Collective behavior of minus-ended motors in mitotic microtubule asters gliding towards DNA
Chaitanya A. Athale, Ana Dinarina, Francois Nedelec, Eric Karsenti
CCollective behavior of minus-ended motors in mitoticmicrotubule asters gliding towards DNA
Chaitanya A. Athale , , Ana Dinarina ‡ , Francois Nedelec and EricKarsenti Div. of Biology, IISER Pune, Sai Trinity, Sutarwadi Road, Pashan, Pune 411021, India. Cell Biology and Biophysics Div., EMBL, Meyerhofstrasse 1, D-69117, Heidelberg,Germany.E-mail: [email protected] ‡ Present address: Thermo Fisher Scientific, Germany. a r X i v : . [ q - b i o . S C ] F e b ACS numbers: 87.16.Ln, 87.16.Ka, 87.16.A, 87.16.Nn
Submitted to:
Physical Biology
Keywords : Centrosomal aster, motility, simulation, tug-of-war, dynein, gliding, asymmetry,chromatin gradient,
Xenopus egg extract.
Abstract.
Microtubules (MTs) nucleated by centrosomes form star-shaped structuresreferred to as asters. Aster motility and dynamics is vital for genome stability, cell division,polarization and differentiation. Asters move either towards the cell center or away fromit. Here, we focus on the centering mechanism in a membrane independent system of
Xenopus cytoplasmic egg extracts. Using live microscopy and single particle tracking, we findthat asters move towards chromatinized DNA structures. The velocity and directionalityprofiles suggest a random walk with drift directed towards DNA. We have developed atheoretical model that can explain this movement as a result of a gradient of MT lengthdynamics and MT gliding on immobilized dynein motors. In simulations, the antagonisticaction of the motor species on the radial array of MTs leads to a tug-of-war purely due togeometric considerations and aster motility resembles a directed random-walk. Additionallyour model predicts that aster velocities do not change greatly with varying initial distancefrom DNA. The movement of asymmetric asters becomes increasingly super-diffusive withincreasing motor density, but for symmetric asters it becomes less super-diffusive. Thetransition of symmetric asters from superdiffusive to diffusive mobility is the result of numberfluctuations in bound motors in the tug-of-war. Overall, our model is in good agreementwith experimental data in
Xenopus cytoplasmic extracts and predicts novel features of thecollective effects of motor-MT interactions.
1. Introduction
Centrosomes nucleating microtubules (MTs) form an aster, with MT minus-ends inside andplus-ends pointing outwards. Smaller asters are also formed in the absence of centrosomesby MT-organizing centers (MTOCs) in plant and animal cells undergoing meiosis. Thesestructures are responsible for a wide range of functions, such as nuclear positioning,fertilization, cell-division and polarization. The positioning of centrosomal asters has beenshown to result from pushing by MT polymerization [1], pulling by depolymerization[2, 3] and pulling by dynein motors in the cytoplasm [4] and on membranes [5, 6]. Inlive cells, asters typically move either towards the center of the cell or away from it [7].The centering movement occurs during fertilization, when male pronuclei are moved bycentrosomal asters towards the center of the egg by cytoplasmic pulling forces acting onthe longest microtubules, as seen in
Clypeaster japonicus [8] and
Caenorhabditis elegans [4].Similar centering movement of centrosomes towards DNA was seen during spindle assembly inmeiotic
Xenopus egg-extracts [9]. Even in the absence of centrosomes, smaller self-organizedMTOCs show directional movement towards DNA in mouse oocyte meiotic spindle assembly210]. Thus, investigating this centering movement is important for a complete understandingof fertilization and meiosis.The mechanical origin of the pulling forces has been identified in
C. elegans as dyneinlocalized on intracellular organelles [11, 12]. Early experiments with
Xenopus egg extractshave also reported dynein dependent movement of microtubules on coverslip surfaces [13, 14].Most recently in vitro work showed that a combination of cortical end-on and surfaceimmobilized dynein motors can center centrosomal asters [15]. However, the inability todistinguish between membrane and cytoplasmic effects and the lack of control over DNAamount and geometry make it difficult to understanding the physical nature of the movement.Theoretical work based on
C. elegans centrosomal movement has validated thehypothesis that pulling forces were necessary for the centering movement of asters [4, 11].However, the study ignored collective motor effects. A subsequent study using anti-paralllelmicrotubule doublets gliding on multiple kinesin molecules, demonstrated that geometricalconsiderations result in a tug-of-war caused by multiple motors of the same kind (i.e. single-motor species) [16]. A motor model with force-dependent detachment and MT lengthasymmetries in the doublet can resolve this tug-of-war. However, for a complete picture MTlength fluctuations characteristic of mitotic spindle assembly i.e. the dynamic instability ofMTs [17], also needs to be included. Additionally, MT dynamic instability has been shownto be biochemically regulated [14] and show spatial inhomogeneity in mitotic [18, 19, 20]and interphasic cells [21, 22]. Thus a theoretical model that includes the physical andbiochemical aspects of centrosome centering motility will provide a more physically realisticunderstanding of the process.Here, we quantify the centering movement of centrosomal asters towards micro-patterned DNA using
Xenopus egg extracts in experiment. We further develop a theoreticalmodel that combines centrosomal MT dynamics with asters gliding on multiple dyneinmotors. As a result of the anti-parallel radial orientation of MTs in asters, multiple dyneinmotors generate antagonistic forces resulting in a 2D single-motor species tug of war fora symmetric aster. A gradient aster asymmetry originating from the DNA resolves thetug-of-war and causes aster to move towards DNA. The movement statistics of the modelqualitatively agree with experimental data and predict novel features of the system.
2. Experimental and data analysis methods
Meiosis II arrested
Xenopus egg cytoplasmic extract was prepared as described before [23, 24]and flowed into a PDMS chamber with an inlet and outlet (Figure 1(a)). The chamberwas sealed at the bottom with a cover glass with regularly spaced circular patterns ofBSA-biotin 30 µm in diameter (Figure 1(b)). This was used to immobilize bis-biotinylatedDNA sandwiched between two layers of paramagnetic beads (DynaBeads, Invitrogen, USA)3Figure 1(c)) as described previously [18, 25]. The presence of DNA on the patterns wasconfirmed by staining with Hoechst (Sigma, Germany) (Figure S8). The bead attachedchambers were pre-incubated with Xenopus egg extract to allow the loading of chromatinproteins. Anti-TPX2 antibodies were added to block chromatin-based nucleation of MTs[18]. Fresh extract containing Cy3-labelled tubulin, anti-Tpx2 antibodies and centrosomeswere flowed into the chamber and observed under the microscope.
Samples were imaged using a Zeiss LSM 5 Live (Carl Zeiss, Jena, Germany) confocalfluorescence microscope. At the start of the experiment still images of the DNA pattern wereacquired in the Hoechst channel (Figure S8). This was followed by time-lapse imaging ofCy3-tubulin every 117 s over the same tiled area (Video SV1). Image acquisition was stoppedwhen most of the asters had coalesced into indistinguishable masses on the chromatin (VideoSV1) after ∼
1h of the start of the reaction. The acquired time series were divided intoregions of interest (ROI) containing asters in the Cy3-tubulin channel with a nearby DNApattern. These time-series of MT asters were tracked using a home built single-particletracking (SPT) program that filters the raw data and detects bright points based on anoptimized intensity threshold. The bright points correspond to the centers of mass of theasters (Figure 1(d)). Aster centers were connected in time (tracking) based on thresholddistances between detected asters in subsequent time frames. The coordinates of circularDNA patterns were automatically extracted by image thresholding and object detection.The image analysis routines were developed in MATLAB (Mathworks Inc., USA), and areavailable upon request from the authors.
3. Theory
The 2D model developed in this study treats the cytoplasmic extract as an aqueous mediumwith an effective viscosity similar to that measured in
Xenopus egg extract cytoplasm( η = 0 . pN · s/µm = 0 . P a · s ) [26, 27]. The DNA pattern is modeled as a static circleof radius 15 µm . This DNA circle is the source of a radially symmetric spatial gradient ofdynamic instability. In our model, microtubule dynamics is treated as a spatial variant of the four-parameterand two-state model of dynamic instability [17, 28]. Rapid transitions between growth andshrinkage are described by the frequencies of catastrophe ( f c : growth to shrinkage) and res-cue ( f r : shrinkage to growth), and velocities of growth ( v g ) and shrinkage ( v s ) [28]. These4arameters affect only the plus-tips of the MT filaments. Initially, all MTs are in a growingstate, at a rate defined by the velocity of growth ( v g ). When an MT shrinks, it shrinks atthe rate given by the velocity of shrinkage ( v s ). Those microtubules that reach length ofzero are instantaneously re-nucleated at the next iteration step. The number of centrosomalnucleation sites is finite and constant (100) based on typical values from vertebrate centro-somes [29]. The frequency of catastrophe ( f c ) and rescue ( f r ) were modeled to be spatiallyinhomogeneous based on previous work that modeled the RanGTP dependent zone of ‘stabi-lization’ around chromosomes [9, 20]. Within the radius of the ‘stabilization’ zone ( r c ), andaround the chromatin of radius ( r n ), the values of dynamic instability enhance MT growth( f minc , f maxr ), while outside it they attain cytoplasmic values ( f maxc , f minr ) [18, 30, 20]. Thesimulation box is made large enough that no MTs leave it (Table 1), and at 1/3 box-distancea boundary is set such that microtubule catastrophes become instantaneous resulting inMTs being inside the box even in the absence of a confining boundary. Thus, the spatialfrequencies of dynamic instabilitiy as a function of the 2D radial distance from chromatinsurface ( r ) are described by: f r ( r ) = f minr + ∆ f r · e ( d − r ) /s e ( d − r ) /s (1) f c ( r ) = f maxc + ∆ f c · (cid:16) − e ( d − r ) /s e ( d − r ) /s (cid:17) (2)where d = r c + r n , the value s is a steepness factor, ∆ f c = f maxc − f minc and ∆ f r = f maxr − f minr .Experimental values were used to fit r n = 15 µm , r c = 20 µm , s = 10 µm , f c and f r (Table 3,[31, 32]) producing an effective gradient as shown in Figure 4(b). A cytoskeleton simulation tool that has been previously described [27] was used to simulatethe movement of microtubules. The simulation uses overdamped Langevin equations torepresent the motion of elastic fibres in a viscous medium. Forces are generated due toBrownian motion and motor mechanics. Motor-MT interactions are modeled as Hookeansprings, where the force exerted by a motor on a point of the fiber is given by (cid:126)f ex = − k ∗ (cid:126)δr ,where k is the stiffness and (cid:126)δr is the motor stretch, projected on the fibre. The vector (cid:126)δr is directed between the position of attachment of the motor base and the end of theprojected stretch. Motors were characterized by their positions, attachment ( r attach ) anddetachment rates ( r detach ), MT-tip detachment rates ( k tipdetach ), speed of movement ( v mot ),probability of stepping, direction of movement, magnitude of the stall force ( f ) and stiffness( k ) of the induced links. Single-molecule experiments with vertebrate dynein have shownthat opposing forces induce increasing rates of detachment [33, 34], reduction in step-sizes (gear-like stepping) [35] and can even induce backward stepping behavior [36]. Wehave modeled detachment rates to be dependent on the magnitude of the parallel force5enerated ( (cid:126)f ex ) based on Kramers theory [37] and used previously in motor-microtubulemodels [16, 30, 38, 39]: r detach = r (cid:48) detach · e ( | f ex | /f ) (3)Using the basal detachment rate r (cid:48) detach = 1 . s − and f = 1 . f ex increasesto three-fold of f , the effective detachment rate increases by an order of magnitude. Theseparameters as well as motor speed ( v mot ), stiffness (k), the maximal distance for motor-MTattachment ( d attach ) and motor-MT attachment rate ( r attach ) are chosen from experimentallymeasured values where possible (Table 2). If MTs and motors are closer than d attach , arandom choice to attach is made by comparing to the probability of binding ( r attach · δt , where δt is the simulation time step). Hence, motor movement was modeled as a probabilisticprocess where the motor may either stay immobile, detach or take a step forwards orbackwards [38]. When the motor is load free, the probability of forwards movement perunit time is p forw = 0 .
9. In order to model the observed gear-like behaviour of dynein[40] we use a piece-wise linear approximation for the probability of forward stepping, whilemaintaining a fixed step size. The first ( f = 5 pN ) and second ( f = 8 pN ) force thresholdsare compared to the parallel force f ex to calculate the probability of forward movement asfollows: p forw = . f ex < f e ( − a · ( f ex − f )) − p back if f ≤ f ex < f p back if f ex ≥ f . (4)where the exponent a = (cid:2) ln(100 · p forw /p back + 1) (cid:3) / ( f − f ) and p back = 0 .
1. The modelof motor behavior when f ex ≥ f is based on observation of backward stepping behavior ofdynein in response to super-stall forces [36]. Simulations were performed with the parameters and box sizes described in Table 1.Centrosomes with 100 nucleation sites (typical for vertebrate centrosomes [29]) wereinitialized at a specific distance away from the chromatin pattern. Microtubules were initiallyof uniform length (5 µm ) and in a growing state. The microtubules fluctuate in lengthaccording to the dynamic instability model and motor binding occurs if an MT filament iswithin the search radius of a single motor and satisfies the rate of attachment. The simulationwas performed for 60 min for independent non-interacting asters. The numbers reported areaverages over all asters under the same initial conditions and parameters.6 . Data Analysis The aster trajectories were analyzed using the following measures:
Tortuosity:
The ratio of the net displacement ( d net ) to path-length ( L ) for a given tra-jectory is the tortuosity χ = d net /L . The value of tortuosity χ is nearly 1 for a straight(directed) movement and it approaches 0 for a random walk. Directional motility coefficient (dmc):
The tendency of the aster to move towards thetarget DNA was quantified by dmc = ∆ d c /d net where ∆ d c is the change in distance fromchromatin from the start of the motion to its end. The value of dmc can vary between -1 to+1. Negative values indicate movement away from the chromatin, 0 can arise from rotationaround the chromosome, a lack of movement or completely random movement, and 1 if it ismoving directly towards the target. Trajectory cos( θ ): The angle made by the points target (DNA), origin of the trajectory( t = 0) and its end ( t = N · δt ) where t is the time, δt is the time between each step and N is the number of steps. If the angle is large ( θ > π/ cos ( θ ) from -1 to 0, but if the angle is small cos ( θ ) ranges from 0 to 1 and the movement isdirected towards the target. Directed random-walk simulation:
In order to test the measures of directionality in asimple directed random-walk scenario, a 2D random-walk simulation was developed. Theparticles were initialized a certain distance away from the origin in the first-quadrant at anangle of π/ r and θ net . The meanradial step is a random number sampled from a normal distribution with mean (cid:104) r (cid:105) = 4 Dt where D = 1 µm /s is the diffusion coefficient and t the time step. The angle of movement( θ net ) is the circular mean[41] given by: θ net = tan − (cid:16) W d · sin( θ d ) + (1 − W d ) · sin( θ r ) W d · cos θ d + (1 − W d ) · cos( θ r ) (cid:17) (5)where W d is the weight of directionality and tan − is calculated from the numerical function arctan2 (Mathworks Inc., USA) to return both the angle and the quadrant (so θ net is between0 and 2 π ). The random angle θ r is a uniform random number between 0 and 2 π and theconstant directed angle θ d was fixed as π/
4. The value of W d was varied between 0 (random)and 1 (directed) in steps of 0.1. The simulation was performed with 100 particles for a totalof 100 s (time step 1 s). The resultant time-projected 2D tracks for increasing weight ofdirectionality ( W d ) show a transition from random to ballistic motion (Figure S1(a)). Thevelocity of the simulated particles is variable at low W d and becomes increasingly uniform7s W d increases, but the population mean remains constant (Figure S1(b)). For such tracksmoving at an average constant velocity, we compare χ (Figure S1(c)), dmc (Figure S1(d))and cos( θ ) (Figure S1(e)) as directionality measures. Mean square displacement to quantify effective diffusion:
The simulated aster movementis evaluated using the expression for the local mean square displacement ( (cid:104) ∆ r (cid:105) , MSD)reported previously [42, 43]: < ∆ r > = (cid:104) (cid:2) r ( t ) − r ( t + δt ) (cid:3) (cid:105) = 4 · D eff · δt α (6)where the displacement between position vectors r at times t and ( t + δt ) are averaged overtime intervals not exceeding 3/4th of the simulated time to avoid artifacts [44, 45]. The MSDas a function of time is fit to the right hand side of Equation 6, where D eff is the effectivediffusion coefficient and α is the coefficient of anomaly. The movement is considered to beeffectively diffusive for α ∼
1, super-diffusive for α > α <
1. TheMSD profiles are fit using the Nelder-Mead simplex algorithm implemented in MATLAB(Mathworks Inc., USA).
5. Results
We observed centrosomal aster dynamics in a PDMS flow-chamber enclosing micron-sizedDNA patterns (Figures 1(a),(b), (c)). Fluorescent asters become visible within ∼ Xenopus extract hasbeen added. These asters move towards DNA (Figure 1(d)) with their lengths fluctuating.The directionality of the aster movement was evaluated using a directional motility coefficient(dmc) measure. The analysis shows that the majority of asters move in the direction ofDNA patterns (n = 45) as seen in Figure 1(e). These asters have dmc > dmc ≤ ∼
90 minutes the asters appeared to aggregate and could not be tracked anymore.
Using the trajectories from SPT, we calculate the frequency distribution of the mean velocity(Figure 2(a)) and fit a log-normal distribution. The frequency distribution of aster velocitiessorted according to movement towards DNA ( dmc >
0) and away ( dmc ≤
0) shows thata smaller number of asters move away as compared to towards. However the velocitiesof the two populations are comparable (Figure 2(b)). The frequency distribution of netdisplacement (i.e. displacement from start point) of the experimental trajectories was fitto a Rayleigh probability distribution ( µ = 21 . µm , σ = 11 . µm and b=17.12) (Figure8(c)), characteristic of a random walk. Considering the net displacement distributions andthe zig-zag movement of the XY tracks of the asters (Figure 1(e)) we analyzed the astermovement further as a modified random-walk model. The path lengths of the trajectoriesmeasured range between 10 to 50 µm with a few outliers (Figure 2(d)). To quantify therandomness, the tortuosity ( χ ) was evaluated. It shows that most trajectories have χ ≥ . χ ∼ . θ ) (Figure 2(f)) and dmc (Figure 2(g)).Taken together the data indicates that the majority of the asters move vectorially towardsthe DNA patch, with a random component to the trajectory. In order to test our interpretation of the measures of directionality, we simulated trajectoriesof particles undergoing random walks with varying degrees of drift. The input weight ofdirectionality ( W d ) determines the degree of randomness (Figure S1(a)). At the same timethe mean velocity of the simulated trajectory is maintained constant (Figure S1(b)). Therange of input parameters from random ( W d = 0) to completely directed motion ( W d = 1)produced a mean tortuosity ( χ ) that also changes from ∼ . χ for a given value of W d is tightly distributed around its mean (Figure S2).In contrast while dmc (Figure S1(d)) and cos( θ ) (Figure 1(e)) also increase with increasing W d , they are more widely distributed. Even for W d ∼ .
4, the dmc (Figure S3) and cos( θ )(Figure S4) measures are highly spread. The simplest explanation for the observed movement of microtubule asters towards chromatinDNA is length dependent pulling of the aster by minus-ended motors in the cytoplasmanchored on the glass coverslip. Previous work has shown that mitotic aster MTs are longerin the direction towards chromatin. The length asymmetry is due to a distance dependenteffect mediated by a reaction-diffusion gradient of RanGTP and downstream components[9, 46, 31, 20, 32]. The effective gradient of microtubule ‘stabilization’ (lengthening) is longrange and step-like. Most net displacement vectors of the trajectories plotted around thecenter of the DNA patch point inwards towards the origin, i.e. the center of the DNA pattern,with no obvious distance-dependence (Figure 3(a)). Further quantification was performed bymeasuring the directionality measures dmc (Figure 3(b)), cos( θ ) (Figure 3(c)) and χ (Figure3(d)) as a function of initial distance of the aster from DNA. The values of dmc and cos( θ )show no obvious distance-dependence although a larger proportion of dmc and cos( θ ) valuesare ∼
1. The values of χ are widely distributed for most values of initial distance. The mean9nstantaneous velocity varies between 0.01 to 0.04 µm/s with no apparent dependence ondistance from DNA (Figure 3(e)). We have developed a minimal model simulation of centrosomal movement by surface-attachedmotors (Figure 4(a)) to test if the experimental data can be reproduced, as well as allowus to consider general properties of the system. In the absence of any directional cues,these asters are expected to move randomly. The tug-of-war that arises out of geometricconsiderations is resolved in our model by MT length asymmetry. Experimentally derivedgradients of f r (Equation 1) and f c (Equation 2) as a function of the distance from theDNA (Figure 4(b)) affect growth of MTs at the plus-tips. Assuming a spatially uniformmotor-density, when MTs come within the field of ‘stabilization’, they become longer. Thelength of MTs attached to motors and consequently asymmetric forces result in directionalmovement. MTs are occasionally observed to bend when multiple motors bind the same fibrein different orientations (Figure 4(a)), Video SV2). The movement of an aster results froma force-asymmetry due to either a length-asymmetry or stochastic fluctuations in boundmotors, thus resolving the tug-of-war. Our model is constrained by experimentally measured parameters of motor mechanicsand the spatial dynamic instability gradient. Hence, to compare the model against ourexperimental measurements, we vary only the (a) initial distance of asters from DNA and (b)the minus-ended motor density. The simulated mean velocity is comparable to the distancebinned data from experiments and shows no change as a function of distance from DNA(Figure 5(a)) over three orders of magnitude of motor density (10 − to 1 motors/µm ).Thedistance binned directionality measures from experiment- cos( θ ) (Figure 5(b)) and dmc(Figure 5(c))- both show an initial increase with distance up to 50 µm , a drop at ∼ µm and a further increase at greater distances. In the simulations, the trend of an initialincrease in directionality is seen at the lowest motor density (10 − motors/µm ), while onlythe calculations at high motor densities (1 motor/µm ) show an increase in directionalityat 75 µm , dropping off again at greater distances. At distances > µm we could onlymeasure 5 experimental tracks, which could potentially skew the distribution. Experimentaltrack lengths are variable as evidenced by the path-length distribution (Figure 2(d)) asa result movement of out of the plane of focus and aster coalescence. In our analysis weconsider distance from DNA based on initial positions alone, so a long track of asters movingdirectionally can skew the distribution. The combination of these effects results in certainregions of the simulation data not matching experiments. Simulations indicate that thedirectionality measures will peak at different distances for different motor densities. Although10ean measures of directionality do show agreement between simulation and experiment,better sampling of experimental data and motor density perturbations are needed to validatethe distance dependence of movement seen in simulations. The initial positions of the simulated asters determines whether the aster will experience anylength asymmetry. Asters located at d=15 µm by virtue of being inside the ‘stabilization’zone and at d=120 µm far from any directional cue are both symmetric. Asters at d=30 to45 µm lie at the boundary of the stabilization zone and are asymmetric. The distributionof motors and their attachment rates ( r attach ) are both spatially homogeneous. Thus, wehypothesize that changes in aster mobility will result from:(i) MT lengths that determine the mean number of bound motors, i.e. asters inside the‘stabilization’ zone will show greater displacement and higher D eff (ii) aster MT asymmetry will result in increasing values of the anomaly parameter ( α )In order to test these predictions, we fit the MSD to the anomalous diffusion model(Equation 6). We find the magnitude of the MSD is greatest for highly asymmetric asters(Figure 6(a),(b)), less for those at the edge of the gradient (Figure , 6(c)) and half for thosefar away from the gradient (Figure 6(d)). The difference in MSD validates hypothesis (i), i.e.the length dependence of motor binding and resultant displacement. The α value indicatesthe extent of deviation from diffusive movement. At d = 45 µm , asters are expected tobe maximally asymmetric and α initially decreases with increasing motor density (up to10 − motors/µm ) and then increases. The initial decrease is due to the increase in diffusivemovement due to stochastic binding (Figure 6(f)). Beyond a critical density of motors( > − motors/ µm ), the anomaly parameter increases based on hypothesis (ii), i.e. ahigh asymmetry with sufficient force generators results in directed motion. The dependenceof α with increasing motor density (Figure 6(e)) for asters completely inside the zone ofasymmetry ( d = 15 µm ) surprisingly shows a peak at 10 − motors/µm dropping off ineither direction. This suggests that a critical density of motors leads to increased super-diffusive movement. Below and above this value of motor density asters either have too fewmotors or too many bound motors, respectively. The anomaly parameter of asters α remains ∼ . d = 30 µm independent of motor density. To our surprise, an increasein motor density for symmetric asters ( d = 120 µm ) leads to a decrease in the anomalyparameter. The mean velocities for a given initial distance from DNA, show little differencewith changing motor density (Figure S7). Thus our model simulations make experimentallytestable predictions of transitions from diffusive to vectorial transport for both symmetricand asymmetric asters, dependent on motor densities.11 . Conclusions We have found that centrosomal MT asters move preferentially towards DNA in experimentswith meiosis II arrested Xenopus egg extracts. Our quantitative study maintains a realisticuniform size and density of DNA and substantially improves on previous work [9]. The2D time-lapse tracks and net displacement distributions suggest a degree of randomness inthe motion. Both qualitatively and using multiple measures of directionality, we find thatasters show a directional bias in their motion. However, aster velocities (Figure 3(e)) anddirectionality measures (Figure 3(b), (c)) lack an obvious distance dependence. The analysisof experimental data is also more detailed than similar work on in vivo movements of smallerMTOC asters in mouse oocytes [10] and results in improved statistics over previous work in
Xenopus oocytes [47] and extracts [9].In our model the centering movement of asters is the result of: (i) a gradient MTdynamic instability, and (ii) MT-length dependent forces generated by molecular motorsimmobilized on the substrate. In this scenario, a tug-of-war emerges from the geometryof the system and asters will move only if the symmetry is broken. Gradients of RanGTPdependent MT dynamic instability in mitotic
Xenopus extracts and cells [32, 31] are know togenerate a length asymmetry in asters [18, 32]. The mechanics of force generation by minus-ended motors is based on experimentally measured single-molecule properties of dyneinwhich include: (a) load-dependent step-size reduction, (b) reversals by super-stall forcesand (c) force dependent detachment rates. Using this setup we have found that aster glidingmovement is 100-fold slower than the intrinsic velocity of single dynein motors (Table 2,Figure 5(a)) as a consequence of the 2D tug-of-war. The two-fold length asymmetry inasters [18, 9] is sufficient to result in directional motion in experiment and simulation (Figure5(b), (c)). In additional calculations, we find that a force independent model of motormechanics can also move asters and qualitatively reproduce the velocity and dmc profilesfrom experiment (Figure S5). However insights from single molecule experiments suggestsuch a simple model is physically unrealistic and do not analyze it further.The qualitative agreement of our simulated velocities with experiments demonstratesthat the expected tug-of-war can be resolved with biologically realistic parameters resultingin small (upto two-fold) length asymmetries. This is in contrast to the extreme degreesof asymmetry (10-100 fold) examined in work by Leduc et al. [16] where they examineda tug-of-war in a 1D anti-parallel MT array gliding on a sheet of kinesin. The simulateddistance dependent velocity profiles are independent of motor densities in the range of 0 . motors/µm . Models and experiments with kinesin based microtubule doublet gliding[16] had used a motor density value orders of magnitude higher. Indeed, work on mitoticaster rotation in sea urchin embryos has shown that a low density of force generators ( < motors/µm ) are sufficient to reorient asters in a length-depdenent manner [48] suggestingour choice of motor densities is in the physiological range.12hile the velocity profiles of experiment and simulation appear to match quite well, thetrend in the distance binned average directionality measures (dmc, cos( θ )) from experimentdiffers from the simulated values. We believe this difference is potentially due to the lowsampling density of the experimental data, combined with the nature of the measures ofdmc and cos( θ ). The frequency distribution of dmc (Figure S3) and cos( θ ) (Figure S4) indirected random-walk simulations is broad, even when the weight of directionality is at 40%( W d ∼ . d = 45 µm ), the anomaly parameter ( α ) increases with increasing motor density asexpected indicating an increasingly vectorial movement (Figure 6(c), (e)). However in thecase of asters very close to DNA ( d = 15 µm ) α is highest at 10 − motors/µm , dropping offon either side (for higher and lower densities). Such a dependence on motor density suggeststhat when the MT lengths in the aster are radially symmetric, (i) the ‘duty ratio’ [49]characteristic of the motor and (ii) the average MT lengths together determine the densityat which α is the highest. This is the critical density of motors which results in super-diffusivemotion. At densities higher than the critical density, the asters are in a mostly bound state,while at lower densities the force generators are very few- both scenarios lead to effectivelydiffusive motion. When the aster is far from any gradient ( d = 120 µm ), the MT lengths arealso radially symmetric [18] but shorter than in asters inside the stabilization zone. Givena constant ‘duty ratio’, the value of α is the highest at 10 − motors/µm and decreaseswith increasing density (Figure 6(e)). This suggests that at low densities of motors, randomimbalances in forces cause vectorial movement. If a motor attaches, it tugs at the astermoving it vectorially. As motor densities increase, the number of binding events increase in alldirections, resulting in more effectively diffusive movement. This suggests that in the absenceof a directional cue, simply modulating the motor density can regulate the qualitative natureof aster movement. Thus, modeling dynein mechanics based on single-molecule experimentalinsights produces new insights into the collective behaviour of aster-motility, as compared toimplicit models of motors in C. elegans aster motility [11, 12]. The dynamics of movementof asymmetric asters at distances of 30 and 45 µm from the DNA center suggest that theyare qualitatively different. Their movement results from a combination of length asymmetryand motor density driven effects. As a result small increases in motor density lead to a slightdrop in directionality, while higher densities lead to an increase. These predictions for bothsymmetric and asymmetric asters can be tested using an in vitro reconstitution approach[15], and would serve as an additional means of understanding the collective behaviour ofdynein.We find that the standard deviation of α is orders of magnitude smaller than the13ifferences between conditions. To examine the heterogeneity in the MSD plots, we examinedthe complete distribution for representative cases of asters at different distances from DNA(Figure S6). Distance from DNA results in differences in magnitude of MSD, a predictionthat can be experimentally tested.The observed convergence of all asters onto the DNA patterns is a result of the anti-TPX2 antibody treatment to inhibit Tpx2 activity. Tpx2 mediates nucleation of MTs fromthe DNA [50] essential for stable spindle assembly [51]. In its absence, the interaction betweencentrosomal MTs and DNA is through a chemical reaction-diffusion gradient of stabilization[32, 31, 19] and kinesins associated with chromosomes, i.e. chromokinesins (reviewed in[52]). Despite the relative simplicity of our model and experimental setup, the qualitativeagreement between theory and experiment suggests is could serve as a module in a morecomplex model of spindle assembly. Such a model would include additional interactionswith Tpx2-nucleated MTs around DNA and chromokinesin activity.In conclusion, we have shown that the centrosomal aster movement in Xenopus extractswith chromatin-DNA is directional. A model of MT asters gliding on dynein motors has beendeveloped that includes detailed motor mechanics and stochastic binding and unbinding.This model can reproduce the qualitative trends in our experimental data. Additionally, themodel also predicts that in the absence of a directional cue, aster movement will undergoa transition from vectorial to effective diffusive movement, based on motor density alone.This tradeoff has relevance to the control of centrosomal and MTOC aster positioning. Theintegration of MT dynamics and motor-mechanics in the model suggests a role for a reaction-diffusion gradient to resolve a tug-of-war resulting from a single motor species.
7. Acknowledgements
CA acknowledges IISER Pune core-funding. We thank Hemangi Chaudhari for re-evaluatingthe single particle tracking data.
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J. Cell Biol. , 158:1005–1015, 2002. . TablesParameter Value Units Time step 0 .
05 sFilament section 0.5 µm Simulation box (2D square) 400x400 µm x µm Total simulated time 60 minThermal energy ( k B T) 4 . pN · nm Microtubule bending modulus 20 pN · µm Fluid viscosity 0.05 pN · s/µm Number of MTs per aster 100 -
Table 1. The simulation parameters.
Motor parameter Value Units Reference
Motor link rigidity ( k ) 0.1 pN/nm [53]Motor stall force ( f ) 1.75 pN [54, 35]Density of motors 0 to 1 motors/µm estimateMotor speed ( v mot ) 2 µm/s [49], [55]Motor attachment rate ( r attach ) 12 1 /s [49], [55]Motor basal detachment rate( r (cid:48) detach ) 1 1 /s [55]Motor detach rate from tips( k tipdetach ) 1 1 /s [49], [55]Motor attachment distance( d attach ) 0.02 µ m [55] Table 2. Molecular motor parameters. T Dynamics pa-rameter Cytoplasm; Chromatin Units Reference
Growth velocity ( V g ) 0.196; 0.196 µm/s [46, 31]Shrinkage velocity( V s ) 0.325; 0.325 µm/s ”Rescue frequency ( f r ) 0.0048 ( f minr ); 0.012 ( f maxr ) 1 /s ”Catastrophe fre-quency ( f c ) 0.049 ( f maxc ); 0.03 ( f minc ) 1 /s ” Table 3. MT dynamic instability parameters. . Figures z yx (a) y x (b) z y biotin-BSAstreptavidin beadsglassbis-biotinylatedDNAstreptavidin beadsbis-biotinylatedDNA (c)(d) (cid:239)(cid:20)(cid:24)(cid:19) (cid:239)(cid:20)(cid:19)(cid:19) (cid:239)(cid:24)(cid:19) (cid:19) (cid:24)(cid:19) (cid:20)(cid:19)(cid:19) (cid:20)(cid:24)(cid:19)(cid:239)(cid:20)(cid:24)(cid:19)(cid:239)(cid:20)(cid:19)(cid:19)(cid:239)(cid:24)(cid:19)(cid:19)(cid:24)(cid:19)(cid:20)(cid:19)(cid:19)(cid:20)(cid:24)(cid:19) X ( (cid:43) m) Y ( (cid:43) m ) (e) Figure 1.
The DNA-micropatterned chamber for aster movement. (a) The PDMS chamberwith vents for the addition of reagents is sealed with a cover glass. (b) A 2D pattern ofregularly spaced circular patches of biotinylated-BSA is made on the cover glass. (c) Thepatterns bind to a layer of paramagnetic streptavidin-coated beads that sandwich a layer ofbis-biotinylated DNA with a second layer of paramagnetic streptavidin coated beads. (d)Two representative images, separated by 2 minutes, depicting a fluorescently labelled astermoving towards the chromatin patch. (e) The XY trajectories of multiple asters movingtowards (black) and away from (gray) the nearest DNA pattern. Circles indicate the initialposition of the asters. .01 0.03 0.05 0.0702550 Velocity ( µ m/s) F r e q u e n c y VelocityLognormal µ =0.026=0.01 σ (a) µ m/s) F r equen cy ToAw (b) µ m) D en s i t y ExptRayleigh (c) (d) (e) (cid:1) cos( (cid:1) ) F r e q u e n c y (f) −1 0 101020 dmc F r e q u e n c y (g) Figure 2.
Movement statistics of asters. (a) A log-normal (mean µ = 0 . µm/s , standarddeviation σ = 0 .
01) function (red-line) is fit to the frequency distribution of instantaneousvelocities (blue-bars). (b) The frequencies of velocities sorted by their movement towards(+ve dmc, black) or away from (-ve dmc, white) DNA. (c) The density distribution of netdisplacement (blue-bars) fit to a Rayleigh distribution (red) with µ = 21 . µm , σ = 11 . µ = 0 .
62 and σ = 0 . θ ) ( µ = 0 . , σ = 0 .
68) and (g) dmc ( µ = 0 . , σ = 0 . µ µ (a) Initial dist. from DNA centre ( µ m) d m c (b) Initial dist. from DNA centre ( µ m) c o s ( θ ) (c) Initial dist. from DNA centre ( µ m) χ T o r t u o s i t y ( ) (d) Initial dist. from DNA centre ( µ m) V e l o c i t y ( µ m / s ) (e) Figure 3.
Distance-dependence of aster motility. (a) The net vectors connecting the startand end-point of each trajectory are plotted around the center of the nearest DNA patch.(b) The directional motility coefficient (dmc), (c) cos( θ ), (d) the tortuosity and (e) meanvelocities (error bars indicate standard error of mean) are plotted as a function of the startpoint of the trajectory. min 15 min45 min 60 min (a) −
100 0 10000.010.020.030.040.05 Dist from DNA centre ( µ m) F r equen cy ( / s ) f r f c (b) Figure 4.
Schematic representation of the simulations. (a) Langevin dynamic simulationsof microtubule asters (gray) initialized 45 µm away from the center of a simulated DNA patch(gray-circle) are run for 60 min ( δt = 0 . s ). The simulation snapshot includes immobilizedmotors (green) and growing (blue-ends) and shrinking (gray-ends) microtubules. The scalebar in the right corner corresponds to 10 µm . (b) The simulated gradients of frequency ofrescue ( f r ) and catastrophe ( f c ) (Equations 1 and 2) plotted as a function of distance fromthe center of DNA.
480 50 150100
Dist. from DNA centre ( μ m) V e l o c i t y ( μ m / s ) - (a) -1010 50 100 150 c o s ( θ ) Dist. from DNA centre (μm) (b)
Dist. from DNA centre ( μ m) d m c (c) Figure 5.
Comparing experiment and simulation. Experimentally measured values (redcircles) of (a) velocity, (b) cos( θ ) and (c) dmc plotted against initial distance from DNA aredistance-binned (blue-bars). The simulated velocity, cos( θ ) and dmc means (10 runs) formotor densities 10 − (blue), 10 − (green), 10 − (red) and 1 (cyan) motors/µm are plottedas lines as a function of initial aster position. Error bars denote standard deviations. Time (s) < r > ( µ m ) d=15 µ m (a) Time (s) < r > ( µ m ) d=30 µ m (b) Time (s) < r > ( µ m ) d=45 µ m (c) Time (s) < r > ( µ m ) d=120 µ m -3 motors/ μ m -2 -1 (d)(e) Motor dens. (motors/ μ m ) D e ff (f) Figure 6.
Analysis of diffusion. MSD profiles of asters initialized at (a) d=15, 6(b) 30,(c) 45 and (d) 120 µm are calculated for increasing motor densities. The motor densities in(a)-(d) correspond to 10 − (red), 10 − (green), 10 − (cyan) and 1 (purple) motors/ µm , asshown in the inset in Figure 6(d). (e) The anomaly parameter α and (f) D eff are plottedas a function of motor density for different initial aster distances from DNA: d=15 (blue),30 (red), 45 (green) and 120 µm (black). All calculations were averaged over 10 asters. upporting Figures d = 0.00 0 100 2000100200 W d = 0.10 50 100 15050100150 W d = 0.20 50 100 15050100150 W d = 0.300 100 2000100200 W d = 0.40 −200 0 200−2000200 W d = 0.50 −100 0 100−1000100 W d = 0.60 −100 0 100−1000100 W d = 0.70−100 0 100−1000100 W d = 0.80 0 50 100050100 W d = 0.90 0 50 100050100 W d = 1.00 (a) d v (b) d χ (c) d d m c (d) d c o s ( θ ) (e) Figure S1. (a) The XY-trajectories (n= 100 particles) for increasing weights of directionalmovement ( W d ). The particles are all initialized at the same position (red) and thetarget is the origin. (b) The magnitude of the velocity, (c) tortuosity ( χ ), (d)directionalmotility coefficient (dmc) and (e) cos( θ ) are plotted against W d . The raw values (black) aresuperimposed with the mean (blue). The error bars indicate the standard error of mean(s.e.m.). χ F r equen cy W d =0.0 0 0.5 1050100 χ F r equen cy W d =0.1 0 0.5 1050100 χ F r equen cy W d =0.2 0 0.5 1050100 χ F r equen cy W d =0.30 0.5 1050100 χ F r equen cy W d =0.4 0 0.5 1050100 χ F r equen cy W d =0.5 0 0.5 1050100 χ F r equen cy W d =0.6 0 0.5 1050100 χ F r equen cy W d =0.70 0.5 1050100 χ F r equen cy W d =0.8 0 0.5 1050100 χ F r equen cy W d =0.9 0 0.5 1050100 χ F r equen cy W d =1.0 Figure S2.
The frequency distributions of tortuosity for increasing values of W d are plottedfor 100 particles based on data from the directed random-walk simulations. F r equen cy W d =0.0 −1 0 1050100 dmc F r equen cy W d =0.1 −1 0 1050100 dmc F r equen cy W d =0.2 −1 0 1050100 dmc F r equen cy W d =0.3−1 0 1050100 dmc F r equen cy W d =0.4 −1 0 1050100 dmc F r equen cy W d =0.5 −1 0 1050100 dmc F r equen cy W d =0.6 −1 0 1050100 dmc F r equen cy W d =0.7−1 0 1050100 dmc F r equen cy W d =0.8 −1 0 1050100 dmc F r equen cy W d =0.9 −1 0 1050100 dmc F r equen cy W d =1.0 Figure S3.
The frequency distributions of dmc for increasing values of W d are plotted for100 particles based on the directed random-walk simulations. θ ) F r equen cy W d =0.0 −1 0 1050100 cos( θ ) F r equen cy W d =0.1 −1 0 1050100 cos( θ ) F r equen cy W d =0.2 −1 0 1050100 cos( θ ) F r equen cy W d =0.3−1 0 1050100 cos( θ ) F r equen cy W d =0.4 −1 0 1050100 cos( θ ) F r equen cy W d =0.5 −1 0 1050100 cos( θ ) F r equen cy W d =0.6 −1 0 1050100 cos( θ ) F r equen cy W d =0.7−1 0 1050100 cos( θ ) F r equen cy W d =0.8 −1 0 1050100 cos( θ ) F r equen cy W d =0.9 −1 0 1050100 cos( θ ) F r equen cy W d =1.0 Figure S4.
The frequency distributions of cos( θ ) for increasing values of W d are plottedfor 100 particles based on the directed random-walk simulations. (a) µ (b) µ (c) Figure S5. (a) A simplified model of motors with a constant attachment ( r attach = 12 s − )and detachment rate ( r detach = 1 . s − ), fixed step sizes and no backward stepping wassimulated and shows aster movement towards chromatin. (b) The mean velocity and (c)dmc profiles for the simulated asters initialized over a range of distances from the chromatincenter with a motor density of 4 · − motors/µm (red) compared to experimental (blackcircles) data show qualitative agreement. The error bars are standard deviations. < (cid:54) r > < (cid:54) r > < (cid:54) r > < (cid:54) r > (a) < (cid:54) r > < (cid:54) r > < (cid:54) r > < (cid:54) r > (b) < (cid:54) r > < (cid:54) r > < (cid:54) r > < (cid:54) r > (c) < (cid:54) r > < (cid:54) r > < (cid:54) r > < (cid:54) r > (d) Figure S6.
The MSD plots of simulated aster trajectories are plotted for all asters (10)simulated for 3600 s for asters initialized at (a) 15, (b) 30, (c) 45 and (d) 120 µm forincreasing motor densities as indicated. These datasets were used to generate the meanMSD profiles in Figure 6. (cid:239) (cid:239) (cid:239) µ m ) V e l o c i t y ( µ m / s ) Figure S7.
The mean velocities of the simulated asters initialized at 15 (blue), 30 (red),45 (green) and 120 µm (black) are plotted as a function of increasing motor densities. ! Figure S8.
A single tiled (4x4) image of the Hoechst labelled DNA micro-patterns wasacquired before imaging the microtubules. The scale-bar corresponds to 100 µm upporting Videos ! Video SV1.
A time-series of 4x4 tiled images (dt = 117 s, 35 frames) was acquired. Thetubulin was labelled with Cy3-Tubulin in centrosome containing
Xenopus extract. The darkregions correspond to DNA patterns (Figure S8). Scale-bar = 100 µm .(a) (b) Video SV2.
Snapshots of the simulation at time (a) 0 s and (b) 59.7 min. The blue circleis the DNA patch (diameter 30 µm ) , the green dots are the minus-ended motors and thegray lines are microtubules of the aster.) , the green dots are the minus-ended motors and thegray lines are microtubules of the aster.