Bistability and oscillations in cooperative microtubule and kinetochore dynamics in the mitotic spindle
aa r X i v : . [ q - b i o . S C ] M a r Bistability and oscillations in cooperativemicrotubule and kinetochore dynamics in themitotic spindle
Felix Schwietert and Jan Kierfeld
Physics Department, TU Dortmund University, 44221 Dortmund, GermanyE-mail: [email protected]
March 2020
Abstract.
In the mitotic spindle microtubules attach to kinetochores via catch bondsduring metaphase, and microtubule depolymerization forces give rise to stochasticchromosome oscillations. We investigate the cooperative stochastic microtubuledynamics in spindle models consisting of ensembles of parallel microtubules, whichattach to a kinetochore via elastic linkers. We include the dynamic instability ofmicrotubules and forces on microtubules and kinetochores from elastic linkers. A one-sided model, where an external force acts on the kinetochore is solved analyticallyemploying a mean-field approach based on Fokker-Planck equations. The solutionestablishes a bistable force-velocity relation of the microtubule ensemble in agreementwith stochastic simulations. We derive constraints on linker stiffness and microtubulenumber for bistability. The bistable force-velocity relation of the one-sided spindlemodel gives rise to oscillations in the two-sided model, which can explain stochasticchromosome oscillations in metaphase (directional instability). We derive constraintson linker stiffness and microtubule number for metaphase chromosome oscillations.Including poleward microtubule flux into the model we can provide an explanationfor the experimentally observed suppression of chromosome oscillations in cells withhigh poleward flux velocities. Chromosome oscillations persist in the presence of polarejection forces, however, with a reduced amplitude and a phase shift between sisterkinetochores. Moreover, polar ejection forces are necessary to align the chromosomesat the spindle equator and stabilize an alternating oscillation pattern of the twokinetochores. Finally, we modify the model such that microtubules can only exerttensile forces on the kinetochore resulting in a tug-of-war between the two microtubuleensembles. Then, induced microtubule catastrophes after reaching the kinetochore arenecessary to stimulate oscillations. The model can reproduce experimental results forkinetochore oscillations in PtK1 cells quantitatively.
Keywords : mitotic spindle, directional instability, microtubule dynamics, kinetochoreoscillations, bistability, stochastic simulation
Submitted to:
New J. Phys. istability and oscillations in cooperative microtubule and kinetochore dynamics
1. Introduction
Proper separation of chromosomes during mitosis is essential for the maintenanceof life and achieved by the mitotic spindle, which is composed of two microtubule(MT) asters anchored at the spindle poles. The spindle contains three types of MTsclassified according to their function [1]: astral MTs interact with the cell membrane toposition the spindle poles, interpolar MTs interact with MTs from the opposite pole tomaintain spindle length, and, finally, kinetochore MTs link to the chromosomes via thekinetochores at the centromere and can apply pulling forces via the linkage. The MT-kinetochore bond is a catch bond [2], i.e., tightening under tension but the molecularnature of the MT-kinetochore link is not exactly known and a complete mechanisticunderstanding of the catch bond is missing [3, 4] but probably involves Aurora B [5];the Ndc80 complexes and Dam1 (in yeast) are believed to play a major role in theMT-kinetochore link. One function of the spindle is to align the chromosomes in themetaphase plate at the spindle equator. It has been observed in several vertebrate cellsthat chromosomes do not rest during metaphase but exhibit oscillations along the poleto pole axis known as directional instability [6, 7, 8, 9, 10, 11, 12], whereas in Drosophilaembryos and Xenopus eggs a directional instability does not occur [13, 14]. If present,these oscillations are stochastic and on the time scale of minutes, i.e., on a much largertime scale than the dynamic instability of single MTs. Both single kinetochores andthe inter-kinetochore distance oscillate; inter-kinetochore or breathing oscillations occurwith twice the frequency of single kinetochore oscillations [11].A quantitative understanding of the underlying mechanics of the MT-kinetochore-chromosome system is still lacking. In the past, several theoretical models have beenproposed that reproduce chromosome oscillations [15, 16, 17, 18, 19, 20, 21]. (see table1 and reviews [22, 23]) These models have in common that they simplify to a quasi-one-dimensional geometry and contain two ensembles of MTs growing from the twospindle poles that connect to one chromosome that is represented by two kinetochoresconnected by a spring (the cohesin bond). Kinetochores follow overdamped motion[16, 17, 18, 19, 20] or are assumed to reach force balance instantaneously under theinfluence of MT depolymerization and polymerization forces (because the friction forceis small) [15, 21].Several MT depolymerization and polymerization forces are included into themodels. The models neglect explicit spindle pole dynamics but possibly include polewardMT flux [16, 20], which describes a constant flux of tubulin from the plus-ends towardsthe spindle pole and is probably driven by plus-end directed kinesin-5 motors pushingoverlapping antiparallel interpolar MTs apart and kinesin-13 proteins that depolymerizeMTs at the centrosome [24]. The main poleward forces on kinetochores are generatedby depolymerization of MTs which builds up and transmits a poleward force via theMT-kinetochore link. Only in the model of Civelekoglu-Scholey et al. [16] the mainpoleward force is generated by MT depolymerization motors at the spindle poles. Inorder to be able to exert poleward pulling forces the MT-kinetochore bond needs to istability and oscillations in cooperative microtubule and kinetochore dynamics et al. [16] proposed a model in which MTs and kinetochores are linked by motor istability and oscillations in cooperative microtubule and kinetochore dynamics et al. [21] take into account this observationby dividing each MT ensemble into a growing and a shrinking sub-ensemble, but alsomake the strong assumption of equal force sharing between the MTs within each sub-ensemble. All other models allow for individual MT dynamics and for different forcesbetween MTs depending on the distances of MTs to the kinetochore.The main mechanism for oscillations differs between models depending on the mainantagonistic AP-directed force that switches a depolymerizing P-directed ensemble backinto AP-directed polymerization. Switching can be triggered by the AP-directed forcethat the other ensemble can exert via the cohesin spring during depolymerization andby AP-directed PEFs if MT catastrophes are suppressed or rescues promoted undertension. In the model by Joglekar and Hunt [15] AP-directed PEFs are essential forswitching. Civelekoglu-Scholey et al. [16] proposed a model in which force is transmittedby motor proteins. By variation of the model parameters they were able to reproducea wide range of chromosome behavior observed in different cell types. In this model,a depolymerizing P-directed ensemble switches because tension in the cohesin springand PEFs promote rescue events. A modified model [19] uses viscoelastic catch bondsand accounts for the observation that in PtK1 cells only chromosomes in the center ofthe metaphase plate exhibit directional instability [11]. They explain this dichotomywith different distributions of PEFs at the center and the periphery of the metaphaseplate. In the model by Shtylla and Keener [17, 18] MT catastrophe-like events are onlytriggered by a chemical feedback such that kinetochore oscillations become coupled tooscillations of the chemical negative feedback system: AP-directed MT polymerizationexerts pushing forces onto the kinetochore but triggers switching into a depolymerizingstate, and MT depolymerization exerts P-directed pulling forces and triggers switchingback into a polymerizing state.Whereas in [15, 16, 19] AP-directed PEFs are present and in the model by Joglekarand Hunt [15] also essential for realistic kinetochore oscillations, Banigan et al. [20]presented a minimal model with simple elastic linkers and neglecting PEFs. Referring istability and oscillations in cooperative microtubule and kinetochore dynamics et al. [21], which aimsto describe kinetochore dynamics in fission yeast, does not rely on PEFs. It uses apermanent MT-kinetochore bond and oscillations result from the interplay between MTdepolymerization and polymerization forces via force-dependence in MT dynamics; alsoin this model MTs can exert pushing forces. Moreover, the model makes the strongassumption of equal force sharing between all growing or shrinking MTs attached to akinetochore. The model also includes kinesin-8 motors that enhance the catastropherate and have a centering effect on the chromosome positions.
Table 1.
Overview of assumptions of models exhibiting stochastic chromosomeoscillations. In the referred sections we discuss how poleward flux, PEFs and theabsence of pushing forces affect kinetochore dynamics in the model used for this work.linker catch equal force-dep. MT pole-Ref. (year) model bonds force MT pushing PEFs wardsharing rescue/cat. forces MT fluxJoglekar [15] (2002) Hill sleeve no no no yes noCivelekoglu [16] (2006) motor no no yes yes yes yesCivelekoglu [19] (2013) viscoelastic yes no yes no yes noShtylla [17, 18] (2010) Hill sleeve yes no yes yes noBanigan [20] (2015) elastic yes no yes yes no yesKlemm [21] (2018) permanent yes yes yes no nothis work elastic yes no yes sec. 8 sec. 7 sec. 6
In all Refs. [15, 16, 17, 18, 19, 20, 21] a sufficient set of ingredients is given for therespective model to exhibit oscillations including a specific choice of parameter values. Itis much harder to give necessary conditions and parameter ranges for oscillations, whichmeans to obtain quantitative bounds on model parameters, than to give a sufficient setof conditions. This is the aim of the present paper within a model that starts from theminimal model by Banigan et al. and generalizes this model in several respects in latersections, see table 1. In this way we discuss the complete inventory of possible forcesacting on the kinetochore and its influence on oscillations.It is also difficult to trace the actual mechanism leading to oscillations. An essentialpart in our quantitative analysis is a mean-field solution of the one-sided minimalmodel of Banigan et al. [20], where a single kinetochore under force is connectedto one or several MTs that experience length-dependent individual loads and feature istability and oscillations in cooperative microtubule and kinetochore dynamics >
16 pN µ m − ; also if MT catastrophes are induced upon reaching the kinetochore,we find oscillations in a similar range of linker stiffnesses. These constraints provideuseful additional information on MT-kinetochore linkers whose molecular nature is notcompletely unraveled up to now.
2. Mitotic spindle model
We use a one-dimensional model of the mitotic spindle (figure 1(a)), similar to the modelfrom [20]. The x -coordinate specifies positions along the one-dimensional model, and we istability and oscillations in cooperative microtubule and kinetochore dynamics x = 0 to be the spindle equator. The spindle model contains a single chromosomerepresented by two kinetochores, which are linked by a spring with stiffness c k andrest length d . Two centrosomes margin the spindle at ± x c . From each centrosome aconstant number M of MTs emerges with their plus ends directed towards the spindleequator. Each MT exhibits dynamic instability [25] and attaches to and detachesfrom the corresponding kinetochore stochastically. Attached MTs are connected tothe kinetochore by a linker, which we model as Hookean polymeric spring with stiffness c and zero rest length. This spring exerts a force F mk = − c ( x m − X k ) on each MT,and each MT exerts a counter force − F mk on the kinetochore, where X k and x m arekinetochore and MT position. c cc k centrosomekinetochore centromerecohesin bondfree MTattached MT xx c − x c X k , l X k , r x m , r ,i x m , l ,i xc F ext x i (a) (b) Figure 1.
One-dimensional model of the mitotic spindle. (a) Two-sided model: M MTs arise from each centrosome and can attach to / detach from the correspondingkinetochore. (b) One-sided model: Left half of two-sided model. The cohesin bondis replaced by the external force F ext . MTs are not confined by a centrosome andpermanently attached to the kinetochore. MT-kinetochore distances x i = x m ,i − X k are the only relevant coordinates. In the following we define all MT parameters for MTs in the left half of the spindlemodel; for MTs in the right half position velocities v and forces F have opposite signs.In the left half, tensile forces on the MT-kinetochore link arise for X k > x m and pull theMT in the positive x -direction, F mk >
0. In [2], the velocities of MT growth v m+ andshrinkage v m − as well as the rates of catastrophe ω c , rescue ω r and detachment ω d ± havebeen measured while MTs were attached to reconstituted yeast kinetochores. They canall be described by an exponential dependence on the force F mk that acts on the MTplus end: v m ± = v ± exp (cid:18) F mk F ± (cid:19) , ω i = ω i exp (cid:18) F mk F i (cid:19) , (1)(for i = r , c , d+ , d − ) with F + , F r , F d+ > F − , F c , F d − < F mk >
0) enhances growth velocity, rescue and detachment ofa growing MT, while it suppresses shrinking velocity, catastrophe and detachment of ashrinking MT (note that we use signed velocities throughout the paper, i.e., v m − < v m+ > istability and oscillations in cooperative microtubule and kinetochore dynamics ω a = ω exp (cid:18) c ( X k − x m ) k B T (cid:19) , (2)according to the MT-kinetochore linker spring energy.The kinetochore motion is described by an overdamped dynamics, v k ≡ ˙ X k = 1 γ ( F kk + F km ) , (3)with the friction coefficient γ , and the forces F kk and F km = − P att . MTs F mk originatingfrom the cohesin bond between kinetochores and the MT-kinetochore linkers of allattached MTs, respectively.We perform simulations of the model by integrating the deterministic equations ofmotion for MTs ( ˙ x m , i = v m ± ,i for i = 1 , ..., M ) and kinetochores (eq. (3)) and includestochastic switching events between growth and shrinking as well as for attachmentand detachment to the kinetochore for each MT. For integration we employ an Euleralgorithm with a fixed time step ∆ t ≤ − s which is small enough to ensure ω i ∆ t ≪ Table 2.
Model parameters.
Transition parameters ω i ω i (s − ) F i (pN)catastrophe ω c . − . ω r .
024 6 . ω d+ .
000 11 3 . ω d − . − . ω a . Velocity parameters v m ± v ± (nms − ) F ± (pN)growth v m+ . . v m − − . − . Other parameters
Symbol Valuecohesin bond stiffness c k
20 pN µ m − estimatedcohesin bond rest length d µ m [7]centrosome position x c . µ m [10]friction coefficient γ µ m − estimatedthermal energy k B T We start with the investigation of the minimal model, i.e. neglecting polewardflux and PEFs and using the same simple spring model for the MT-kinetochore linkeras Banigan et al. where the MT plus ends are able to “overtake” the kinetochore( x m > X k , again for MTs in the left half of the spindle) and thereby exert pushingforces F km > istability and oscillations in cooperative microtubule and kinetochore dynamics v m ± . PEFs, which push the kinetochore away from the pole [27], will beincluded in a second step as external forces, which depend on the absolute positions ofthe kinetochores. Finally, we will take account of the hypothesis that MTs are not ableto apply pushing forces on the kinetochore [7, 26] by modifying the model such that thegrowth of a MT is stalled or that the MT undergoes a catastrophe when it reaches thekinetochore.At the centrosome, MTs are confined: It is reasonable to assume that they undergoa forced rescue and detach from the kinetochore if they shrink to zero length. If themean distance of MTs from the spindle equator is sufficiently small, |h x m i| ≪ | x c | , wecan also consider the MTs as unconfined ( | x c | → ∞ ). Then both MT and kinetochoredynamics solely depend on their relative distances and not on absolute positions, whichsimplifies the analysis.
3. Mean-field theory for bistability in the one-sided model
We first examine the one-sided model of Banigan et al. [20], which only consists ofthe left half of the two-sided model with an external force applied to the kinetochore(figure 1(b)). In simulations of this one-sided spindle model, kinetochore movementexhibits bistable behavior as a function of the applied force, i.e., within a certain forcerange there are two metastable states for the same external force: In one state the MTspredominantly grow and the kinetochore velocity is positive while in the other statethe kinetochore has a negative velocity as a consequence of mainly shrinking MTs. Itdepends on the history which of these two states is assumed: When the system entersthe bistable area in consequence of a force change, the kinetochore velocity will maintainits direction (following its current metastable branch) until the force is sufficiently largethat the system leaves the bistable area again (the current metastable branch becomesunstable). Later we will show that this hysteresis of the one-sided model can explainstochastic chromosome oscillations in metaphase if two one-sided models are coupled inthe full two-sided model.In the following, we present a Fokker-Planck mean-field approach that lets usderive bistability analytically and obtain constraints for the occurrence of bistability.We obtain a system of Fokker-Planck equations (FPEs) for the M MT-kinetochoredistances x i ≡ x m ,i − X k ( i = 1 , ..., M ) and decouple the MT dynamics in a mean-fieldapproximation, which neglects kinetochore velocity fluctuations.We make two assumptions. First we assume that all M MTs are always attachedto the kinetochore. Because the MT-kinetochore links are catch bonds this assumptionis equivalent to assuming that these links are predominantly under tension. We willcheck below by comparison with numerical simulations to what extent this assumptioncan be justified. Secondly, we neglect that MTs are confined by a centrosome. Then, istability and oscillations in cooperative microtubule and kinetochore dynamics x i , which measure the extension of the i -th linker.The MTs are coupled because they attach to the same kinetochore: each MTexperiences a force F mk ,i = − cx i from the elastic linker to the kinetochore, which isunder tension (compression) for x i < x i > F km = c P i x i . Therefore, the kinetochore velocity v k is a stochasticvariable depending on all distances x i , on the one hand, but determines the velocities˙ x i = v m ± ( x i ) − v k of MTs relative to the kinetochores, on the other hand. The equationscan be decoupled to a good approximation because the one-sided system assumes asteady state with an approximately stationary kinetochore velocity v k after a short time(rather than, for example, a cooperative oscillation as for an MT ensemble pushingagainst an elastic barrier [38]). In our mean-field approximation we then assume aconstant kinetochore velocity v k ≡ h v k i and neglect all stochastic fluctuations aroundthis mean. This mean value is determined by the mean linker extension h x i via v k = 1 γ ( F ext + cM h x i ) . (4)Fluctuations around the mean value are caused by fluctuations of the force F km = c P i x i around its mean h F km i = M c h x i , which become small for large M (following the centrallimit theorem).If v k is no longer a stochastic variable, the dynamics of the MT-kinetochoreextensions x i decouple. Then, the probability distribution for the M extensions x i factorizes into M identical factors p ± ( x i , t ), where p ± ( x, t ) are the probabilities to findone particular MT in the growing (+) or shrinking ( − ) state with a MT-kinetochorelinker extensions x . We can derive two FPEs for the dynamical evolution of p ± ( x, t ), ∂ t p + ( x, t ) = − ω c ( x ) p + ( x, t ) + ω r ( x ) p − ( x, t ) − ∂ x ( v + ( x ) p + ( x, t )) , (5) ∂ t p − ( x, t ) = ω c ( x ) p + ( x, t ) − ω r ( x ) p − ( x, t ) − ∂ x ( v − ( x ) p − ( x, t )) , (6)where v ± ( x ) denotes the relative velocity of MT and kinetochore, v ± ( x ) ≡ v m ± ( x ) − v k = v ± exp (cid:18) − cxF ± (cid:19) − v k , (7)and ω c ,r ( x ) = ω ,r exp ( − cx/F c ,r ). The velocity v k is no longer stochastic but self-consistently determined by (4). We note that these FPEs are equivalent to single MTFPEs with position-dependent velocities, catastrophe and rescue rates [39, 40, 41, 42].We will now obtain the force-velocity relation of the whole MT ensemble by firstsolving the FPEs (5) and (6) in the stationary state ∂ t p ± ( x, t ) = 0 and then calculatingthe mean linker extension h x i for given kinetochore velocity v k using the stationarydistribution p ± ( x ). The external force that is necessary to move the kinetochore withvelocity v k then follows from (4), F ext = γv k − cM h x i ( v k ) . (8)The MT-kinetochore distance x is limited to a maximal or a minimal value x max or x min for a given kinetochore velocity v k > <
0, respectively, see table 3. These istability and oscillations in cooperative microtubule and kinetochore dynamics x has adjusted: First we consider v k < x > x <
0) because the relative velocityis negative, ˙ x = v − ( x ) <
0. The MT-kinetochore distance x continues to decreaseuntil ˙ x = v − ( x min ) = 0 in (7), where the shrinking velocity of the MTs is the same asthe prescribed kinetochore velocity ( v m , − = v k ). Further shrinking to x < x min is notpossible but distances x > x min can always be reached if MTs are rescued. If v k < x , as the relative velocity˙ x = v + ( x ) is always positive; x starts to grow into the compressive regime x > v + ( x ) ≈ − v k > v k > x grows until ˙ x = v + ( x max ) = 0, and smallerdistances can be reached by catastrophe but there is no lower bound on x for shrinkingMTs. Linker extensions x max ( x min ) are reached as stationary states if catastrophes(rescues) are suppressed (for example, because of large forces), such that MTs can grow(shrink) for sufficiently long times. If the external force F ext is prescribed rather than akinetochore velocity, all MTs reach a stationary state with the same velocity ˜ v ± givenby (8), F ext = γ ˜ v ± − cM x max , min . In this stationary state, both MT-tips and kinetochoremove with the same velocity˜ v ± ≡ M F ± γ W (cid:18) γv ± M F ± exp (cid:18) F ext M F ± (cid:19)(cid:19) , (9)where W () denotes the Lambert-W function (defined by x = W ( x ) e W ( x ) ). Table 3.
Maximal or a minimal value x max or x min of the stationary linker extensiondistribution p ( x ) from conditions v − ( x min ) = 0 and v + ( x max ) = 0. The distance x min ( x max ) is a function of the prescribed kinetochore velocity v k and has to be specifiedseparately depending on the direction of v k ; x min ( x max ) is approached if the MTsshrink (grow) for a sufficiently long time.MT shrinks MT grows v k > v − ( x ) < − v k always v + ( x ) > x < x max x min = −∞ x max = ( F + /c ) ln (cid:0) v /v k (cid:1) v k < v − ( x ) < x > x min v + ( x ) > v k always x min = ( F − /c ) ln (cid:0) v − /v k (cid:1) x max = ∞ v k = 0 v − ( x ) < v + ( x ) > x min = −∞ x max = ∞ In the complete absence of stochastic switching between growth and shrinkingby catastrophes and rescues, the MT ensemble reaches stationary states with peakeddistributions p + ( x ) ∝ δ ( x max − x ) and p − ( x ) ∝ δ ( x − x min ). Stochastic switchingshifts and broadens these peaks, and the FPEs (5) and (6) lead to a distribution istability and oscillations in cooperative microtubule and kinetochore dynamics p ± ( x, t ) of linker extensions x in the growing and shrinking states with statistical weight p ± ( x, t ) > x min ≤ x ≤ x max . At the boundaries x min and x max ofthis interval, the probability current density j ( x, t ) ≡ v + ( x, t ) p + ( x, t ) + v − ( x, t ) p − ( x, t ) (10)has to vanish. Furthermore, in any stationary state ( ∂ t p ± ( x, t ) = 0) the current densityis homogeneous, as can be seen by summing up the FPEs (5) and (6):0 = ∂ x ( v + ( x ) p + ( x ) + v − ( x ) p − ( x )) = ∂ x j ( x ) . (11)Together with j = 0 at the boundaries this implies that j = 0 everywhere in a steadystate. The resulting relation v + ( x ) p + ( x ) = − v − ( x ) p − ( x ) can be used to reduce thestationary FPEs to a single ordinary differential equation with the solution [41] p ± ( x ) = ±N v ± ( x ) exp (cid:18) − Z (cid:18) ω c ( x ) v + ( x ) + ω r ( x ) v − ( x ) (cid:19) d x (cid:19) (12)for the stationary distribution of linker extensions x in the growing and shrinking states.The normalization constant N must be chosen so that the overall probability density p ( x ) ≡ p + ( x ) + p − ( x ) satisfies R x max x min p ( x )d x = 1. Obviously, p ± ( x ) = 0 for x > x max and x < x min . The stationary probability densities p ± ( x ) from (12) can then be used tocalculate the mean distance h x i as a function of the kinetochore velocity v k , which entersthrough (7) for v ± ( x ). The integral in the exponent in (12) as well as the normalizationcan be evaluated numerically to obtain an explicit h x i ( v k )-relation, which is shown infigure 2(a).At this point it should be noticed that in the mean-field theory the h x i ( v k )-relationis independent of the MT number M . Therefore, we call it master curve henceforth. Infigure 2(a) we compare the mean-field theory result to stochastic simulations and findthat the mean-field approach becomes exact in the limit of large M , where fluctuationsin the kinetochore velocity around its mean in (4) can be neglected.The master curve is a central result and will be the basis for all further discussion.Together with the force-balance (8) on the kinetochore, the master curve will give theforce-velocity relation for the MT-kinetochore system. A positive slope of the mastercurve, as it can occur for small v k ≈ δv k > δ h x i >
0. According to the force-balance(8), a compression δ h x i > δv k > h x i in this unstable regime. This is confirmed by stochastic simulation results in figure2(a), which show that the unstable states are only assumed transiently for very shorttimes. Therefore, occurrence of a positive slope in the master curve in figure 2(a) is theessential feature that will give rise to bistability in the one-sided model and, finally, tooscillations in the full two-sided model.Now we want to trace the origin of this instability for small v k ≈
0. If the MTsare growing (shrinking) for a long time, all linker extensions assume the stationary istability and oscillations in cooperative microtubule and kinetochore dynamics − − − v k (cid:0) µ m s − (cid:1) − − h x i ( µ m ) (a) .
003 0 . − . − . . M = 5 M = 20 M = 50 M = 200 FPE ˜ v + ˜ v − − − − − v k (cid:0) µ m s − (cid:1) − . . − . . − − F ext /M (pN) − − − − v k (cid:0) µ m s − (cid:1) − . . − − F ext /M (pN) − . . (b) Figure 2.
Mean-field results compared to stochastic simulations of the one-sidedmodel. (a) The master curve h x i ( v k ) from the mean-field approach (red line) agreeswith simulation results for different MT-numbers M = 5 , , , x min , max ( v k ) from table 3. We run simulations with constant external forces andaverage over 80 simulations for each force. Initially, the MT-kinetochore distance iseither x min or x max while all MTs grow or shrink with velocity ˜ v ± , respectively. Thesystem then enters a (meta-)stable state, in which we measure the mean kinetochorevelocity and MT-kinetochore distances. The marker size depicts the time the systemrests in this state on average, which is a measure for its stability (maximum marker sizecorresponds to t rest ≥ h x i ( v k )-relation is independentof the MT-number M . (b) Resulting force-velocity relations for different MT-numbers M = 5 , , , v k ≈ ˜ v ± given by(9). We used a linker stiffness of c = 20 pN µ m − both in (a) and (b). values x ≈ x max ( v k ) ( x ≈ x min ( v k )) from table 3, where the MT-velocity adjusts tothe kinetochore velocity, v k ≈ v m ± ( x ). If the kinetochore velocity increases in thesestates by a fluctuation (i.e., δv k > δx < x max ( v k ) (for v k >
0) and x min ( v k ) (for v k < x max and x min , but we expect also the mastercurve h x i ( v k ) to have a negative slope for a wide range of velocities v k . Figure 2(a) showsthat this is actually the case for kinetochore velocities v k around the force-free growth orshrinking velocities v ± of the MTs, i.e., if the imposed kinetochore velocity v k roughly“matches” the force-free growing or shrinking MT velocity. Then a small mismatchcan be accommodated by small linker extensions x , which do not dramatically increasefluctuations by triggering catastrophe or rescue events.The situation changes for small negative or small positive values of the kinetochorevelocity around v k ≈
0. For v k .
0, MT-kinetochore linkers develop logarithmically istability and oscillations in cooperative microtubule and kinetochore dynamics x min (see table 3) corresponding to a slow kinetochoretrailing fast shrinking MTs that strongly stretch the linker. Likewise, for v k & x max corresponding to a slow kinetochore trailing fast growing MTs that strongly compressthe linker. Around v k ≈
0, the system has to switch from large negative x to largepositive x because the resulting tensile force F mk = − cx on the shrinking MT willdestabilize the shrinking state and give rise to MT rescue at least for x < − F r /c .Therefore, also the mean value h x i switches from negative to positive valuesresulting in a positive slope of the master curve if the stationary distributions p − ( x )and p + ( x ) remain sufficiently peaked around the linker extensions x min and x max , alsoin the presence of fluctuations by catastrophes and rescues. In the supplementarymaterial, we show that the stationary distributions assume a power-law behavior p + ( x ) ∝ ( x max − x ) α + [ p − ( x ) ∝ ( x − x min ) α − ] around x max [ x min ] for v k > v k < α ± that depend on the MT-kinetochore stiffness c as α ± + 1 ∝ /c inthe presence of fluctuations. It follows that distributions are peaked (i.e., have a largekurtosis) and bistability emerges if the MT-kinetochore linker stiffness c is sufficientlylarge such that deviations of the MT velocity from the kinetochore velocity becomesuppressed by strong spring forces. This is one of our main results. We also find that α ± + 1 ∝ ( | v k /v ± | ) − −| F ± /F c , r | such that the distributions become also peaked around x min , max in the limit of large velocities | v k | . Then the velocity approaches v k ≈ ˜ v ± ( F ext )for a prescribed external force such that ˜ v ± from (9) represents the large velocity andlarge force limit of the force-velocity relation of the kinetochore (see figure 2(b)).In the unstable regime around v k ≈
0, the linker length distribution p ( x ) is typicallybroad without pronounced peaks and has a minimal kurtosis (as a function of v k ) in thepresence of catastrophe and rescue fluctuations. In this regime the system assumes astate with a heterogeneous stationary distribution of growing and shrinking MTs, i.e.,the total probabilities to grow or shrink become comparable, R p + ( x )d x ∼ R p − ( x )d x .If the kinetochore velocity is increased, δv k >
0, the system does not react by δx < switching
MTs from the shrinking to the growing state (on average), which then evenallows to relax the average linker tension.Using the force-balance (8) on the kinetochore, the master curve is converted to aforce-velocity relation for the MT-kinetochore system; the results are shown in figure2(b). The bistability in the master curve directly translates to a bistability in the force-velocity relation of the MT ensemble, and we obtain a regime with three branches ofpossible velocities for the same external force. The upper and the lower branches agreewith our simulation results and previous simulation results in [20], and our mean-fieldresults become exact in the limit of large M , see figure 2(b). These branches correspondto the two stable parts of the master curve with negative slope, that are found forkinetochore velocities v k roughly matching the force-free growth or shrinking velocities v ± of the MTs. The mid branch corresponds to the part of the master curve withpositive slope, where the system is unstable. Also figure 2(b) demonstrates that this istability and oscillations in cooperative microtubule and kinetochore dynamics identical extensions x i ≈ x and all attached MTs are in the samestate (growing or shrinking), is exact for a single MT ( M = 1) by definition but notsufficient to obtain a bistable force-velocity relation for MT ensembles ( M >
1) (seesupplementary material). The same assumption of identical MT positions has alreadybeen used to study an ensemble of MTs that are connected to the same kinetochorevia Hill sleeve like linkers [17, 29]. The model of Klemm et al. [21] divides each MTensemble into a growing and a shrinking sub-ensemble, and assumes equal load sharingonly between MTs within each sub-ensemble. We can show that, together with a force-sensitive rescue force, this is sufficient to obtain a bistable force-velocity relation in acorresponding one-sided model.
4. Bistability gives rise to oscillations in the two-sided model
As already worked out by Banigan et al. [20], the bistability in the force-velocity relationof the one-sided MT ensemble can be considered to be the cause for stochastic oscillationsin the two-sided model. Each ensemble can be either on the lower branch of the force-velocity relation, where it mainly depolymerizes and exerts a P-directed pulling force( v k <
0) or on the upper branch where it mainly polymerizes and exerts an AP-directedpushing force ( v k > F kk = c k ( X k , r − X k , l − d ) of the cohesin bond in the full model with astiffness c k and rest length d , see table 2. Since the cohesin force is a linear functionof the inter-kinetochore distance, the force-velocity relation can be treated as distance-velocity (phase space) diagram for the two kinetochores (see figure 3(a)), where bothkinetochores move as points on the force-velocity relation. The cohesin bond alwaysaffects the two kinetochores in the same way because action equals reaction: if thecohesin spring is stretched, both kinetochores are pulled away from their pole (AP), ifit is compressed, both kinetochores are pushed polewards (P). Thus, the kinetochoresalways have the same position on the F kk -axis in the F kk - v k diagram in figure 3(a), if F kk on the horizontal axis is defined as the force on the kinetochore in AP-direction (i.e., F kk , l ≡ F kk and F kk , r ≡ − F kk for the left/right kinetochore). Likewise, we define v k onthe vertical axis as the velocity in AP-direction (i.e., v k , l ≡ ˙ X k , l and v k , r ≡ − ˙ X k , r for theleft/right kinetochore). The upper/lower stable branch of the force-velocity relation isdenoted by v ± k ( F kk ). Typically, a kinetochore on the upper (lower) branch has v +k > v − k <
0) and, thus moves in AP-(P-)direction. Using F kk = c k ( X k , r − X k , l − d ) forthe spring force, we find ˙ F kk = − c k ( v k , r + v k , l ), i.e., kinetochores move with the sum oftheir AP-velocities along the force-velocity curve in the F kk - v k diagram.Oscillations arise from the two kinetochores moving through the hysteresis loop ofthe bistable force-velocity relation as described in figure 3(a). Three states are possible(see figure 3(b)). In state 0, both kinetochores move in AP-direction (i.e., in oppositedirections) relaxing the F kk -force from the cohesin bond, i.e., on the upper branch and istability and oscillations in cooperative microtubule and kinetochore dynamics − F kk (pN) − − − v k (cid:0) µ m s − (cid:1)
01 22 33 ′ ′ ′ ′ ′ v + k , AP v − k , P F − F min F max v l v r ′ F kk ′ ′ (b) − − X k ( µ m ) X k,r X k,l t (s) ∆ X k ( µ m ) ′ ′ ′ Figure 3.
Bistability gives rise to oscillations in the two-sided model. (a,b) Differentstates of sister kinetochore motion can be deduced from the bistability of the force-velocity relation: either both kinetochores are in the upper branch (0) or one is in theupper and the other one in the lower branch (2, 2 ′ ). In the first case, both kinetochoresmove away from their pole (AP) towards each other. Thus, the spring force F kk decreases until it reaches F min . Since the upper branch is not stable anymore below F min , either the left (1) or the right (1 ′ ) kinetochore switches to the lower branch andchanges direction to poleward movement (P). The system is then in state 2 or 2 ′ , whereboth kinetochores move into the same direction: the leading kinetochore P, the trailingkinetochore AP. As P- is much faster than AP-movement (MT shrinking is muchfaster than growth), the inter-kinetochore distance and the spring force are increasing.Above F max only AP-movement is stable, which is why the leading kinetochore changesdirection (3, 3 ′ ) and the system switches to state 0 again. (c) Solution of the equationsof motion (13) for c = 20 pN µ m − and M = 25 with an imposed periodic order ofstates (0 − − − ′ − − ... ). The initial condition is F kk = F max (both kinetochoresat the right end of the upper branch). For an animated version see video S1 in thesupplementary material. to the left in the v k - F kk -diagram with velocity ˙ F kk = − c k v +k <
0. After reaching thelower critical force F min of the hysteresis loop, one of the two kinetochores reversesits direction and switches to the lower branch resulting into states 2 or 2 ′ where onekinetochore continues in AP-direction with v +k > v − k < v k - F kk -diagram,this results in a motion to the right with velocity ˙ F kk = c k ( − v − k − v +k ) > − v − ≫ v , see table 2). Moving onopposite P- and AP-branches increases the kinetochore distance and builds up F kk -forcein the cohesin bond. After reaching the upper critical force F max of the hysteresis loop,it is always the kinetochore on the lower branch moving in P-direction which switchesback and state 0 is reached again. This behavior is in agreement with experimentalresults [11]. The system oscillates by alternating between state 0 and one of the states2 or 2 ′ (which is selected randomly with equal probability).For each of the states 0, 2 and 2 ′ depicted in figure 3(ab) the two branches v ± k = v ± k [ F kk ] provide deterministic equations of motion for the kinetochore positions.Inserting F kk = c k ( X k , r − X k , l − d ) we obtain both kinetochore velocities as functions istability and oscillations in cooperative microtubule and kinetochore dynamics X k , l = v +k [ c k ( X k , r − X k , l − d ) ] > , ˙ X k , r = − v +k [ c k ( X k , r − X k , l − d ) ] < , state 2: ˙ X k , l = v − k [ c k ( X k , r − X k , l − d ) ] < , ˙ X k , r = − v +k [ c k ( X k , r − X k , l − d ) ] < , state 2 ′ : ˙ X k , l = v +k [ c k ( X k , r − X k , l − d ) ] > , ˙ X k , r = − v − k [ c k ( X k , r − X k , l − d ) ] > . (13)Solving these equations gives idealized deterministic trajectories of the sisterkinetochores, when we also assume that the left and the right kinetochore pass the lowerbranch alternately such that the order of states is a periodic sequence 0 − − − ′ − − ... as shown in the example in figure 3(c). Then single kinetochores oscillate with halfthe frequency of inter-kinetochore (breathing) oscillations, just as observed in PtK1cells [11]. Moreover, we can obtain numerical values of the frequencies directly fromthe trajectories. For a MT-kinetochore linker stiffness c = 20 pN µ m − and 20–25 MTsper kinetochore, which is a realistic number for mammalian cells [43], we get periods of206–258 s and 103–129 s for kinetochore and breathing oscillations, respectively. Thesevalues coincide with experimental results of 239 s and 121 s measured in PtK1 cells [11].The calculated trajectories are idealized since they neglect stochastic fluctuationsthat occur in simulations of the two-sided model and have two main effects onthe kinetochore dynamics which already arise in simulations that comply with theassumptions behind the mean-field theory (no confinement ( x c → ∞ ) and permanentbonds ( ω d = 0)): Firstly, the sister kinetochores do not pass the lower branch alternatelybut in random order. Therefore, we observe phases where one kinetochore moves in AP-direction for several periods, while the other one changes its direction periodically butmoves polewards on average (figure 4(a)). Since this does not influence the trajectoryof the inter-kinetochore distance, breathing oscillations still occur in a more or lessregular manner, which allows us to measure their frequencies by Fourier analysis. Wewill show below that additional polar ejection forces suppress this random behaviorand force the kinetochores to pass the lower branch alternately. As a second effectof the stochastic character of the simulation, kinetochores do not change the branchinstantaneously after crossing the critical forces F max or F min . Instead, they tend tomaintain their primary state for a while (figure 4(b)) and follow the metastable statesthat we also observe in the one-sided model (figure 2(b)). Hence, the frequencies wemeasure in the simulations are smaller than those we calculate from the Fokker-Planckmean-field approach (figure 4(c)). The latter effect vanishes in the limit of many MTs(large M ): the switching points approach the theoretical values F max and F min , and thesimulated breathing frequencies converge to our mean-field predictions.So far we have demonstrated that the mean field theory correctly describeskinetochore dynamics in simulations of the unconfined model where we suppressdetachment in order to prevent unattached MTs from shrinking towards infinity. Asshown in figure 5(ab), kinetochore oscillations also survive in simulations of the confined istability and oscillations in cooperative microtubule and kinetochore dynamics (a) − − X k ( µ m ) M = 25 X k , r X k , l t (s) ∆ X k M = 100 t (s) −
25 0 25 50 75 F kk (pN) − − − v k (cid:0) µ m s − (cid:1) M = 25 −
50 0 50 100 150 F kk (pN) M = 100 F kk (pN) M = 500 (c) M − − − − f ( H z ) FPEconfined ω det = 0 unconfined(d)0 2 4 f (mHz) M = 25 f (mHz) M = 100∆ ˜ X k ˜ X k , r Figure 4.
Oscillations in stochastic simulations of the unconfined model comparedto mean-field results. (a) Kinetochore trajectories and breathing oscillations in thetwo-sided model without confinement ( x c → ∞ ) and detachment ( ω d = 0). Thekinetochores behave as described in figure 3 with a random order of states 2 / ′ . Thebreathing oscillations are regular enough to assign a frequency by Fourier analysis, see(d). With less MTs oscillations are more fluctuative. (b) Kinetochore velocity againstcohesin force in simulations of the unconfined two-sided model without detachment(green). For many MTs the velocity follows very precisely the predicted hysteresisfrom the mean-field approach (red). For animated versions see videos S2 ( M = 25)and S3 ( M = 500) in the supplementary material. (c) Double-logarithmic plot offrequencies of breathing oscillations as a function of MT number M : calculated fromthe mean-field approach according to figure 3 (red) and measured in simulations ofthe unconfined (green diamonds) as well as the confined model with detachable catchbonds (blue circles) and with permanent attachment (orange triangles). Confinementbecomes relevant for large MT numbers. In the presence of detachable catch bonds only75 % of the MTs are attached on average, which corresponds to a simple shift of thecurve to lower MT numbers. (d) Trajectories from (a) in Fourier space. While ˜ X k , r hasits maximum at f = 0 due to the random order of states in figure 3, ∆ ˜ X k has a distinctpeak that becomes sharper for large M indicating regular breathing oscillations. Forall simulations the MT-kinetochore linker stiffness was c = 20 pN µ m − . model independently of whether the MTs are able to detach from the kinetochore, i.e.,to rupture the catch bond. However, confinement by the centrosome influences thekinetochore dynamics in the limit of large M : since more MTs exert a higher force onthe kinetochore, it is possible that one of the two sisters gets stuck at the centrosomefor a while (see figure 5(ab)). Hence, the frequencies measured in the confined two-sidedmodel deviate from the frequencies in the unconfined case above M ≈ istability and oscillations in cooperative microtubule and kinetochore dynamics (a)(c) (b)(d) − x c − x c X k ( µ m ) M = 25 X k , l X k , r t (s) ∆ X k t (s) M a tt M attl M attr − x c − x c X k ( µ m) M = 25 M att M = 25 − x c − x c M = 100 t (s)
024 0 2500 5000 7500 10000 t (s) − x c − x c X k ( µ m) M = 100 X k , l X k , r
20 60 100 M att M = 100 M attl M attr Figure 5.
Dynamics in the confined model with detachable MTs. (a) Kinetochorepositions X k and inter-kinetochore distance ∆ X k over time in simulations with a totalnumber of M = 25 and M = 100 MTs per spindle pole. Oscillations as describedin figure 3 are recognizable. With 100 MTs one kinetochore can get stuck to thecentrosome for a while. (b) Distribution of kinetochore positions. The kinetochoresare not aligned to the spindle equator and for M = 100 they are most likely to befound near the centrosomes. (c) Number of attached MTs M att over time. MTs aremore likely to be attached when the correspondent kinetochore is near the centrosomesince the free MTs can reattach to the kinetochore faster in that case. (d) Distributionof M att . On average 75 % of the MTs are attached independently of the total MTnumber M . MTs are more likely to reattach to the kinetochore the closer it is to the centrosome.Moreover, on average, about 75 % of the MTs are attached independently of the totalMT number (see figure 5(cd)). Therefore, the catch bond nature of the link leads toan effective behavior similar to a system without detachment but with less MTs, whichexplains the difference in frequencies between the confined models with and withoutdetachment in figure 4(c). We conclude that detachment does not play a major role forthe occurrence of kinetochore oscillations in cells with many MTs as despite detachmentthere are always enough MTs attached to justify our mean-field approximation. Hence,(periodic) changes in the number of attached MTs as they can be seen in figure 5(c)are rather a passive consequence than an active source of kinetochore oscillations. Thisargumentation may not be tenable, if just a few MTs are attached to a kinetochore, sothat even detachment of a single MT effects the total force acting on the kinetochoresignificantly. Then, detachment can be the primary cause of directional instability asworked out by Gay et al. [44], who modeled the mitotic spindle of fission yeast.Taking into account the results of the last paragraph, we will mainly investigatethe unconfined model with permanently attached MTs in the following sections. Thisprocedure is reasonable as we do not lose any qualitative key features of kinetochore istability and oscillations in cooperative microtubule and kinetochore dynamics et al. [21] divides each MT ensemble into a growing and a shrinking sub-ensemble, and assumes equal load sharing only between MTs within each sub-ensemble.Together with a force-sensitive rescue force, this is sufficient to obtain oscillations.
5. Constraints on linker stiffness and MT number for bistability andoscillations
We already argued above in Sec. 3 that bistability (and thus oscillations) can only emergeif the MT-kinetochore linker is sufficiently stiff. To analyze the influence of the linkerstiffness c and the MT number M on bistability quantitatively, the transformation fromthe master curve to the force-velocity relation is visualized in figure 6(a) as search forthe intersections of the master curve with linear functions h x i = 1 cM ( γv k − F ext ) . (14)In the limit of large M these linear functions have zero slope. Bistable force-velocityrelations with three intersection points are only possible if the master curve has positiveslope for intermediate v k resulting in a maximum and minimum. The extrema of themaster curve vanish, however, in a saddle-node bifurcation if the linker stiffness dropsbelow c bist = 7 .
737 pN µ m − , which is, therefore, a lower bound for the occurrence ofbistability. In the case of finite MT numbers M , bistable force-velocity relations canonly be found if the slope in the inflection point of the master curve exceeds γ/cM (theslope of the linear function (14)). This allows us to quantify a bistable regime in theparameter plane of linker stiffness c and MT number M as shown in figure 6(b). We showed in Sec. 4 that bistability of the one-sided model is a necessary condition foroscillations in the two-sided model. Now we show that bistability in the one-sided modelis, however, not sufficient for oscillations in the full model. If the force-velocity relation istability and oscillations in cooperative microtubule and kinetochore dynamics − − − − v k (cid:0) µ m s − (cid:1) − − h x i ( µ m ) ± ∆ F ext M ≫ µ m − µ m −
10 pN µ m −
20 pN µ m − M = 1 (a) 7.5 c bist c (cid:0) pN µ m − (cid:1) M bistable(b) Figure 6.
Constraints for bistability in the one-sided model. (a) Master curvesfor different linker stiffnesses c and linear functions according to (14). In the limit oflarge M the linear functions have zero slope and bistability occurs if the master curvehas two extrema, which is the case for c > c bist . For finite M bistable solutions arepossible if the linear functions have a smaller slope than the inflection point of themaster curve. (b) Resulting bistable regime in the parameter plane of linker stiffness c and MT number M . is interpreted as phase space diagram for the two kinetochores, kinetochores only switchbranches in the v k - F kk -diagram if their velocity changes its sign at the turning points F min and F max . If this is not the case and one of the two branches crosses v k = 0 (e.g. theright branch for c = 10 pN µ m − in figure 6(a), which transforms to the upper branchof the force-velocity relation), the intersection point is a stable fixed point in the phasespace diagram (see figure 7(a)). At this fixed point kinetochore motion will relax tozero velocity and just exhibit fluctuations around an equilibrium distance instead ofoscillations.As a sufficient condition for oscillations we have to require – besides bistability – astrictly positive velocity in the upper and a strictly negative velocity in the lower branchin the v k - F kk -diagram. Based on this condition we quantify an oscillatory regime in theparameter plane of linker stiffness c and MT number M in figure 8(a). In the limit ofmany MTs the sufficient condition for oscillations can be formulated in terms of themaster curve: the maximum of the master curve has to be located at a positive and theminimum at a negative velocity. This is the case for c > c osc = 15 .
91 pN µ m − , whichis, therefore, a lower bound for the occurrence of oscillations. This constraint on thelinker stiffness for metaphase chromosome oscillations provides additional informationon MT-kinetochore linkers whose molecular nature is not known up to now.Because of stochastic fluctuations, the transition between oscillatory and non-oscillatory regime is not sharp in our simulations. In the non-oscillatory regimekinetochores fluctuate around a fixed point of inter-kinetochore distance, where theupper branch crosses v k = 0. However, these fluctuations can be large enough forthe inter-kinetochore distance to shrink and leave the upper branch on the left side, istability and oscillations in cooperative microtubule and kinetochore dynamics c . c osc F kk v k c < c osc (a) −
14 pN µ m − (c) 0 3000 6000 9000 t (s) − X k ( µ m )
10 pN µ m − t (s) ∆ X k v k F kk (pN) − − v k (cid:0) µ m s − (cid:1)
10 pN µ m − F kk (pN)14 pN µ m − Figure 7.
Kinetochore dynamics in the non-oscillatory regime. (a) Schematicexplanation of kinetochore motion in the non-oscillatory regime based on the force-velocity relation. Where the upper branch crosses zero velocity, inter-kinetochoredistance has a fixed point, around which it fluctuates. With higher linker stiffnesses c the fixed point comes closer to the left turning point F min . When c is just slightlysmaller than c osc , fluctuations can be large enough for the kinetochore distance toleave the upper stable branch. Then, one of the two sister kinetochores passes oncethrough the lower branch. (b,c) This behavior can be observed in simulations. While at c = 10 pN µ m − kinetochores just fluctuate around the fixed point, at c = 14 pN µ m − the kinetochores occasionally pass through the hysteresis loop. Simulations wereperformed with an unconfined system and 100 MTs on each side. c osc c (cid:0) pN µ m − (cid:1) M oscillatory(a) 0 5 10 15 20 25 30 c (cid:0) pN µ m − (cid:1) h ∆ X k i ( µ m ) (b) Figure 8.
Constraints for oscillations in the two-sided model. (a) Oscillatory regimein the parameter plane of linker stiffness c and MT number M . (b) Mean inter-kinetochore distance according to (16) (red) and measured in simulations (blue) with M = 100. Below c osc = 15 .
91 pN µ m − (dashed line) both results match, whereas inthe oscillatory regime mean inter-kinetochore distance diverges from the fixed point,and its standard deviation increases notably. especially for stiffnesses c slightly below c osc . If that happens, one kinetochore passesonce through the lower branch of the force-velocity relation just as in an oscillation. istability and oscillations in cooperative microtubule and kinetochore dynamics c < c osc and c . c osc . Moreover, the force-velocity relations as well asthe kinetochore trajectories measured in corresponding simulations are shown.In the non-oscillatory regime, the fixed point should determine the mean inter-kinetochore distance h ∆ X k i = h X k,r − X k,l i . Solving the FPEs for v k = 0, we computethe (external) force F that has to be applied to one kinetochore to stall its motion: F = γv k − cM h x i = − cM h x i ( v k = 0) . (15)In the two-sided model this force is applied to the kinetochores by the cohesin bondat the fixed point. With F kk = c k (∆ X k − d ) we compute the corresponding meaninter-kinetochore distance: h ∆ X k i = F c k + d = − cMc k h x i ( v k = 0) + d . (16)Figure 8(b) shows that simulations agree with this result in the non-oscillatory regime.At c osc the transition to the oscillatory regime can be recognized, where the mean inter-kinetochore distance deviates from the fixed point (16). Moreover, the variance of ∆ X k increases significantly at c osc due to the transition to the oscillatory regime.In order to provide an overview and to make orientation easier for the reader, wesummarize in figure 9 where the stochastic simulations from the last three sections andthe master curves in figure 6(a) are located in the parameter plane of linker stiffness c and MT number M , and which regime they are part of. c bist c osc
20 25 30 c (cid:0) pN µ m − (cid:1) M bistable oscillatory figure 2figure 4(ab)figure 4(c)figure 5figure 6(a)figure 7figure 8(b) Figure 9.
Locations in c - M -parameter plane of the master curves from figure 6(a)and the simulations from figures 2, 4, 5, 7 and 8.
6. Poleward microtubule flux suppresses oscillations
An effect we have not included so far is poleward MT flux, which was observed inseveral metazoan cells (table 4). It describes the constant flux of tubulin from the plus-ends towards the spindle pole and is probably driven by plus-end directed kinesin-5 istability and oscillations in cooperative microtubule and kinetochore dynamics
Table 4.
Metaphase poleward flux velocities v f and occurrence of directionalinstability. For a more detailed review of poleward flux measurements see [45]Cell type v f (nm s − ) Directional instabilityLLC-PK1 (porcine) 8 . . . . Poleward flux can be easily included in our model by subtracting a constant fluxvelocity v f from the MT velocity. Then, the relative MT-kinetochore velocities (7)become v ± ( x ) = v ± exp (cid:18) − cxF ± (cid:19) − v f − v k . (17)Hence, the flux velocity can be treated as an offset to the constant kinetochore velocityin the solution of the stationary FPEs. The final effect is a shift of both the mastercurves and the force-velocity relations by v f towards smaller kinetochore velocities v k as shown in figure 10(a). If the shift is so large that the left turning point F min ofthe force-velocity hysteresis is located at a negative velocity, poleward flux suppressesdirectional instability because a fixed point emerges, and we expect similar behavior asfor intermediate linker stiffnesses in the previous section (see figure 7). In the limit ofmany MTs, the maximum flux velocity that still allows directional instability is given bythe velocity in the maximum of the master curve, which provides the boundary of theoscillatory regime in the parameter plane of linker stiffness c and poleward flux velocity v f (figure 10(b)). Phase space diagrams (figure 10(c)) and kinetochore trajectories(figure 10(d)) from simulations with appropriate flux velocities confirm our argumentsexhibiting similar behavior as for intermediate linker stiffnesses in figure 7. For smallflux velocities the boundary of the oscillatory regime in figure 10(b) approaches ourabove result c osc = 15 .
91 pN µ m − . For increasing flux velocities the oscillatory regimeshrinks, and its boundary has a maximum at c ≈
50 pN µ m − with v f ≈ .
11 nm s − .We conclude that kinetochore oscillations can be suppressed by moderate flux velocitiesindependently of the linker stiffness.Our theory also agrees with and explains simulation results in [20], where, for largeflux velocities, suppression of kinetochore oscillations were observed but at the same timemaintenance of bistability. Moreover, our results explain the experimentally observed istability and oscillations in cooperative microtubule and kinetochore dynamics (d)(c)0 50 100 150 F kk (pN) − − − v k (cid:0) µ m s − (cid:1) − F kk (pN)4 nm s − − − − X k ( µ m ) − t (s) ∆ x k − − t (s) F ext v k v f > v min F ext (a) v f < v min fixed pointno fixed point10 c osc
100 1000 c (cid:0) pN µ m − (cid:1) v f (cid:0) n m s − (cid:1) oscillatory(b) v min Figure 10.
Poleward flux suppresses oscillations. (a) Due to (17), the force-velocityrelation is shifted by the amount of the flux velocity v f towards smaller kinetochorevelocities. If the flux is slower than the kinetochore velocity v min in the left turningpoint F min , the kinetochores still oscillate. For larger flux velocities, a fixed pointarises on the upper branch and the kinetochores behave as described in figure 7. (b)Oscillatory regime in the parameter plane of c and v f in the limit of many MTs.Fast poleward flux suppresses kinetochore oscillations for arbitrary linker stiffnesses c .(b,c) Phase space diagrams and MT trajectories from simulations of the unconfinedtwo-sided model with c = 20 pN µ m − and M = 100. While at v f = 2 nm s − thesystem is still in the oscillatory regime, where hysteresis is recognizable in phase space,at v f = 4 nm s − kinetochores show fluctuative motion as described in figure 7. correlation between flux velocity and directional instability. Kinetochore oscillationshave been observed in the mitotic vertebrate cells listed in table 4 (LLC-PK1, PtK1/2,newt lung, U2OS) which have poleward flux velocities not exceeding 10 nm s − , whereasin the mitosis of a Drosophila embryo as well as in meiosis of a Xenopus egg, whereflux velocities are three to four times higher, chromosomes do not exhibit directionalinstability.
7. Polar ejection forces provide an alternating oscillation pattern andchromosome alignment at the spindle equator
So far, we have not included polar ejection forces (PEFs). They originate from non-kinetochore MTs interacting with the chromosome arms and pushing them therebytowards the spindle equator, either through collisions with the chromosome arms or viachromokinesins [27], and provide additional pushing forces on kinetochores. Therefore,they can be included into the model by adding forces F PEF , r ( X k , r ) and F PEF , l ( X k , l ) actingon kinetochores, which depend on the absolute position of the kinetochores [19]. Dueto the exponential length distribution of free MTs as well as the spherical geometry of istability and oscillations in cooperative microtubule and kinetochore dynamics x = 0), where oppositePEFs compensate each other. This assumption is supported by the monotonic PEFdistribution that has been measured in vivo by Ke et al. [51]. Here, we will only discusslinearized PEFs F PEF , l ( X k , l ) = − kX k , l , F PEF , r ( X k , r ) = kX k , r , (18)where the spring constant k defines the strength of the forces, and the signs are chosenso that a positive force acts in AP-direction. We show in figure S3 in the supplementarymaterial that other force distributions do not differ qualitatively in their influence onthe kinetochore dynamics.To determine kinetochore trajectories of the two-sided model in the presence ofPEFs, we can start from the same force-velocity relations as for the basic one-sidedmodel. In the presence of PEFs, the total forces F k , l and F k , r that act on the left andthe right kinetochore in AP-direction depend on the absolute kinetochore positions X k , l and X k , r : F k , l = F kk (∆ X k ) + F PEF , l ( X k , l ) , (19) F k , r = F kk (∆ X k ) + F PEF , r ( X k , r ) . (20)We can investigate the motion of kinetochores in the full two-sided model again by usinga phase space diagram; in the presence of PEFs we use a v k - F k -diagram with the totalforce F k in AP-direction on the horizontal axis and the velocity v k in AP-direction onthe vertical axis. Because the total forces contain the external PEFs they are no longerrelated by action and reaction and, thus, the two kinetochores no longer have the sameposition on the F k -axis, but they still remain close to each other on the F k -axis as longas the cohesin bond is strong enough.A kinetochore on the upper/lower branch moves in AP-/P-direction with v ± k ( F k )if v +k > v − k < F min at the same time so that it is always the leading kinetochore that switches to the lowerbranch. Since in general the absolute P-velocity is much larger than the AP-velocity( − v − for the lower branch is much larger than + v + for the upper branch), the AP-directed PEF contribution to the total force increases faster on the lower branch thanon the upper one. As a result, the P-moving kinetochore overtakes its sister on the F k -axis before switching back to the upper branch such that the leading kinetochoreautomatically becomes the trailing kinetochore in the next oscillation period (again,“leading” and “trailing” in terms of phase space positions). This periodic change of istability and oscillations in cooperative microtubule and kinetochore dynamics (a) (b) (c) − F k (pN) − − − v k (cid:0) µ m s − (cid:1) F − F min F max ′ ′ ′ ′ ′ ′ X k − X k ( µ m ) t (s) ′ ′ ∆ X k ( µ m ) X k,r X k,l Figure 11.
Kinetochore motion in the presence of PEFs. (a,b) At the beginning ofstate 1 the left kinetochore (green) has just switched from P- to AP-movement, so thatboth kinetochores are on the upper branch. Both kinetochores move in AP-direction,which means that both the cohesin force and the PEFs decrease and both kinetochoresmove left in the force-velocity diagram. Due to different PEFs, the right kinetochore(red) reaches the left turning point F min first and switches to the lower branch, whichmarks the start of state 2. This state is dominated by the fast P-movement of theright kinetochore, which causes a steep increase of both F kk and F PEF , r . Therefore,the right kinetochore moves to the right in the force-velocity diagram. Meanwhile, theleft sister still moves in AP-direction and F k , l increases slightly as the increase of F kk is larger than the decrease of F PEF , l . Since ˙ F k , r > ˙ F k , l , the right kinetochore overtakesits sister on the F k -axis before it reaches the right turning point and switches to theupper branch. The then following states 1 ′ and 2 ′ are the exact opposite to 1 and 2with swapped kinetochores. (c) Solution of the corresponding equations of motion for c = 20 pN µ m − , k = 10 pN µ m − and M = 25. For an animated version see video S4in the supplementary material. The alternating oscillation pattern robustly survives in stochastic simulations inthe presence of moderate PEFs ( k ∼
10 pN µ m − ) as we demonstrate in figure 12(a) bymeans of the kinetochore trajectories in real space. In figure 12(b), emergence of regularoscillations is illustrated in Fourier space: Whereas for rather small values of k singlekinetochore oscillations are still irregular resulting in a nearly monotonic decreasingFourier transform, for k = 10 pN µ m − single kinetochore motion has a distinct peakin the Fourier space indicating a regular shape of oscillations in real space. Moreover,frequency doubling of breathing compared to single kinetochore oscillations can directlybe recognized by comparing the corresponding Fourier transforms. As a consequence ofregular oscillations, the kinetochores stay near the spindle equator and can not get stuckto one of the centrosomes as in the basic model, see histograms of kinetochore positionsin figure 12(c). We conclude that PEFs are necessary to assure proper chromosomealignment in the metaphase plate at the spindle equator. This is consistent with an istability and oscillations in cooperative microtubule and kinetochore dynamics . − X k ( µ m ) k = 1000 pN µ m − t (s) − ∆ X k − X k ( µ m ) k = 10 pN µ m − (a)0 3000 6000 900005 ∆ X k k = p N µ m − M = 25 p N µ m − X k ( µ m) p N µ m − (c)0 2 4 k = p N µ m − M = 25∆ ˜ X k ˜ X k , r p N µ m − f (mHz) p N µ m − M = 100 f (mHz) (b) − X k , l X k , r M = 25∆ X k − − ∆ X k ( µ m) Figure 12.
Kinetochore dynamics under the influence of PEFs. (a) Kinetochoretrajectories with different PEF constants k from simulations with M = 100, c =20 pN µ m − and without confinement at the spindle poles. The PEFs force thekinetochores to oscillate regularly and to stay near the spindle equator. For k =10 pN µ m − kinetochores oscillate as described in figure 11. Since with strong PEFskinetochores tend to switch to the lower branch simultaneously when reaching F min inthe phase space at the same time, for k = 1000 pN µ m − oscillations are in antiphasedue to symmetric initial conditions before the system equilibrates at t ≈ k = 10 pN µ m − ) and S6( k = 1000 pN µ m − ) in the supplementary material. (b) Single (right) kinetochoreand breathing oscillations in Fourier space. For weak PEFs ( k = 1 pN µ m − ) singlekinetochore oscillations are still irregular and ˜ X k , r has its maximum at f = 0. If k =10 pN µ m − , ˜ X k , r has a distinct peak at half the breathing frequency, indicating regularoscillations as described in figure 11 and frequency doubling of breathing comparedto single kinetochore oscillations. With sufficiently strong PEFs ( k &
100 pN µ m − )frequency doubling is lost as a consequence of antiphase oscillations and the peaks of˜ X k , r and ∆ ˜ X k coincide with each other. (c) Histograms of kinetochore positions andinter-kinetochore distances for the realistic case of M = 25. Chromosomes are alignedat the spindle equator despite missing confinement at the centrosome. The range ofkinetochore positions is narrower and the distances smaller if PEFs are stronger. Moreover, PEFs reduce the amplitude and increase the frequency of oscillations.The amplitude decreases for increasing PEF strength k as the kinetochores have to covera smaller distance between the turning points at F min and F max . The increase of the istability and oscillations in cooperative microtubule and kinetochore dynamics k , which can be deduced from the linear increase of | ˙ F k | : | ˙ F k , l | = | c k ( v k , r + v k , l ) + kv k , l | , (21) | ˙ F k , r | = | c k ( v k , r + v k , l ) + kv k , r | (22)(defining v k , l ≡ ˙ X k , l and v k , r ≡ − ˙ X k , r as the velocities in AP-direction as before).Since PEFs do not have any influence on the underlying master curves and force-velocity relations, they do not affect the kinetochore velocities v k and never completelysuppress kinetochore oscillations in the deterministic Fokker-Planck model, but onlyreduce their amplitude and increase their frequency. For strong PEFs, however, thisgives rise to kinetochore motion with a fluctuative character, see figure 12 (see alsovideo S6 in the supplementary material). The same observation was made in the modelof Civelekoglu-Scholey et al. [19]. Additionally, we detect sister kinetochore oscillationsbeing in antiphase if PEFs are strong enough ( k &
100 pN µ m − ), see figure 12(a). Thisfollows from the phase space velocities ˙ F k being dominated by the strong PEFs comparedto inter-kinetochore tension: Imagine, both kinetochores are in the upper branch of thephase space and reach the turning point F min at nearly the same time. When now oneof the two kinetochores switches to the lower branch and starts moving polewards, itssister does not change its direction in phase space as in state 2 / ′ in figure 11(a) butcontinues moving left since the decrease of PEFs due to its poleward motion can not becompensated by the increasing AP-directed cohesin tension if k ≫ c k . As a consequence,the kinetochore will switch to the lower branch just after its sister and both kinetochorespass the lower branch simultaneously, i.e. move apart from each other, finally resultingin antiphase oscillations. While the antiphase behavior vanishes after a certain time ofequilibration in the deterministic model, in stochastic simulations periods of antiphaseoscillations can be observed over and over again regardless of whether the system hasbeen equilibrated before. A characteristic of antiphase oscillations is the loss of frequencydoubling which also appears in the Fourier space where the peaks of single kinetochoreand breathing motion coincide with each other if PEFs are strong, see figure 12(b). Sinceantiphase kinetochore oscillations have not been observed experimentally, we concludethat in vivo PEFs are weak compared to the inter-kinetochore tension but strong enoughto assure chromosome alignment at the spindle equator. Compared to experimentalresults [6, 7, 10, 11, 12, 19], in our model, k = 10 pN µ m − seems a reasonable choiceas it assures regular oscillations with frequency doubling, keeps the inter-kinetochoredistance within a suitable range of (1 . ± . µ m, and aligns kinetochores in a realisticmaximum distance of 3 µ m from the spindle equator with a standard deviation of 0 . µ min the lifelike case of M = 25.
8. Catastrophe promotion at the kinetochore is required to stimulatedirectional instability if microtubules can not exert pushing forces
So far, we assumed that MTs are also able to exert pushing forces on the kinetochore.During oscillations we find, on average, slightly less (48%) MT-kinetochore links istability and oscillations in cooperative microtubule and kinetochore dynamics x = x m − X k is limited to x ≤ x = 0 from belowin the one-sided model. Different choices for the corresponding catastrophe rate ω kinc at x = 0 are possible: (i) A reflecting boundary, i.e., ω kinc = ∞ , where a catastropheis immediately triggered if the MT plus-end reaches the kinetochore. (ii) A “waiting”boundary condition, where the relative velocity v + = v m+ − v k = 0 stalls if the MTreaches x = 0 (in the simulation, we set the MT velocity to v m+ = v k ). In contrast tothe reflecting boundary condition, the catastrophe rate ω kinc at the kinetochore is finitesuch that the MT waits at the kinetochore until it undergoes a catastrophe for a meanwaiting time 1 /ω kinc , as similarly observed in metaphase of PtK1 cells [36]. Because x = 0also results in F mk = 0, the force-free catastrophe rate seems a natural choice, ω kinc = ω [see (1)], which should be realized in the absence of any additional catastrophe regulatingproteins at the centromere. (iii) If catastrophes are promoted by regulating proteins, butnot immediately as for (i), we obtain intermediate cases of waiting boundary conditionswith ω < ω kinc < ∞ . In mammalian cells, such regulating mechanisms could be providedby the kinesin MCAK, which is localized at the centromere during metaphase [53] andhas been reported to increase the catastrophe rate of MTs roughly 7-fold [54]. Therefore,waiting boundary conditions with an increased catastrophe rate appear to be the mostrealistic scenario. We introduce a numerical catastrophe enhancement factor n ≥ ω kinc = nω . Within this general scenarioreflecting boundary conditions (i) are recovered for n = ∞ and (ii) waiting boundaryconditions with the zero force catastrophe rate for n = 1. We will discuss the generalcase (iii) in the following.In our basic model, where MTs can exert pushing forces on kinetochores, thepushing phases where x > n > istability and oscillations in cooperative microtubule and kinetochore dynamics x max to zero, where it is positivein the basic model. When x max is negative in the basic model (for v k > v , seetable 3), confining boundary conditions do not modify the basic model, since the MTsare not able to reach the fast kinetochore. For negative kinetochore velocities v k < v − ,the minimum distance x min becomes positive while x max is zero. Then, all confiningboundary conditions fix the MT tips to the kinetochore position as they do not shrinkfast enough to move away from the poleward-moving kinetochore after a catastropheresulting in h x i = 0 and F ext = γv k . All in all, confinement leads to the followingmaximal and minimal values for the MT-kinetochore distance x modifying table 3: x confmax = ( , v k < v x max , v k ≥ v , x confmin = ( , v k < v − x min , v k ≥ v − . (23)We calculate the master curves h x i ( v k ) for all three types of confining boundaryconditions (see figure 13(a)). Because x confmax ≤ h x i <
0, i.e., the complete master curves lie in the regime of tensile MT-kinetochorelinker forces reflecting the fact that pushing forces are strictly suppressed. Therefore, theMT-kinetochore catch bond is on average under tension establishing a more firm MT-kinetochore connection during the stochastic chromosome oscillations in metaphase.Oscillations then become a tug-of-war, in which both sets of MTs only exert pullingforces onto each other.With a waiting boundary condition at the kinetochore, the probability densities p ± ( x, t ) have to be supplemented with the probability Q ( t ) to find a MT at thekinetochore ( x = 0). Besides the FPEs (5) and (6) for the probability densities, wealso have to solve the equation for the time evolution of Q ( t ): ∂ t Q ( t ) = v + (0) p + (0 , t ) − ω kinc Q ( t ) . (24)The analogous model for a free MT that grows against a rigid wall has already beensolved in [55, 41]. In the stationary state, (24) leads to Q = p + (0) v + (0) /ω kinc . For theprobability densities p ± ( x ) we get the same solution as for the basic model withoutconfinement, except for the normalization constant. The overall probability density canthen be written as p ( x ) = p + ( x ) + p − ( x ) + Qδ ( x ) and has to satisfy R x confmax x confmin p ( x )d x = 1.From the overall probability density p ( x ) we obtain the master curves, which weshow in figure 13(a) for n = 1 , , , , , ∞ and a linker stiffness of c = 20 pN µ m − .Again we can analyze the master curves for extrema to obtain constraints on linkerstiffness c and catastrophe enhancement factor n = ω kinc /ω for the occurrence ofbistability and oscillations. The results of this analysis are shown in figure 13(b) ascolored regions. It turns out that extrema in the master curve and, thus, bistabilityoccur if the linker stiffness is sufficiently high c > c bist . For the zero force catastropherate n = 1 we find a high threshold value c bist = 178 pN µ m − , in the limit of a reflectingboundary n = ∞ a very low threshold c bist = 1 .
218 pN µ m − .We remind that a sufficient condition for oscillations is the absence of a stable fixedpoint, where one of the two branches in the v k - F kk -diagram crosses v k = 0. In contrast tothe basic model, the maxima of the master curve are now located at a positive velocity istability and oscillations in cooperative microtubule and kinetochore dynamics − − v k (cid:0) µ m s − (cid:1) − − h x i ( µ m ) (a) ω ω ω ω ω reflecting 1 10 100 1000 c (cid:0) pN µ m − (cid:1) ω k i n c / ω c oscillatory oscil.bistable (b)100 200 300 F kk (pN) − − v k (cid:0) µ m s − (cid:1) ω (c)100 200 300 400 F kk (pN)50 ω − X k ( µ m ) ω (d)0 15000 30000 t (s) ∆ X k − ω t (s) Figure 13.
Microtubule confinement at the kinetochore. (a) Master curves of asystem with a waiting boundary condition for various ω kinc = n ω and c = 20 pN µ m − .(b) Regimes in the parameter plane of c and ω kinc in the limit of many MTs. Outsidethe blue region, the master curve is bistable. In the orange region, the left branch ofthe master curve and, therefore, the lower branch of the v k - F kk -diagram cross v k = 0,which leads to a fixed point suppressing oscillations (see text), whereas in the red regionoscillations are possible. In stochastic simulations, kinetochores already oscillate atmuch smaller ω kinc than predicted by the master curves. Additionally, a new kind offixed point, which is depicted in (c), emerges in the shaded region. (c,d) Phase spacediagrams and kinetochore trajectories from simulations of the unconfined two-sidedmodel with c = 20 pN µ m − and M = 100. The blue dots mark the new kind of fixedpoint, where the leading kinetochore in the lower branch moves with the same velocityas the trailing kinetochore in the upper branch. Then the inter-kinetochore distanceremains constant, while the center of mass moves with a constant velocity as in (d)for ω kinc = 20 ω at t ≈
25 000 s. In the presence of PEFs, these fixed points are absentand the shaded region in (b) does not apply. for n >
1. Therefore, oscillations are suppressed by a fixed point v − k = 0 on the lowerbranch in the v k - F kk -diagram, which occurs if the velocity is positive in the minimum ofthe master curve. In general, oscillations occur if the linker stiffness is sufficiently high c > c osc . Again we find a high threshold value c osc = 280 pN µ m − for n = 1 and a lowthreshold c osc = 1 .
237 pN µ m − for a reflecting boundary condition ( n = ∞ ).For n <
10 the threshold values remain high. Moreover, at such high linkerstiffnesses and for for small n , the simulations of the two-sided model do not showthe expected behavior. For n = 1 and high linker stiffnesses in the oscillatory regimethe kinetochore trajectories do not exhibit regular oscillations. Naively, one could arguethat kinetochore oscillations are suppressed due to the lack of a pushing force and can be istability and oscillations in cooperative microtubule and kinetochore dynamics n = 1)the waiting time 1 /ω kinc ∼ − n &
20 over a wide range of linker stiffnesses from c = 10 pN µ m − to c = 200 pN µ m − .For n >
1, at the boundary between bistable and oscillatory regime in figure 13(b), afixed point v − k = 0 on the lower branch of the v k - F kk phase space diagrams appears,which can suppress oscillations. This fixed point is, however, less relevant becausethe kinetochores will only occasionally pass the lower branch simultaneously, which isnecessary to reach this fixed point. Furthermore, this fixed point is located near theright turning point F max so that the kinetochores can easily leave the fixed point by astochastic fluctuation (as in figure 7). For these two reasons, in stochastic simulations,oscillations already occur for n &
5, that is at a much lower n than the deterministicallypredicted n &
20, but not for n = 1, i.e., in the absence of a catastrophe promotingmechanism.The fixed point analysis of the v k - F kk phase space diagrams reveals that also anew type of fixed point corresponding to a non-oscillatory motion emerges for n . F kk = − c k ( v k , r + v k , l ) = 0, and the inter-kinetochore distance remains constant, whilethe center of mass moves with a constant velocity (see figure 13(d)). In the presence ofPEFs, however, this new type of fixed point does not survive because for the P- movingkinetochore the AP-directed PEFs increase, whereas they decrease for an AP-movingkinetochore. Then the upper blue dot in figure 13(c) moves to the left, while the lowerblue point moves to the right such that this new type of fixed point is unstable inthe presence of PEFs. Therefore, in the entire shaded region in figure 13(b) PEFs areessential to re-establish oscillations.We conclude that both the linker stiffness c >
10 pN µ m − and the catastrophe rate ω kinc at the kinetochore ( n &
20 or n & n , which still enable oscillations, might be advantageousin the cellular system. We note that poleward flux can influence existence and positionsof fixed points: An intermediate flow velocity can eliminate a fixed point on the lowerbranch by moving it into the unstable area of the phase space diagram. If flux issufficiently large it can establish additional fixed points on the upper branch of thephase space diagrams, which suppress oscillations as in the basic model.Moreover, the linker stiffness has to be sufficiently high to give linker extensionscompatible with experimental results. An important part of the MT-kinetochore linkage istability and oscillations in cooperative microtubule and kinetochore dynamics ◦ with a broad distribution [57]. This bending corresponds to linker length changes of | x | ∼
50 nm. Moreover, fluorescent labeling showed total intra-kinetochore stretchesaround 100 nm [58] or 50 nm [12]. Therefore, we regard linker extensions x .
100 nmas realistic values. For large n ≫
20 only a small linker stiffness is necessary to enableoscillations. At the small threshold stiffness, the average linker length |h x i| is typically1 µ m in this regime. Increasing the linker stiffness leads to a decreasing linker length |h x i| . We conclude that, for n ≫
20, experimental observations of linker extensions | x | .
100 nm put a stronger constraint on linker stiffness than the experimentalobservations of oscillations. Linker stiffnesses significantly above 5 pN µ m − and, thus,far above c osc are necessary to obtain a realistic linker length.For n ∼ −
20, which is compatible with the experimental result n ∼ c = 20 pN µ m − , the increasedcatastrophe rate at the kinetochore leads to a realistic behavior with linker extensions x ∼
100 nm, which are also compatible with the experimental results [56, 57, 58, 12](see figure 13(a)). This parameter regime is within the shaded regions in figure 13(b)and PEFs are necessary to establish oscillations. The linker extension is independent ofPEFs.For an increased catastrophe rate around n ∼ −
20 and a linker stiffness c = 20 pN µ m − , the more realistic model with waiting boundary conditions at thekinetochore exhibits a similar behavior as our basic model because pushing phases where x >
9. Model parameters can be adjusted to reproduce kinetochore oscillationsin PtK1 cells
So far, we took the experimentally measured parameters for MT transitions andvelocities from table 2 for granted in order to analyze the effects of poleward flux,PEFs and confinement at the kinetochore by means of our mean-field theory. Thesevalues stem from experiments with yeast kinetochores [2], which can only bind oneMT [59], whereas the mean-field theory is only correct if the kinetochores are attachedto multiple MTs as in metazoan cells. Moreover, in budding yeast, the Ndc80 fibrilsare connected to MTs via ring-like Dam1 complexes, which do not appear in metazoancells [60]. In this section, we demonstrate that by adjusting the parameters of MTdynamics our model can reproduce experimental data of metazoan spindles using theexample of PtK1 cells.Our model exhibits a large difference of P versus AP-velocity ( ∼
100 vs. ∼ − ,see figure 8) which is the origin of frequency doubling and also appears in PtK1 cellsbut not in this extent ( ∼
19 vs. ∼
16 nm s − ) [11]. As a consequence, in our model bothkinetochores move towards each other in AP-direction (state 0 in figure 3) most of the istability and oscillations in cooperative microtubule and kinetochore dynamics / ′ in figure 3). In a first step we will respectthese results by adjusting the master curve (or force velocity relation) in a way thatthe two stable branches fit the experimentally measured velocities. This objective willbe achieved by modifying the force-free MT velocities v ± (shifting the upper / lowerbranch up- or downwards) and the corresponding characteristic forces F ± (altering theslope of the upper / lower branch). Moreover, as a last parameter of MT dynamics,we will change the rescue rate ω in order to adjust the MT-kinetochore distance to arealistic value. In a second step we will fit the measured frequencies and amplitudes byvarying the parameters that do not affect the master curves ( c k , k ).Using the model with confinement at the kinetochore, we assume a ten timesincreased catastrophe rate ω kinc = 10 ω according to experimental results [54]. Weset the linker stiffness to c = 20 pN µ m − and keep it unchanged henceforth since thisvalue results in strongly bistable master curves and the manifold consequences that afurther modification of c has on kinetochore dynamics are hard to handle. The fluxvelocity is v f = 8 nm s − (see table 4). The force-free MT growth velocity v has to begreater than v f for two reasons: Firstly, detached MTs would not have a chance to reachthe kinetochore again, otherwise. Secondly, this choice prevents a fixed point at theupper branch, as the left turning point in phase space (maximum of the master curve)is located at v − v f , when the MTs are confined at the kinetochore. We increase theforce-free growth velocity roughly four-fold to v = 20 nm s − , so that the minimum AP-velocity v − v f = 12 nm s − in the left turning point F min lies below the observed meanvelocity of ∼
16 nm s − . In order to adjust the maximum AP-velocity, we reduce thecharacteristic force in MT growth to F + = 5 pN, which leads to a steeper upper branchin the phase space diagram. The force-free shrinking velocity v − should be smallerthan the observed P-velocity since the lower, P-directed branch always lies above it.Analogously to the upper branch and F + , also the slope of the lower branch can beadjusted by varying the characteristic force F − : An increase of F − , i.e. a decrease of itsabsolute value, steepens the lower branch and thereby slows down the poleward motion.It turns out that it is a good choice to keep the values for v − and F − from table 2unchanged. Finally, we reduce the rescue rate ω , which lets MTs shrink to smallerlengths x m (the minimum of the master curve is shifted downwards) and increases theMT-kinetochore distance | x | = | X k − x m | to a realistic value.Since we enable detachment in this section, we set M = 35 as it results in amean number of ∼
20 attached MTs. Finally, we adjust the strength of PEFs k andthe cohesin bond stiffness c k to the following conditions: Firstly, the PEFs have to bestrong enough to assure proper chromosome alignment at the equator as well as a regularoscillation pattern, but should not dominate compared to the inter-kinetochore tensionin order to prevent antiphase oscillations. Secondly, k and c k affect the amplitude andthe frequency of kinetochore oscillations which should resemble experimental results inthe same manner: An increase of both k and c k decreases the amplitude and increasesthe frequency. We find that k = 20 pN µ m − and c k = 20 pN µ m − fulfill both conditions. istability and oscillations in cooperative microtubule and kinetochore dynamics Table 5.
Parameters to reproduce of kinetochore oscillations in PtK1 cells.Parameters not listed here have been unchanged compared to table 2.Description Symbol Valuezero force rescue rate ω .
012 s − zero force MT growth velocity v
20 nm s − characteristic force of MT growth F + ω kinc .
019 s − MT flux velocity v f − PEF coefficient k
20 pN µ m − cohesin bond stiffness c k
20 pN µ m − MT-kinetochore linker stiffness c
20 pN µ m − number of MTs M The resulting kinetochore dynamics is shown in figure 14. The simulatedkinetochore trajectories in figure 14(a) are very similar to the experimental results in[11, 19] as they exhibit frequency doubling of breathing compared to single kinetochoreoscillations and move predominantly in phase, i.e. there is a leading P- and a trailingAP-kinetochore (state 2 / ′ in figure 3). The motion of the inter-kinetochore distance israther fluctuative, resulting in a broad Fourier transform, in which the maximum at thebreathing frequency is hardly recognizable, see figure 14(d). This is the only significantdifference to the real kinetochore motion. The distributions of kinetochore positionsas well as inter-kinetochore and MT-kinetochore distances (figure 14(e-g)) are in goodagreement with experimental results [19].In table 6, we list several characteristic quantities of kinetochore oscillations thathave also been determined experimentally for PtK1 cells. Comparison with our modelresults shows quantitative agreement. In particular, the large discrepancy in the P- andAP-velocities is eliminated. Table 6.
Characteristic quantities of model kinetochore oscillations compared toexperimental results in PtK1 cells.Description Model Experimentmean P velocity 21 . − . − [11]mean AP velocity 15 . − . − [11]single kinetochore frequency 4 .
27 mHz 4.14–4 .
23 mHz [11]breathing frequency ∼ . .
25 mHz [11]mean inter-kinetochore distance (1 . ± . µ m (1 . ± . µ m [19]mean MT-kinetochore distance (0 . ± . µ m (0 . ± . µ m [19]standard deviation of kinetochore position 0 . µ m 0.5–1 . µ m [19]mean number of attached MTs 21.4 20–25 [43] istability and oscillations in cooperative microtubule and kinetochore dynamics − X k ( µ m ) (a) X k , l X k , r t (s) ∆ X k t (s) M a tt (b) M attl M attr F k (pN) − v k ( n m s − ) (c)0 5 10 15 f (mHz) (d) ˜ X k , r ∆ ˜ X k − − X k ( µ m) (e) X k , l X k , r ∆ X k ( µ m) (f) 0.0 0.1 0.2 | x | ( µ m) (g)10 20 30 M att (h) M attl M attr Figure 14.
Reproduction of kinetochore oscillations in PtK1 cells. (a) Kinetochorepositions and inter-kinetochore distance over time. Although the breathing oscillationsare rather fluctuative, frequency doubling is recognizable. (b) Number of attached MTsover time. (c) Kinetochore motion in phase space (green) compared to the mean-fieldforce-velocity relation (red, calculated with the mean number of attached MTs). For ananimated version see video S7 in the supplementary material. (d) Position of the rightkinetochore and inter-kinetochore distance in Fourier space. Fluctuative breathingoscillations lead to a Fourier transform with broad maxima, which are almost onlyrecognizable in the smoothed curve (dark blue). (e-h) Distributions of kinetochorepositions X k , inter-kinetochore distance ∆ X k , MT-kinetochore distance | x | , and thenumber of attached MTs M att .
10. Discussion
We provided an analytical mean-field solution of the one-sided spindle model introducedby Banigan et al. [20], which becomes exact in the limit of large MT numbers. The mean-field solution is based on the calculation of the mean linker extension h x i as a functionof a constant kinetochore velocity v k (the master curve). Together with the equationof motion of the kinetochore we obtained the force-velocity relation of the one-sidedmodel from the master curve. Our solution clearly shows that the force feedback oflinkers onto the MT depolymerization dynamics is essential for a bistable force-velocityrelation within the minimal model. The shape of the distribution p ± ( x ) of linker lengths(12) is governed by this force feedback, and we traced the bistability to the peakedness(kurtosis) of this distribution.Bistability of the force-velocity relation in the one-sided model is a necessary (butnot sufficient) condition for oscillations in the two-sided model. Interpreting the bistableforce-velocity relation as phase space diagram, we mathematically described kinetochoreoscillations as an emergent result of collective dynamics of coupled MTs that exhibitdynamic instability individually. Our theory becomes exact in the limit of large MTnumbers M . This interpretation of oscillations is underpinned by the experimental istability and oscillations in cooperative microtubule and kinetochore dynamics ∼
20 MTs per kinetochore [43, 66]. Moreover, we were able to deduce idealizedkinetochore oscillations, whose periods conform with experimental results [11]. Fora MT-kinetochore linker stiffness c = 20 pN µ m − and 20–25 MTs per kinetochore,we get periods of 206–258 s and 103–129 s for kinetochore and breathing oscillations,respectively. Our approach reproduced the frequency doubling of breathing comparedto single kinetochore oscillations, observed in the experiment [11]. Both in the modeland in the experiment this doubling originates from the different velocities of AP- andP-moving kinetochores, which ensure that a P-to-AP switch (3 / ′ in figure 3) alwaysfollows an AP-to-P switch of the same kinetochore (1 / ′ in figure 3). In the modelthe velocity difference is, however, much larger. As a consequence, in our model with20–25 MTs an AP-to-P switch follows 96–119 s after a P-to-AP switch of the sisterkinetochore, which is 93 % of a breathing period, whereas in PtK2 cells a mean intervalof merely 6 s has been measured [12]. In other words, in our model, most of the timeboth kinetochores move towards each other in AP-direction (state 0 in figure 3), whereasin the experiment, mostly one kinetochore moves in P- while the trailing sister is movingin AP-direction (state 2 / ′ in figure 3). In our model, different AP- and P-velocitiesare based on the fact that the MT shrinkage is much faster than growth. The modelparameters for MT dynamics were taken from experimental measurements with yeastkinetochores [2], which, however, are distinct from metazoan kinetochores in two mainpoints: firstly, they can only attach to one MT [59]; secondly, the Ndc80 fibrils areconnected to MTs via ring-like Dam1 complexes, which do not appear in metazoancells [60]. We show in section 9 that this discrepancy can be eliminated by adjustingsome MT parameters and, moreover, the model can reproduce kinetochore oscillationsin PtK1 cells quantitatively.In experiments with HeLa cells Jaqaman et al. [67] observed an increase ofoscillation amplitudes and periods when they weakened the cohesin bond. In our model,a smaller cohesin stiffness c k has the same two effects as the inter-kinetochore distancehas to be larger to reach the turning points F min and F max of the hysteresis loop, and thephase space velocity ˙ F kk = c k ( v k , r + v k , l ) and, therefore, the frequencies are proportionalto c k .Our analytical approach also allowed us to go beyond the results of [20] and quantifyconstraints on the linker stiffness c and the MT number for occurrence of bistabilityin the one-sided model and for the occurrence of oscillations in the full model. Wefound that bistability requires linker stiffnesses above c bist ≃ µ m − . Bistabilityis, however, not sufficient for oscillations. Our phase space interpretation showed thatbistability only leads to directional instability if the two branches of the force-velocityrelation are also separated by the zero velocity line. This condition quantifies theoscillatory regime in the parameter plane of c and M . We predict that oscillations should istability and oscillations in cooperative microtubule and kinetochore dynamics c osc ≃
16 pN µ m − . Ourmodel can thus provide additional information on the MT-kinetochore linkers whosemolecular nature is unknown up to now. Several Ndc80 fibrils, which cooperativelybind to the MT, are an important part of the MT-kinetochore link and the stiffnessof this Ndc80 link has been determined recently using optical trap measurements [68].These experiments found stiffnesses above ∼
20 pN µ m − , which are compatible withour bounds. Moreover, they found a stiffening of the link under force, which could beincluded in our model in future work.The derivation of the lower bound for the stiffness for the occurrence of oscillationsis based on the occurrence of a new zero AP-velocity fixed point in the force-velocitydiagram of the kinetochores, which suppresses oscillations upon decreasing the stiffness.Also the influence of poleward flux to the system could be analyzed by a fixed pointanalysis of the force-velocity diagram. Since poleward MT flux shifts the force-velocitytowards smaller AP-velocities of the kinetochore, the upper branch may cross zerovelocity establishing again a zero velocity fixed point suppressing oscillations. Thisexplains why high flux velocities suppress directional instability and rationalizes thecorrelation between kinetochore oscillations and poleward flux observed in several cells(table 4). It has been observed in newt lung cells that oscillations are occasionally(11 % of time) interrupted by phases in which the kinetochores pause their motion [6]analogously to resting in the fixed point in our model. This indicates that the spindleof newt lung cells operates near the boundary between the oscillatory and the non-oscillatory regime.Also experimental results in [69, 70, 71, 72] on the effects of phosphorylation ofHec1, which is part of mammalian Ndc80 complex, onto kinetochore dynamics can berationalized by our force-velocity diagram of the kinetochores. Dephosphorylation leadsto hyper-stable MT-kinetochore attachments, increases the inter-kinetochore distance,damps or completely suppresses oscillations, and lets the kinetochores more often befound in a “paused state”. The increase of the inter-kinetochore distance can beexplained with the hyper-stable MT-kinetochore attachments: in the oscillatory regime,the bistable area of the force-velocity relation increases if more MTs are attached tothe kinetochore (figure 2(b)); in the non-oscillatory regime, the mean distance h ∆ X k i is a linear function of M ((16)). However, the suppression of oscillations and thefrequent appearance of paused states, which are both effects of leaving the oscillatoryregime in our model, can not be explained with an increasing number of attachedMTs. Instead, we suggest three additional effects of Hec1 phosphorylation: Firstly, itis imaginable that Hec1 is a catastrophe factor that is activated by phosphorylation,i.e., if phosphorylation is suppressed, the catastrophe rate at the kinetochore ω kinc decreases. Secondly, phosphorylation of Hec1 could stiffen the Ndc80 complex so thatdephosphorylation suppresses oscillations by decreasing the linker stiffness c . Since thestiffness of the Ndc80 complex has been measured in a recent experiment [68], this secondoption might be testable. The third possible explanation is based on the observationof Umbreit et al. [73] that phosphorylation of Hec1 suppresses rescues. Following the istability and oscillations in cooperative microtubule and kinetochore dynamics α − that defines the leading order of p ( x ) near x min is a linear function of ω ( α − + 1 ∝ ω , see supplementary material), theprobability density p ( x ) becomes sharper for negative kinetochore velocities if rescue issuppressed, finally leading to a bistable master curve that allows for oscillations. In [71],besides suppression, Hec1 phosphorylation has also been enforced on up to four sites.As a result, the number of attached MTs and the periods of kinetochore oscillationsdecreased, which is consistent with our model (figure 4(c)). Moreover, kinetochoreoscillations were supported but became more erratic just like in our model, wherekinetochore motion is more fluctuative if less MTs are attached (figure 4(ab)). Thisexperimental result reinforces our point of view that regular kinetochore oscillations arean emergent phenomenon that results from the collective behavior of stochastic MTdynamics.Furthermore, we added linearly distributed PEFs, which depend on the absolutekinetochore positions. Their main effect is a phase shift between the sister kinetochoresin their phase space trajectories, which leads to regularly alternating kinetochoreoscillations and, finally, forces the kinetochores to stay near the spindle equator.Consistently, experimental results show that a proper formation of the metaphase plateis not assured when PEFs are suppressed [52]. Since the PEFs do not affect the mastercurves and phase space diagrams, deterministically, they never completely suppressoscillations but only reduce their amplitude and increase their frequency, while thekinetochore velocities v k are unchanged. This is consistent with experiments of Ke etal. [51], who observed an increase in amplitude but no influence on the occurrence ofoscillations and the velocity of chromosomes after severing the chromosome arms andthereby weakening the PEFs. In stochastic simulations, the kinetochore oscillations aremore fluctuative in the presence of PEFs, see figure 12. A similar observation was madein the model of Civelekoglu-Scholey et al. [19]. Moreover, in stochastic simulations,sister kinetochores tend to oscillate in antiphase and frequency doubling of breathingcompared to single kinetochore oscillations is lost if PEFs are strong compared to theinter-kinetochore tension ( k ≫ c k ). Since to our knowledge such antiphase oscillationshave not been observed in vivo, we conclude that the inter-kinetochore tension is thedominating force for directional instability.Consistently with experimental observations in both fission yeast [74, 75] andhuman cells [76], kinesin-8 motors investigated in the model of Klemm et al. [21]have a similar centering effect as the PEFs in our model. Since fission yeast doesnot contain chromokinesins [77], the Klemm model does not include PEFs, whereasour model does not include kinesin-8. It remains an open question whether and howthe similar effects of PEFs and kinesin-8 cooperate if both are present. As kinesin-8 depolymerizes MTs in a length-dependent manner [78, 79], it could be included inour model by a catastrophe rate ω c that depends on the MT length x m , While suchMT length-dependent catastrophe rates can easily be implemented in the stochasticsimulations, they are difficult to include into our mean-field theory, which is based on istability and oscillations in cooperative microtubule and kinetochore dynamics ω kinc can,in principle, range from the force-free MT catastrophe rate ω , which is realistic inthe absence of any catastrophe promoting proteins up to infinity if a catastrophe isimmediately triggered. In the presence of the centromere-associated regulating proteinMCAK increased catastrophe rates ω kinc = 7 ω are expected [54]. We found that both the linker stiffness c and the catastrophe rate ω kinc at the kinetochore have tobe sufficiently large to obtain bistability and oscillations. We find, in particular, thatthe force-free MT catastrophe rate is not sufficient to lead to oscillations, which showsthat catastrophe-promoting proteins are essential to induce oscillations. In the presenceof PEFs, oscillations can be recovered also for relatively small catastrophe rates: For ω kinc /ω ∼
5, we found no oscillations in the absence of PEFs; for ω kinc /ω < ω kinc /ω = 20 and a linker stiffness of c = 20 pN µ m − , we found realistic behavior. Ourresults can explain experimental observations in [80], where PtK2-cells were observedunder depletion of centromeric MCAK, which decreases ω kinc . Then, in accordance to ourresults (see figure 13(cd)), the turning point F max of the hysteresis loop decreases. Asa result the oscillation frequency increases and the mean centromere stretch decreases,while the “motility rates”, i.e., the velocities do not change.Kinetochore motion in the non-oscillatory regime can be described as fluctuationsaround a fixed point with constant inter-kinetochore distance. This is exactly thebehavior of peripheral kinetochores in PtK1 cells [11, 19], while the central kinetochoresdo exhibit directional instability. Civelekoglu-Scholey et al. [19] explained thisdichotomy with different distributions of polar ejection forces in the center and theperiphery of the metaphase plate. However, the model kinetochore trajectories in thepresence of strong PEFs, which they declare to be representative for the motion ofperipheral kinetochores (figure 6C in [19]), still have a regular oscillating shape withonly a reduced amplitude and an increased frequency, in agreement with the results ofour model in figure 12. The experimental trajectories for peripheral kinetochores from[11, 19], on the other hand, are very fluctuative, hardly show any regular oscillations,and are very similar to the trajectories in the non-oscillatory regime of our model. For aclear characterization of the experimentally measured motion of peripheral kinetochoresas either stochastic fluctuations or regular oscillations its representation in Fourier spacewould be helpful as already provided for the central kinetochores by Wan et al. [11] andas provided in figure 12 for our model. If the Fourier transforms do not have any distinctpeaks, differences in PEFs are ruled out as a possible explanation for the dichotomy inPtK1 cells according to both our model and the one of Civelekoglu-Scholey et al. Instead, our results suggest differences in linker stiffness or catastrophe promotion istability and oscillations in cooperative microtubule and kinetochore dynamics ω kinc and suppress oscillations of peripheralkinetochores. Differences in poleward flux might be another possible explanation for thedichotomy according to our results. However, Cameron et al. [81] observed that the fluxvelocities in PtK1 cells do not depend on the chromosome to which a MT is attached.In conclusion, the minimal model can rationalize a number of experimentalobservations. Particularly interesting are constraints on the MT-kinetochore linkerstiffness that are compatible with recent optical trap measurements [68]. The predictedresponses to the most relevant parameter changes are summarized in table 7 and suggestfurther systematic perturbation experiments, for example, by promoting catastrophesat the kinetochore. Table 7.
Summary. Effect of an increase of the parameter in the first column onoccurrence, frequency, and amplitude of kinetochore oscillations.parameter symbol occurrence frequency amplitude additional effectslinker stiffness c stimulation decrease increase decrease of inter-kinetochore distance innon-oscillatory regimepoleward flux v f suppression decrease nonepolar ejection forces k none increase decrease PEFs force kineto-chores to oscillatealternately and tostay near the spindleequatorcatastrophe rate of stalled MTs ω kinc stimulation decrease increasecohesin bond stiffness c k none increase decreaseMT number M (stimulation) decrease increase Acknowledgments
We acknowledge support by the Deutsche Forschungsgemeinschaft (grant No.KI 662/9-1). We acknowledge financial support by Deutsche Forschungsgemeinschaftand TU Dortmund University within the funding programme Open Access Publishing.
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