Collective cargo hauling by a bundle of parallel microtubules: bi-directional motion caused by load-dependent polymerization and depolymerization
aa r X i v : . [ q - b i o . S C ] J a n Collective cargo hauling by a bundle of parallel microtubules: bi-directional motioncaused by load-dependent polymerization and depolymerization
Dipanwita Ghanti and Debashish Chowdhury ∗ Department of Physics, Indian Institute of Technology Kanpur, 208016
A microtubule (MT) is a hollow tube of approximately 25 nm diameter. The two ends of thetube are dissimilar and are designated as ‘plus’ and ‘minus’ ends. Motivated by the collectivepush and pull exerted by a bundle of MTs during chromosome segregation in a living cell, we havedeveloped here a much simplified theoretical model of a bundle of parallel dynamic MTs. The plus-end of all the MTs in the bundle are permanently attached to a movable ‘wall’ by a device whosedetailed structure is not treated explicitly in our model. The only requirement is that the deviceallows polymerization and depolymerization of each MT at the plus-end. In spite of the absenceof external force and direct lateral interactions between the MTs, the group of polymerizing MTsattached to the wall create a load force against the group of depolymerizing MTs and vice-versa;the load against a group is shared equally by the members of that group. Such indirect interactionsamong the MTs gives rise to the rich variety of possible states of collective dynamics that we haveidentified by computer simulations of the model in different parameter regimes. The bi-directionalmotion of the cargo, caused by the load-dependence of the polymerization kinetics, is a “proof-of-principle” that the bi-directional motion of chromosomes before cell division does not necessarilyneed active participation of motor proteins.
I. INTRODUCTION
A microtubules (MT) is a hollow tube of approximately25 nm diameter. It is one of major components of thecytoskeleton [1] that provides mechanical strength to thecell. Each MT is polar in the sense that its two endsare structurally as well as kinetically dissimilar. One ofthe unique features of a MT in the intracellular environ-ment is its dynamic instability [2]. The steady growthof a polymerizing MT takes place till a “catastrophe”triggers its rapid depolymerization from its tip. Oftenthe depolymerizing MT is “rescued” from this decayingstate before it disappears completely and keeps growingagain by polymerization till its next catastrophe. Thus,during its lifetime, a MT alternates between the states ofpolymerization (growth) and depolymerization (decay).One of the key structural features of a MT is thatduring its depolymerization the tip of this nano-tubeis curved radially outward from its central axis. Whilepolymerizing, the growing MT can exert a pushing forceagainst a transverse barrier thereby operating, effectively,as a nano-piston [3, 4]. Although lateral cross linkingbetween the MTs and their unzipping can have interest-ing effects [5, 6], no such cross link is incorporated inour model here because of the different motivation of ourwork. Similarly, the splaying tip of a depolymerizing MTcan pull an object in a manner that resembles the oper-ation of a nano-hook [7, 8]. Thus, a MT can performmechanical work by transducing input chemical energy.In analogy with motor proteins that transduce chemi-cal input energy into mechanical work, force generatingpolymerizing and depolymerizing MTs are also referredto as molecular motors [9–11]. ∗ Email: [email protected]
In this paper we consider a bundle of parallel MTs thatare not laterally bonded to each other. Such bundles havebeen the focus of attention in recent years because of thenontrivial collective kinetics and force generation by abundle in spite of the absence of any direct lateral bondbetween them [3]. For example, while driving chromoso-mal movements in a mammalian cell before cell division,the members of a bundle that is attached to a singlekinetochore wall, undergo catastrophe and rescue in asynchronized manner [12]. The type of collective phe-nomenon under consideration here is of current interestalso in many related or similar contexts in living systems.For example, there are similarities between the motion ofa chromosomal cargo pushed/pulled by a group of poly-merizing/depolymerizing MTs and that of a membrane-bounded vesicular cargo hauled by the members of twoantagonistic superfamilies of molecular motors along afilamentous track [13–15]. But, as we explain here, thereare crucial differences between the two systems. Otherexamples of similar collective dynamics include (stochas-tic) oscillations in muscles, flagella (a cell appendage) ofsperm cells, etc. [9, 10].Here we develop a theoretical model for the collectivekinetics of a bundle of parallel MTs that interact with amovable wall. The model is based on minimum numberof hypotheses for the polymerization kinetics of individ-ual MTs. More specifically, the model takes into accountthe force-dependence of the rates of catastrophe/rescueand polymerization/depolymerization. These hypothesesare consistent with in-vitro experiments on single MT re-ported so far in the literature. Numerical simulation ofthe model demonstrates synchronized growth and shrink-age of the MT bundle in a regime of model parameters.We classify the states of motion of the wall that is col-lectively pushed and pulled by the MT bundle. In one ofthe states thus discovered the wall executes bi-directional motion and it arises from a delicate interplay of load-force-dependent rates of MT polymerization and depoly-merization. This work also provides direct evidence thatbidirectional motion of MT-driven cargo is possible evenin the absence of any motor protein [9, 10]; the “proof-of-principle” presented here will be elaborated later else-where [16] in the context of modeling chromosome segre-gation.
II. THE MODEL !" , -( , .( FIG. 1: (Color online) Our model is depicted schematically.The wall to which the MTs are attached is represented by theblue vertical slab. The polymerizing (growing) and depoly-merizing (shrinking) MTs are represented by the horizontalgreen cylinders (with dashed outlines) and red cylinders (withsolid outlines), respectively. The grey short cylinders, intowhich the plus end of the MTs are inserted, represent the“couplers” that couple the MTs with the wall. The directionsof polymerization and depolymerization are indicated by thedouble arrows. The directions of the load force (the absolutevalue of which is F + ) on the polymerizing MTs and load ten-sion (the absolute value of which is F − ) on the depolymerizingMTs are indicated by the single arrows. As we stated in the introduction, our study has beenmotivated partly by collective force generation by MTsduring chromosome segregation. The model is depictedschematically in fig.1. At present the identity of themolecules that couple MTs to the chromosomal cargo andthe detailed mechanism of force generation at the MT-cargo interface are under intense investigation [12, 17].Plausible scenarios for the coupling suggested in the lit-erature include, for example, mechanisms based on (i)biased diffusion of the coupler [18], (ii) power stroke onthe coupler [19], (iii) attachment (and detachment) oflong flexible tethers [20]. Moreover, the mechanisms ofthe MT-cargo coupling may also vary from one speciesto another. Therefore, in order to maintain the genericcharacter of our model, we neither make any explicit mi-croscopic model for the coupler nor postulate any de-tailed mechanism by which the individual members ofthe MT bundle couple with the chromosomal cargo. This is similar, in spirit, to the kinetic models of hauling ofmembrane-bounded cargo where no explicit molecularmechanism is assumed for the motor-cargo coupling [15].The only assumption we make about the MT-wall cou-pling is that none of the MTs detaches from the wallduring the period of our observation; catastrophe andrescue of a MT merely reverse the direction of the forcethat it exerts on the wall.We assume that during the period of our observationneither nucleation of new MTs take place nor does any ofthe existing MTs disappear altogether because of its de-polymerization. Therefore, the total number of MTs, de-noted by N , is conserved. We use the symbols n + ( t ) and n − ( t ) to denote the MTs in the growing and shrinkingphases, respectively, at time t so that n + ( t )+ n − ( t ) = N .Thus, at time t , n + ( t ) MTs push the cargo while, simul-taneously, n − ( t ) MTs pull the same cargo in the oppositedirection.As we show in this paper, the distribution of the MTsin the two groups, namely n + and n − , is a crucial de-terminant of the nature of the movements of the wall.By carrying out computer simulations, we also monitorthe displacement and velocity distribution of the wall indifferent regimes of the model parameters to character-ize its distinct states of motility. The observed states ofmotility are displayed on a phase diagram. A. Parameters and equations: force-dependence
The force-dependent rates of catastrophe and rescue ofa MT are denoted by the symbols c ( F ) and r ( F ), respec-tively, while c and r refer to the corresponding valuesin the absence of load force. The experimental data col-lected over the last few years [21] and very recent mod-eling [22] have established the dependence of the rates ofcatastrophe and rescue on the load force. Accordingly,we assume c ( F ) = c exp ( | F | /F ∗ c ) (1)where | F | is the absolute value of the load force opposingpolymerization, and F ∗ c is the characteristic force thatcharacterizes the rapidity of variation of the catastropherate with the load force. Similarly, we assume r ( F ) = r exp ( | F | /F ∗ r ) (2)where | F | is the absolute value of the load tension oppos-ing depolymerization and F ∗ r is the characteristic forcethat characterizes the rapidity of variation of the depoly-merization rate with the load force.We denote the load-free velocities of polymerizationand depolymerization of a single MT by the symbols v g and v d , respectively. For the load-dependence of these ve-locities we follow the standard practice in the literaturefor modeling the load-dependence of the velocity of mo-tor proteins. Specifically, we assume linear force-velocityrelation v + = v g (cid:18) − | F + | F s + (cid:19) for | F + | ≤ F s + (3)for the polymerizing MTs, and v − = v d (cid:18) − | F − | F s − (cid:19) for | F − | ≤ F s − (4)for the depolymerizing MTs, where | F + | and | F − | denotethe absolute values of the respective load forces opposingpolymerization and depolymerization, respectively, of thecorresponding MT. F s + and F s − are the absolute valuesof the stall forces corresponding to the polymerizing anddepolymerizing MTs.In the spirit of many earlier theoretical models of bidi-rectional transport of membrane-bounded cargo [15] wealso make the following assumptions:(i) push on the wall by polymerizing MTs generates theindirect load tension on the depolymerizing MTs whilethe pull of the latter on the same wall simultaneouslycreates the load force against the polymerizing MTs. (ii)the load force against polymerization is shared equally bythe n + ( t ) MTs, and the load tension against depolymer-ization is also shared equally by the n − ( t ) MTs. At anyarbitrary instant of time t , each of the n + ( t ) polymeriz-ing MTs attached the chromosomal cargo experiences aload force F + and exerts a force − F + . Similarly, each of n − ( t ) depolymerizing MTs feels a load tension − F − andoffers a load force F − . Therefore, force balance on thewall attached simultaneously by n + ( t ) and n − ( t ) MT is n + F + = − n − F − ≡ F w ( n + , n − ) . (5)Equation (5), which defines the symbol F w ( n + , n − ), isjust a mathematical representation of Newton’s thirdlaw: the force exerted by the wall on each MT is equaland opposite to that exerted by the same MT on the wall.According to our choice of the signs, the load force on thepolymerizing MTs are positive. Based on the assumption(ii) above, we now have | F + ( n + , n − ) | = | F w ( n + , n − ) | /n + | F − ( n + , n − ) | = | F w ( n + , n − ) | /n − . (6)When n + ( t ) and n − ( t ) are the numbers of MTs in thepolymerizing and depolymerizing phases, respectively,the corresponding catastrophe and rescue rates are c n + ,n − ( F ) = c exp ( | F + ( n + , n − ) | /F ⋆c ) r n + ,n − ( F ) = r exp ( | F − ( n + , n − ) | /F ⋆r ) , (7)respectively.Since all the n + ( t ) polymerizing MTs and n − ( t ) de-polymerizing MTs are, by definition, attached to the wallat time t , their velocities of growth and decay, respec-tively, must be identical to the wall velocity v w , i.e., v w ( n + ( t ) , n − ( t )) = v + ( F w /n + ( t )) = − v − ( − F w /n − ( t ))(8) Now substituting eqs.(6) into eqs.(3) and (4) we get v + = v g (cid:18) − F w n + F s + (cid:19) (9) v − = v d (cid:18) − F w n − F s − (cid:19) (10)Imposing the constraint (8) on eqs. (3) and (4) we getthe force F w ( n + , n − ) = µn + F s + + (1 − µ ) n − F s − (11)and velocity v w ( n + , n − ) = n + F s + − n − F s − n + (cid:18) F s + v g (cid:19) + n − (cid:18) F s − v d (cid:19) (12)for the wall where µ − = 1 + ( n + F s + v d )( n − F s − v g ) .At every instant of time t the state of the system ischaracterized by n + ( t ) and n − ( t ). The probability offinding the system in the state n + ( t ) , n − ( t ) at time t isdenoted by P ( n + , n − , t ) and its time evolution is gov-erned by the master equation dP ( n + , n − , t ) dt =+ c n + +1 ,n − − P ( n + + 1 , n − − , t )+ r n + − ,n − +1 P ( n + − , n − + 1 , t ) − ( c n + ,n − + r n + ,n − ) P ( n + , n − , t ) (13)where c n + ,n − and r n + ,n − are given by the equations (7)while c n + +1 ,n − − ( F ) = c exp ( | F + ( n + + 1 , n − − | /F ⋆c ) r n + − ,n − +1 ( F ) = r exp ( | F − ( n + − , n − + 1) | /F ⋆r ) . (14)Note that the rate constants in equation (13) depend onthe time-dependent quantities n + ( t ) and n − ( t ). There-fore, the rate constants on the right hand side of eqnation(13) take their instantaneous values at time t . B. Comparison with other similar phenomena andmodels
We can now compare the model system under studyhere with the models of transport of membrane-boundedcargo (vesicles or organelles) by antagonistic motor pro-teins (e.g., kinesins and dyneins) that move on a fila-mentous track (e.g., a MT) [9, 10]. The polymerizingand depolymerizing MTs are the analogs of kinesins anddyneins. However, one crucial difference between thetwo situations is that unlike a MT, that can switch frompolymerizing to depolymerizing phase (because of catas-trophe) and from depolymerizing to polymerizing phase(because of rescue), interconversion of plus and minus- end directed motor proteins is impossible.A systematic comparison of these two systems is pre-sented in table I.
Object or property Multi-MT system Multi-motor systemCargo kinetochore (kt) vesicle or organelleAntagonistic force generators Polymerizing MT, depolymerizing MT kinesin, dyneinInter conversion of force generators Polymerizing MT ⇋ Depolymerizing MT: possible Kinesin ⇋ Dynein: impossibleTotal number conservation N + ( t ) + N − ( t ) = N =constant N + + N − = N =constantNumbers of + and - force generators Time-dependent N + ( t ), N − ( t ) N + =constant, N − =constantNumber attached to cargo n + ( t ) = N + ( t ) , n − ( t ) = N − ( t ) n + ( t ) ≤ N + , n − ( t ) ≤ N − Track No analog MTRate of attachment to track No analog ω a Rate of detachment from track No analog ω d TABLE I: Comparison of cargo hauling by bundle of parallel dynamic MTs and by antagonistic cytoskeletal motor proteins.
III. RESULTS AND DISCUSSION: MOTILITYSTATES
We have simulated the time evolution of n + ( t ) and n − ( t ), that is described by the master equation (13) usingGillespie algorithm [23]. We get the instantaneous veloc-ity v w ( n + , n − ) by substituting the corresponding valuesof n + ( t ) and n − ( t ) into equation (12). This data is usednot only for plotting the velocity distribution but alsofor computing the displacements of the wall as a functionof time. Moreover, the instantaneous force F w ( n + , n − )evaluated using (11) is used for updating n + and n − inthe next time step. Finally, all the distinct states ofmotility of the wall are characterized by the statisticsof n + , n − . For the simulation we have assumed thatinitially half of the MTs are in the the state of poly-merization while the remaining half are in the state ofdepolymerization. For the presentation of our data, itwould be convenient to reduce the number of parameters.Therefore, in this paper we report the results only for thecases where v g = v d , c = r = γ , F s + = F s − = F s . Themagnitudes of v g = v d = 50 nm/s are identical in all thefigures plotted below. (A)
0 2 4 6 8 0 2 4 6 8 0 0.1 0.2 0.3 0.4 P r obab ili t y n+ n- P r obab ili t y (B) P r obab ili t y Velocity (nm/s) -200-100 0 100 0 20 40 60 D i s t an c e ( n m ) Time (s)
FIG. 2: Characteristics of “bidirectional motion” (+-) state.(A) There are two distinct maxima in the 3D plot of theprobability against the numbers of growing ( n + ) and shrink-ing ( n − ) MT; the peak near n − = 0 occurs when majority ofthe MTs are growing whereas the peak near n + = 0 indicatesmajority of the shrinking MTs. (B) The long stretches of al-ternate positive and negative displacements are clearly visiblein the typical trajectory displayed in the inset. The two dis-tinct and sharp peaks at the velocities ± F s = 6 pN, v g = v d =50 nm/s, c = r = 10s − , and F ∗ c = F ∗ r = 3 . (A)
0 2 4 6 8 0 2 4 6 8 0 0.1 0.2 0.3 P r obab ili t y n+ n- P r obab ili t y (B) -20-10 0 10 20 0 20 40 60 D i s t an c e ( n m ) Time (s) (C) P r obab ili t y Velocity (nm/s)
FIG. 3: Characteristics of “no motion” (0) state. (A) In the3D plot of the probability against the numbers of growing( n + ) and shrinking ( n − ) MTs peaks are seen along the di-agonal, i.e., n + = n − . (B) The small random positive andnegative displacements of the wall recorded with the passageof time reflects the fluctuations in the position of the wallas a consequence of the tug-of-war between the growing andshrinking MTs. (C) The sharp dominant peak at zero veloc-ity in the distribution of velocities is a signature of the ‘nomotion’ state; the small non-zero probabilities at other veloc-ities correspond to the small random excursions of the wallabout its mean position. Numerical value of the parameters,used in the simulation are F s = 0 . v g = v d =50 nm/s, F ∗ c = F ∗ r = 10 . c = r = 2000 s − . • Bi-directional motion in the ‘Plus and MinusMotion (+-)’ state
First we simulated the model for the parameter val-ues c = r = 10 s − , and F ∗ c = F ∗ r = 3 . V = ±
50 nm/s in the distribution of velocities of thewall. The physical origin of this bi-directional motion[15] can be inferred from the nature of the distribution P ( n + , n − ) which now exhibits two maxima. The max-imum at n + > , n − ≃ n − > , n + ≃ n + ( t ) , n − ( t ) and thedependence of c ( F ) , r ( F ) on n + ( t ) , n − ( t ). Although ini-tially, n + (0) = N/ n − (0), stochastic nature of catas-trophe and rescue causes population imbalance in spiteof the equal values of the corresponding rate constants.Because of the appearance of n + ( t ) , n − ( t ) in eq.(7) thepopulation imbalance grows further. Thus, if n + ( t ) keepsincreasing with time t , the wall continues to move for-ward; eventually, if n − vanishes, v w would become iden-tical to v g . However, by that time, because of the non-vanishing c , the polymerizing MTs start suffering catas-trophe that, in turn, increases population of depolymer-izing MTs which has a feedback effect. Once the majorityof the MTs are in the depolymerizing state the wall re-verses its direction of motion. The velocity of the wallalternates between positive and negative values, corre-sponding to the alternate segments of the trajectory, thatcharacterizes the bi-directional motion of the wall. (A1) (B1)
0 2 4 6 8 0 2 4 6 8 0 0.2 0.4 0.6 0.8 P r obab ili t y n+ n- P r obab ili t y
0 2 4 6 8 0 2 4 6 8 0 0.2 0.4 0.6 0.8 P r obab ili t y n+ n- P r obab ili t y (A2) (B2) P r obab ili t y Velocity (nm/s) -3000 0 0 20 40 60 D i s t an c e ( n m ) Time (s) P r obab ili t y Velocity (nm/s) D i s t an c e ( n m ) Time (s)
FIG. 4: Characteristics of only minus and only plus motion.Numerical value of the parameters, used in the simulationare F s = 5 . v g = v d =50 nm/s, c = r = 10 s − . Toget only minus or only plus motion we take F ∗ c = 3 . F ∗ r = 6 . • Tug-of-war in the ‘No Motion (0)’ state
Next we chose the parameter values F ∗ c = F ∗ r =10 . c = r = 2000 s − . Note that that catastro-phe and rescue rates are much higher than those in theprevious case. Consequently, each MT very frequentlyswitches between the polymerizing and depolymerizingstates. Moreover, because of the much smaller values of F s , a MT can exert very small force on the wall in bothcases. The results of simulation are plotted in fig.3. Thevelocity distribution exhibits a single peak at zero ve-locity which is a signature of the ‘No Motion (0)’ state[15]. This is a consequence of the tug-of-war betweenthe growing and shrinking MTs that is evident in the single maximum at n + = n − in the probability distri-bution P ( n + , n − ). Tug-of-war does not mean completestall; small fluctuations around the stall position visiblein the trajectory gives rise to the non-zero width of thevelocity distribution around zero velocity. The underly-ing physical processes in this case are almost identical tothose in the case of bi-directional motion except for cru-cial difference arising from the relatively higher valuesof c = r ; even before the wall can cover a significantdistance in a particular direction the MTs reverse theirvelocities. Because of the high frequencies of catastropheand rescue the wall remains practically static except forthe small fluctuations about this position. • Only plus (+) motion and only minus (-) mo-tion
Finally, for the sake of completeness, we exploredasymmetric behavior of the model by choosing F ∗ c < F ∗ r in one case and F ∗ c > F ∗ r in the other so that in one casethe wall exhibits only minus motion whereas in the othercase it exhibits only plus motion. The simulation resultsare plotted side-by-side for these two cases in fig.4. Inthis case the typical trajectories indicate motion eitheronly in the plus direction or only in the minus direction.This observation is also consistent with the single sharppeaks at V = +50 nm/s and V = −
50 nm/s in thedistribution of velocities of the wall. The maximum at n − > , n + ≃ n + > , n − ≃ • Phase diagram f k (-)(+) (+-)(+0-) FIG. 5:
All the distinct states of motility of the wallfor the case F ∗ c = F ∗ r are displayed on the f − k phasediagram (see text for the definitions of f and k ).The phases are labelled by the corresponding sym-bols (see text for the meaning of these symbols). There-entrance phenomenon exhibited for f . . is alsoexplained in the text. The movement of the wall depends crucially on two im-portant dimensionless parameters, namely, k = c / ( c + r ) and f = F s / ( F s + F ⋆ ). For the sake of simplicity andfor reducing the number of parameters, here we have as-sumed F ∗ c = F ∗ r = F ∗ . We have varied these modelparameters over wide range of values and, for each setof values, we have identified the state of movement ofthe wall. The corresponding observations for the sym-metric case F ∗ c = F ∗ r = F ∗ , are summarized in fig.5 inthe form of a phase diagram on the f − k plane. Sinceno new motility state, other than those observed in thesymmetric case F ∗ c = F ∗ r are observed in the more gen-eral asymmetric case F ∗ c = F ∗ r , we have not drawn thehigher-dimensional phase diagram.For high force ratio ( f & .
23) we observe transition from (+) to (-) region through the (+-) region in between,as the ratio k is varied. In contrast, for f . .
23 thesystem exhibits “re-entrance”; as k increases from 0 . .
6, first a transition from the motility state (+-) tothe state (+0-) takes place and, at a somewhat highervalue of k a re-entrance to the state (+-) occurs. Thisre-entrance phenomenon disappears at f & . IV. SUMMARY AND CONCLUSION
In this paper we have introduced a theoretical modelfor studying the cargo-mediated collective kinetics ofpolymerization and depolymerization of a bundle of par-allel MTs that are not bonded laterally with one another.Carrying out computer simulations of the model we haveidentified and characterized the motility states of a hardwall-like cargo that is pushed and pulled by this MT bun-dle that is oriented perpendicular to the plane of the wall.Among these motility states, one corresponds to “no mo-tion” (except for small fluctuations); it arises from the“tug-of-war” between the polymerizing and depolymer-izing MTs. The wall exhibits a bi-directional motion inanother motility state. The qualitative features of thecharacteristics of this bi-directional state of motility ofthe wall are similar to those observed in bi-directionalmotion of vesicular cargo driven by antagonistic motorproteins. But, as we have argued here, the physicalorigin of the bidirectional motion in these two systemsare completely different. In the case of motor protein-driven vesicular cargo the bi-directional motion is causedby an interplay of the stall force of the motors and theload-dependent detachment of the motor from its track[15]. In contrast, in our model of MT-driven wall, the bi-directional motion arises from a subtle interplay of thestall force of the MTs and the load-dependent depoly-merization of the MTs. This result should be regardedalso as a “proof-of-principle” that for bi-directional mo-tion of the chromosomal cargo active participation mo-tor proteins is not essential; in contrast to past claims inthe literature (see ref.[24] for a very recent claim), thepolymerization / depolymerization kinetics of the MTsare adequate to cause bidirectional motion provided theload-dependence of their rates are properly taken intoaccount. The latter principle will be elaborated furtherelsewhere [16] in the context of chromosome segregationwith a more detailed theoretical model which also treatsthe kinetochore wall as a soft elastic object [25].
Acknowledgement
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