gopalakrishnan, Physical Biology ,046003 (2012).[9] F. Posta, M. R. D’Orsogna, and T. Chou, Physical Chem-istry Chemical Physics , 4851 (2009).[10] S. Guha, S. Ghosh, I. Pagonabarraga, and S. Muhuri,EPL (Europhysics Letters) , 58003 (2019).[11] S. Ghosh, V. N. S. Pradeep, S. Muhuri, I. Pagonabarraga,and D. Chaudhuri, Soft Matter , 7129 (2017).[12] M. J. I. M¨uller, S. Klumpp, and R. Lipowsky, Proceed-ings of the National Academy of Sciences , 4609(2008).[13] D. Ando, M. K. Mattson, J. Xu, and A. Gopinathan,Scientific Reports , 1 (2014).[14] S. Muhuri and I. Pagonabarraga, Physical Review E ,021925 (2010).[15] A. Kunwar, S. K. Tripathy, J. Xu, M. K. Mattson, P.Anand, R. Sigua, M. Vershinin, R. J. McKenney, C. Y.Clare, A. Mogilner, and S. P. Gross, Proceedings of theNational Academy of Sciences , 18960 (2011).[16] B. Guo and W. H. Guilford, Proceedings of the NationalAcademy of Sciences , 9844 (2006).[17] P. Puri, N. Gupta, S. Chandel, S. Naskar, A. Nair, A.Chaudhuri, M. K. Mitra, and S. Muhuri, Physical ReviewResearch , 023019 (2019).[18] A. Nair, S. Chandel, M. K. Mitra, S. Muhuri, and A.Chaudhuri, Physical Review E , 032403 (2016).[19] W. O. Hancock, Nature Reviews Molecular Cell Biology , 615 (2014).[20] Z. Lansky, M. Braun, A. L¨udecke, M. Schlierf, P. R. tenWolde, M. E. Janson, and S. Diez, Cell , 1159 (2015).[21] S. Klumpp and R. Lipowsky, Proceedings of the NationalAcademy of Sciences , 17284 (2005).[22] O. Camp`as, J. Casademunt, and I. Pagonabarraga, EPL(Europhysics Letters) , 48003 (2008).[23] S. Muhuri, I. Pagonabarraga, and J. Casademunt, EPL(Europhysics Letters) , 68005 (2012).[24] S. L. Reck-Peterson, A. Yildiz, A. P. Carter, A. Genner-ich, N. Zhang, and R. D. Vale, Cell , 335 (2006).[25] M. J. Schnitzer, K. Visscher, and S. M. Block, NatureCell Biology , 718 (2000).[26] V. Belyy, M. A. Schlager, H. Foster, A. E. Reimer, A.P. Carter, and A. Yildiz, Nature Cell Biology , 1018(2016).[27] R. Mallik, D. Petrov, S. Lex, S. King, and S. Gross, Cur-rent Biology , 2075 (2005).[28] S. Toba, T. M. Watanabe, L. Yamaguchi-Okimoto, Y. Y.Toyoshima, and H. Higuchi, Proceedings of the NationalAcademy of Sciences , 5741 (2006).[29] C. Leduc, O. Camp`as, K. B. Zeldovich, A. Roux, P.Jolimaitre, L. Bourel-Bonnet, B. Goud, J.-F. Joanny,P. Bassereau, and J. Prost, Proceedings of the NationalAcademy of Sciences , 17096 (2004).[30] H. Khataee and J. Howard, Physical Review Letters ,188101 (2019).[31] D. Johann, D. Goswami, and K. Kruse, Physical ReviewLetters , 118103 (2015).[32] J. Pecreaux, J. -C. R¨oper, K. Kruse, F. J¨ulicher, A. A.Hyman, S. W. Grill, and J. Howard, Current Biology16