A hydrodynamic instability drives protein droplet formation on microtubules to nucleate branches
Sagar U. Setru, Bernardo Gouveia, Raymundo Alfaro-Aco, Joshua W. Shaevitz, Howard A. Stone, Sabine Petry
AA hydrodynamic instability drives protein dropletformation on microtubules to nucleate branches
Sagar U. Setru † , , Bernardo Gouveia † , , Raymundo Alfaro-Aco , JoshuaW. Shaevitz ∗ , , , Howard A. Stone ∗ , , Sabine Petry ∗ , Lewis-Sigler Institute for Integrative Genomics Department of Chemical and Biological Engineering Department of Molecular Biology Department of Physics Department of Mechanical and Aerospace EngineeringPrinceton University, Princeton, NJ 08544, USA † These authors contributed equally. ∗ To whom correspondence should be addressed; E-mail: [email protected],[email protected], [email protected].
Liquid-liquid phase separation [1, 2] occurs not only in bulk liquid, butalso on surfaces. In physiology, the nature and function of condensates oncellular structures remain unexplored. Here, we study how the condensedprotein TPX2 behaves on microtubules to initiate branching microtubule nu-cleation [3–5], which is critical for spindle assembly in eukaryotic cells [6–10].Using fluorescence, electron, and atomic force microscopies and hydrodynamictheory, we show that TPX2 on a microtubule reorganizes according to theRayleigh-Plateau instability, like dew droplets patterning a spider web [11,12].After uniformly coating microtubules, TPX2 forms regularly spaced dropletsfrom which branches nucleate. Droplet spacing increases with greater TPX2 a r X i v : . [ phy s i c s . b i o - ph ] S e p oncentration. A stochastic model shows that droplets make branching nu-cleation more efficient by confining the space along the microtubule wheremultiple necessary factors colocalize to nucleate a branch. Branching microtubule nucleation plays a major role in spindle assembly and chromo-some segregation in dividing eukaryotic cells, where it is required to generate microtubulesin the spindle for kinetochore fiber tension, spindle bipolarity, and cytokinesis [6–10]. Itsmalfunction has been linked to a worse prognosis in cancer [13, 14]. The nucleation ofa new microtubule from the side of a preexisting microtubule requires TPX2, the aug-min complex, and the γ -tubulin ring complex ( γ -TuRC) [3]. The first component tobind to the preexisting microtubule is TPX2 [4], which forms a liquid-like condensate onthe microtubule; the condensate recruits tubulin and increases branching nucleation effi-ciency [5]. Other proteins also form condensed phases when associated with microtubules,such as Tau [15, 16] and BugZ [17]. Yet, how these proteins behave on the microtubulesurface and how this behavior translates to biological function remain unexplored. Here,we investigate the dynamics of condensed TPX2 on the microtubule. We find that the hy-drodynamic Rayleigh-Plateau instability causes TPX2 to form regularly spaced dropletsalong the microtubule. Then, microtubule branches nucleate from these droplets.We first studied the dynamics of TPX2 binding to microtubules in vitro using total in-ternal reflection fluorescence (TIRF) microscopy (Fig. 1a, Fig. 1b, Methods). GFP-TPX2at a concentration of 1 µ M formed an initially uniform coating along individual micro-tubules within seconds. This coating then broke up into a periodic pattern of dropletsover tens of seconds with size 0 . ± . µ m and spacing 0 . ± . µ m (mean ± standarddeviation, N = 35 microtubules) along individual microtubules (Supplementary Movie1, Supplementary Fig. 1). Similar patterns of condensed protein have also been previ-ously observed on single microtubules for TPX2 [5] and microtubule bundles for Tau [15],2 TPX2, MT a Initial film Droplets Merge dFig. 1
20 s 190 s 190 s20 s 420 s 420 s20 s 420 s 420 s c Mother MT, TPX2, Augmin Branched MTs, TPX2, Augmin
Montage_tubulin_newMts_Substack (12,13,16,18)-rotated-cropped.pdf88 x 47
80 s 82 s 88 s 92 s60 s b DropletsMerge110 s60 s
Figure 1: TPX2 uniformly coats microtubules and then forms periodicallyspaced droplets that can nucleate branches. (a) Initial films and subsequent dropletsof TPX2 on microtubules visualized using TIRF microscopy (Supplementary Movie 1).1 µ M GFP-TPX2 was spiked onto a passivated glass surface coated with Alexa568-labeledmicrotubules. Scale bars are 1 µ m. (b) Large field of view of a TIRF experiment afterdroplets have formed along microtubules. GFP-TPX2 concentration is 1 µ M. Micro-tubules with a droplet pattern are marked with a number. Scale bars are 5 µ m. (c)TPX2 droplets on microtubules imaged using electron microscopy. 0 . µ M GFP-TPX2was incubated with microtubules bound to a carbon grid. Scale bars are 100 nm. (d)Branched microtubules nucleating from TPX2 droplets formed along the initial mothermicrotubule, assembled in vitro as in [18] (Supplementary Movie 2). γ -TuRC purified from X. laevis meiotic cytosol and recombinant GFP-augmin were included. Arrows indicatebranched microtubules. Scale bars are 5 µ m (top) and 1 µ m (bottom). Only the solubleCy5-tubulin channel (magenta) was imaged over time to enable a higher frame rate. TheGFP-TPX2 and GFP-augmin channel (cyan) and the Alexa568 template microtubulechannel (red) were only imaged at the start at 60 s.3ugZ [17], and LEM2 [19]. We next performed the same experiment at a lower, physi-ological concentration of TPX2, 0 . µ M [5, 20]. We observed a uniform coating but novisible droplet formation (Supplementary Fig. 2a). In contrast, at higher resolution, elec-tron microscopy (Methods) revealed that regularly spaced droplets do form at 0 . µ M,with size 0 . ± . µ m and spacing 0 . ± . µ m (mean ± standard deviation, N = 2microtubules) (Fig. 1c). This indicates that these droplets can exist below the diffrac-tion limit of visible light. We then reconstituted branching microtubule nucleation invitro using purified proteins [18] (Methods) and observed that branches originate fromTPX2 droplets colocalized with augmin and γ -TuRC (Fig. 1d, Supplementary Movie 2).75 ±
18% of TPX2 droplets nucleated branches (mean ± standard deviation, N = 7microtubules, Supplementary Table 1). Droplet formation always happened before thenucleation of a branch, and no branches nucleated from areas that did not have droplets.Finally, in meiotic cytosol (Methods), microtubules also nucleate from a TPX2-coatedmicrotubule to form a branched network (Supplementary Fig. 2b, Supplementary Movie3). These results suggest that droplet formation from condensed TPX2 may be importantfor branching microtubule nucleation.We wished to study the dynamics of droplet formation of TPX2 alone on microtubulesat higher spatial resolution than available by fluorescence microscopy and with temporalresolution not accessible by electron microscopy. Therefore, we turned to atomic force mi-croscopy (AFM) to measure the topography of the initial coating and subsequent beadingup of TPX2 on microtubules (Methods). By scanning the AFM tip over the sample every (cid:39) (cid:39) (cid:39) ± ± standard deviation) (Fig. 2a − . ± . µ M (estimated4 H e i gh t ( n m ) -3 -2 Inverse wavelength 1/ (nm -1 ) -2 -1 S pe c t r a l po w e r ( n m ) UncoatedInitially coatedAfter droplet formationMean = 260 20 nm c Uncoated MT, -5 minTPX2-coated MT, 0 min30 min60 min
Fig. 2ab
Length along microtubule (nm) H e i gh t ( n m ) UncoatedInitially coatedAfter droplet formation
Uncoated MT, -5 minTPX2-coated MT, 0 min30 min ° Figure 2: AFM measure-ments reveal condensedTPX2 dynamics on micro-tubules. (a) Time-lapse AFMheight topographies of TPX2uniformly coating and thenforming regularly spaced dropletson a microtubule. During dataacquisition, GFP-TPX2 at0 . ± . µ M (estimated range)was spiked onto microtubulesadhered to a mica surface. Shownis the entire measured span ofthe microtubule in the top leftof Supplementary Movie 4. Thetopography was smoothed usinga 40 nm ×
40 nm median filter.White carets mark droplets.Scale bar is 100 nm. (b) Heightprofiles centered on the micro-tubule long axis before coating(black), just after coating (blue),and when droplets have formedon the microtubule surface (red).Height profiles were smoothedusing a moving-average windowof 40 nm. The raw heightprofiles and their power spectraare shown in SupplementaryFig. 3. (c) Averaged powerspectra calculated from theraw, unsmoothed height profilesacross many microtubules foruncoated (black, N = 22 mi-crotubules), uniformly coated(blue, N = 23 microtubules), anddroplet-patterned microtubules(red, N = 17 microtubules). Apeak is seen only in the data fordroplet-patterned microtubules,corresponding to a droplet spac-ing of 260 ±
20 nm (mean ± standard deviation). Shadedregions represent 95% bootstrapconfidence intervals.5ange). After addition of TPX2, the height signal uniformly increased to 41 ± ± standard deviation) as the condensed protein coated the microtubule (Fig. 2a 0 min,Fig. 2b blue line). The film of TPX2 then proceeded to bead up into a periodic patternof droplets along the microtubule with spacing 250 ±
35 nm (mean ± standard error ofthe mean) (Fig. 2a 60 min, Fig. 2b red line, Supplementary Movie 4). The white caretsin the 60 minute topography in Fig. 2a mark the droplets. The longer time scale to formdroplets and the different spacings between droplets in AFM experiments compared tofluorescence and electron microscopy experiments is due to the different biochemical con-ditions and components used in each experimental method (Supplementary Table 2). Theemergent periodicity of the condensate is evident in the power spectra of the raw heightprofile along the microtubule averaged over many samples (Fig. 2c). Power spectra relyon the Fourier transform to identify the frequency components of a signal buried in noise(Methods). Peaks in a power spectrum indicate the presence of a periodic pattern amidstnoise; a monotonic power spectrum is expected for data that lacks periodicity. The powerspectra (Fig. 2c) show no characteristic length scale before and immediately after coatingwith TPX2, whereas a peak with wavelength 260 ±
20 nm (mean ± standard deviation)has emerged by 60 minutes. Thus, the topography of condensed TPX2 on microtubulesexhibits systematic emergent periodicity.Fluids that coat a solid fiber are known to form droplets via the Rayleigh-Plateauinstability [11]. Surface tension causes the fluid film to be unstable due to the curvatureof the filament surface and the surface area is minimized by forming periodically spaceddroplets along the fiber [12, 22–24]. Following Goren [22], but working directly at lowReynolds number as is appropriate for our experimental system, we solved a linear stabilityproblem for the growth rate σ of the droplet pattern as a function of the wave number k =2 π/λ , where λ is the pattern wavelength (Fig. 3a, Supplementary Fig. 4, Supplementary6heory). We find that for a given ratio of the microtubule radius to the outer film radius, r i /r o , there is a wavelength λ max that grows with the largest growth rate σ max (Fig. 3b).This wavelength will grow exponentially faster than all other wavelengths, leading to aperiodic interface with wavelength λ max . Thus, we identify the thicker regions of the AFMheight profiles as droplets formed by this instability.We tested the ability of this theory to explain droplet formation on microtubulesby measuring film thicknesses and subsequent droplet spacings at different bulk con-centrations of TPX2 (Supplementary Fig. 5). The radius of the microtubule is fixed at r i = 25 nm. However, the thickness of the initial TPX2 film depends on its bulk concentra-tion in solution and the density of microtubules (Supplementary Fig. 6, SupplementaryTheory). Capitalizing on this experimental fact, we changed the initial film thicknessfrom h = 13 ± . ± . µ M TPX2 to h = 22 ± . ± . µ M TPX2(mean ± standard deviation and estimated range, respectively) for a fixed microtubuledensity. The lower concentrations are physiological in healthy cells [5, 20]. The higherconcentrations may reflect overexpression in cancer tumor cells, in which TPX2 often hashigher genetic copy number [25] and transcript and protein expression [13, 26], and canbe a negative prognostic indicator [14]. TPX2 formed regularly spaced droplets alongmicrotubules with consistently larger spacings λ max as its bulk concentration increased,following theory (Fig. 3c, Supplementary Fig. 7a, Supplementary Table 3).The theory is purely geometric and has no free parameters. The predicted wavelengthdoes not depend on the material properties of the TPX2 condensate such as viscosity orsurface tension, which only set the timescale for pattern formation. We note that thehigher microtubule density used in AFM experiments (Fig. 2, Fig. 3c) leads to thinnercondensed films than in the EM and reconstitution experiments (Fig. 1) and hence smallerdroplet sizes, even at similar TPX2 concentrations (Supplementary Table 2, Supplemen-7
10 20 30
Film thickness h (nm) D r op l e t s pa c i ng m a x ( n m ) max = f(h eff ) max = f(h)0.1 0.05 M0.2 0.1 M0.6 0.3 M0.8 0.4 M F r a c t i on o f m a x i m u m g r o w t h r a t e Film thickness h (nm) D r op l e t s pa c i ng m a x ( n m ) max = f(h eff ) max = f(h)0.1 0.05 M0.2 0.1 M0.6 0.3 M0.8 0.4 M Wave number 2 r o / G r o w t h r a t e r o / ( x ) A s pe c t r a t i o r i / r o Fig. 3cab
Figure 3: Hydrodynamictheory predicts TPX2droplet formation on a micro-tubule surface. (a) Schematicof the Rayleigh-Plateau insta-bility. TPX2 initially coatsthe microtubule uniformly withthickness h = r o − r i . This filmbreaks up into droplets withspacing λ max due to capillaryforces on a time scale r o µ/γ ,where µ is the condensate viscos-ity and γ is the surface tension.(b) Dispersion relation showingthe growth rate versus wavenumber for different aspect ratios r i /r o . The most unstable mode(black circles and line) growsmost quickly and correspondsto the observed droplet spacing λ max . (c) λ max = f ( h eff ) (solidorange line) and λ max = f ( h )(dotted orange line). Overlaidare AFM measurements of thefilm thickness and droplet spac-ing for many microtubules overa range of TPX2 concentra-tions. λ max = f ( h eff ) uses aneffective height h eff , which iscalculated to compensate for thereduced volume of condensatethat coats the microtubule whenthe microtubule is resting on asurface. λ max = f ( h ) is the exactprediction by theory. There areno fit parameters in either case.The shaded area encompassesthe wavelengths that grow within25% of the maximum growthrate for each h eff . Error bars arestandard error of the mean. Foreach concentration, we also plotthe corresponding average powerspectra (Supplementary Fig. 7a).8ary Theory). We also note that our theory systematically predicts a larger wavelengththan we measure (Fig. 3c, dotted orange line). This is due to the adjacent surface under-neath the microtubules during AFM imaging. The surface reduces the volume of fluid thatcan coat the microtubule; since λ max ∼ h ∼ V / , a smaller volume produces a smallerwavelength. Using trigonometry, we estimate this lost volume and plot the wavelength asa function of an effective height, λ max = f ( h eff ) (Fig. 3c, solid orange line). We see thatthis effective curve compensates for the systematic offset that the data have with respectto the exact prediction by theory.Even with this correction, Fig. 3c shows a large spread in the measured λ max . Weidentify two sources for the spread in our measurements. First, the dispersion relation wecalculate (Fig. 3b) has a broad peak, which means that wavelengths near the maximumgrowth rate λ max will grow nearly as fast (Fig. 3c, shaded area). Therefore, spread inthe measured λ max is a natural consequence of the hydrodynamic theory. Second, low-force (25-40 pN), nanometric AFM in fluid is invariably susceptible to thermal noise.This is apparent in the raw height profiles (Supplementary Fig. 3a) and power spectra(Supplementary Fig. 3b) of the microtubule shown in Fig. 2a and Fig. 2b, as well as thepower spectra of individual microtubules across TPX2 concentrations (SupplementaryFig. 5, right column).In addition, we measured the growth rate of λ max to be exponential at early times, asexpected for a linear instability (Supplementary Fig. 7b). At later times, the periodicityhas already been selected as the droplet pattern has set in. Thus, the spectral powerversus time stops changing. We also see that the time to form droplets is orders ofmagnitude greater than the time to grow the initial film. The film grows more quicklybecause the timescale for its growth is set by fast diffusion of protein in the bulk, whereasthe timescale for droplet formation is limited by the slow capillary velocity γ/µ of the9ondensate (Supplementary Theory). As a control, kinesin-1, a motor protein that doesnot exist as a condensed phase in any known physiological context and whose binding siteon the microtubule is structurally known [27, 28], did not exhibit hydrodynamic behavioron the microtubule as measured by AFM (Supplementary Fig. 7a).How might TPX2 droplets facilitate branching microtubule nucleation? Noting thatthis process requires the coordinated action in time and space of at least two additionalfactors, augmin and γ -TuRC, we first imaged the localization of γ -TuRC on microtubulesin the presence of TPX2 and augmin using electron microscopy. We found that the ratioof γ -TuRC on microtubules to γ -TuRC on the grid surface was 0 . ± .
05 without TPX2and augmin (mean ± standard deviation, N = 3 microtubules, Supplementary Fig. 8,Supplementary Table 4). With TPX2 and augmin, this ratio was 0 . ± .
04 (mean ± standard deviation, N = 4 microtubules, Supplementary Table 5), confirming that TPX2and augmin preferentially localize γ -TuRC to microtubules. We observed that multiple γ -TuRCs cluster inside TPX2 droplets spaced 0 . ± . µ m (mean ± standard deviation, N = 4 microtubules) apart along microtubules (Fig. 4a), consistent with a recent report[18]. The ratio between the number of γ -TuRCs inside TPX2 droplets to the number onbare regions of the same microtubules was 4 . ± . ± standard deviation, N = 3microtubules). Although this is an underestimate, given the difficulty of counting γ -TuRCs in TPX2 droplets, these results demonstrate that γ -TuRC preferentially localizesto TPX2 droplets along microtubules.The first step in branching is the binding of TPX2 to the microtubule, which thenlocalizes the other factors [4]. As such, we hypothesized that regularly spaced TPX2droplets lead to more efficient colocalization of factors than a uniform coating (Fig. 4b).For a uniform coating, multiple factors must search a greater length along the microtubulebefore finding each other to nucleate a new branch. With regularly spaced droplets, the10 Total number of each factor N C o l o c a li z a t i on t i m e k on l FilmDroplets N -1 k on l bcFig. 4a Factor 1Factor 2
Figure 4: A stochastic modelpredicts that TPX2 dropletsenhance the efficiency ofbranching microtubulenucleation. (a) Electronmicroscopy images show ring-shaped γ -TuRCs (black arrows)localizing to regularly spacedTPX2 droplets along the micro-tubule in the presence of augmin.Scale bars are 100 nm. (b)Schematic of branching factorsbinding to a microtubule coatedwith a uniform TPX2 layerversus periodic TPX2 droplets.(c) Monte Carlo simulationsshow that droplets colocalize twonecessary factors faster than auniform coating. Here, k on l/k off = 10. These results are notsensitive to this parameter choice(Supplementary Fig. 9). Eachdatapoint is the average of 10 independent simulations.11xplored distance is shorter, which reduces the search time. We performed kinetic MonteCarlo simulations [29] for two factors binding to (with rate k on ) and unbinding from(with rate k off ) a microtubule of length l with a uniform TPX2 coating and a periodicpattern of TPX2 droplets (Supplementary Theory). These results show that the time τ to colocalize on the microtubule, and hence the minimum time for nucleation of anew branch, is smaller for droplets than for a uniform layer (Fig. 4c, SupplementaryFig. 9). As a negative control for this model, we used AFM to measure the topographyof a C-terminal fragment of TPX2 on microtubules. This fragment is known to be lessefficient at nucleating branches in cytosol [5]. Consistent with our model, it did not formdroplets on microtubules (Supplementary Fig. 7a). Thus, synergistic with TPX2’s abilityto recruit tubulin [5] and its high concentration as a condensate, its organization intodroplets partitions the microtubule so that multiple factors can more easily find eachother. Taken together, TPX2’s phase behaviour enhances reaction kinetics via dropletpatterning, condensate concentration, and tubulin recruitment.It is important to think about our model in a cellular context. During cell division,TPX2 is released as a gradient in the vicinity of chromosomes [30]. The typical TPX2concentration in X. laevis is 90 nM [20] and the typical gradient length is ∼ µ m [30].This gives ∼ × TPX2 molecules that are available to condense on microtubules nearchromosomes, assuming a spherical volume. We estimate the concentration of TPX2 inthe condensed phase to be 10 µ m − using our AFM data (Methods). There are then ∼
80 TPX2 molecules needed to form a 10-nm-high condensed film on a typical 7- µ m-long, 25-nm-diameter microtubule [31]. Therefore, TPX2 can coat ∼ × microtubulesduring cell division. Given that the density of microtubules in the metaphase spindle is ∼ µ m − within 10 µ m of chromosomes [31], ∼ × microtubules lie in the vicinityof chromosomes. Thus, TPX2 can coat ∼
40% of the metaphase microtubule mass near12hromosomes at this film thickness. We hypothesize that the Rayleigh-Plateau instabilityis most relevant during early spindle assembly in order to accelerate the generation ofmicrotubules, as TPX2 is responsible for creating most of the spindle microtubules viabranching nucleation [32], in particular during early stages of spindle assembly [8].As the study of liquid-like protein condensates has intensified [1, 2], the physical phe-nomenology has been dominated by optical observations of droplets in solution or onmicrotubules. Here, we quantitatively demonstrate the emergence of non-trivial hydro-dynamic features such as films and spatiotemporal periodic instabilities that arise whena condensate interacts with a filament. In future work, it will be interesting to explorehow multiple proteins, such as TPX2 and BuGZ [17], condense on the microtubule, asmulti-protein condensates in solution have been reported [33]. We suspect that interfa-cial physics could manifest itself in other ways when condensates interact with cellularfilaments, such as via elastocapillary effects [34] that could produce forces between cy-toskeletal filaments or other semi-flexible macromolecules such as RNA and DNA [35].13 cknowledgements
We thank Drs. Stephanie Lee, Tseng-Ming Chou, and Matthew Libera at Stevens Insti-tute of Technology for access to their AFM; Drs. Ian Armstrong and Samrat Dutta atBruker for access to and support for their AFM; Drs. Matthew King, Benjamin Bratton,Mohammad Safari, Matthias Koch, Pierre Ronceray, and Ned Wingreen for helpful dis-cussions; Dr. Akanksha Thawani for purification of TPX2; Henry Ando, Caroline Holmes,physiology students Dr. Valentina Baena, Davis Laundon, and Linda Ma, and the Phys-iology Course at the Marine Biological Lab for assisting with the first AFM trials; andPrincetons Imaging and Analysis Center, which is partially supported by the PrincetonCenter for Complex Materials, an NSF-MRSEC program (DMR-1420541).B.G. was supported by PD Soros and NSF GRFP. S.U.S. was supported by NIHNCI NRSA 1F31CA236160 and NHGRI training grant 5T32HG003284. This work wasfunded by NIH NIA 1DP2GM123493, Pew Scholars Program 00027340, Packard Founda-tion 201440376, and CPBF NSF PHY-1734030.
Author contributions
S.U.S., B.G., J.W.S., H.A.S., and S.P. conceptualized the project. B.G. and S.U.S. per-formed fluorescence microscopy, and B.G. performed analysis of fluorescence microscopydata. B.G. performed TPX2-only electron microscopy with assistance from R.A. andS.U.S., theory, and simulations. S.U.S. performed atomic force microscopy with assistancefrom B.G., associated data analysis, and meiotic cytosol experiments. R.A. performedbranching reconstitution and multiple-protein electron microscopy, and S.U.S. performedanalysis of branching reconstitution data. S.U.S. and B.G. wrote the paper with assis-tance from J.W.S., H.A.S, and S.P. J.W.S., H.A.S., and S.P. supervised the research. All14uthors discussed and interpreted results and revised the paper.
Competing interests
The authors declare no competing interests.
Ethics
Data, code, and materials used are available upon request.Animal care was done in accordance with recommendations in the Guide for the Careand Use of Laboratory Animals of the NIH and the approved Institutional Animal Careand Use Committee (IACUC) protocol 1941-16 of Princeton University.15 hydrodynamic instability drives protein dropletformation on microtubules to nucleate branches:supplementary information
Sagar U. Setru † , , Bernardo Gouveia † , , Raymundo Alfaro-Aco , JoshuaW. Shaevitz ∗ , , , Howard A. Stone ∗ , , Sabine Petry ∗ , † These authors contributed equally. ∗ To whom correspondence should be addressed; E-mail: [email protected],[email protected], [email protected] upplementary Figures
Droplet spacing ( m) C oun t Inverse wavelength 1/ ( m -1 ) S pe c t r a l po w e r ( a . u . ) max = 0.6 0.1 m ab Droplet size ( m) C oun t Supplementary Fig. 1: Statistics of droplet patterned microtubules imagedwith TIRF microscopy. (a) Histogram of droplet sizes and spacings for TIRF exper-iments at 1 µ M GFP-TPX2. N = 35 microtubules were analyzed with a mean size of0 . ± . µ m and spacing of 0 . ± . µ m (mean ± standard deviation). (b) Average powerspectrum of GFP-TPX2 fluorescence intensities of droplet patterns for TIRF experimentsat 1 µ M GFP-TPX2 ( N = 35 microtubules). The peak indicates the emergence of aperiodic pattern with wavelength λ max = 0 . ± . µ m (mean ± standard deviation),in agreement with the histogram analysis. The shaded region represents 95% bootstrapconfidence intervals. 17 ig. S1
20 s 400 s 400 s440 s 440 s
TPX2, MT C o m po s i t e_ t b_eb1_ f r . pd f x C o m po s i t e_ t b_eb1_ f r . pd f x C o m po s i t e_ t b_eb1_ f r . pd f x t p x f r . pd f x t p x f r . pd f x t p x f r . pd f x C o m po s i t e_a ll f r . pd f x C o m po s i t e_a ll f r . pd f x C o m po s i t e_a ll f r . pd f x TPX2EB1, MT Merge
16 s ab Supplementary Fig. 2:TPX2 on the microtubulecan appear uniform whenimaged via optical mi-croscopy. (a) TIRF mi-croscopy time lapses showingthat a 0.1 µ M TPX2 coatingdoes not break up into visibledroplets like the 1 µ M TPX2coating does. (b) Branchingmicrotuble nucleation visualizedby TIRF microscopy in
X. laevis meiotic cytosol at 0.1 µ M TPX2,indicating that branching canoccur from diffraction limiteddroplets.18 -2 Inverse wavelength 1/ (nm -1 ) -3 -2 -1 S pe c t r a l po w e r ( n m ) UncoatedInitially coatedAfter droplet formation max = 248 nm
Length along microtubule (nm) H e i gh t ( n m ) UncoatedInitially coatedAfter droplet formation
Length along microtubule (nm) H e i gh t ( n m ) UncoatedInitially coatedAfter droplet formation
Raw height profilesSmoothed height profiles ab Supplementary Fig. 3: Rawand smoothed AFM heightprofiles, and power spectra ofraw height profiles. (a) Rawheight profiles of the topographiesin Fig. 2a. The smoothed pro-file from Fig. 2b is shown againfor reference. (b) Power spec-tra of the raw height profiles in(a). The red curve shows themean ± standard error of themean over nine topographies ofthe microtubule after the dropletpattern had formed. The fre-quency f at which the peak in thered curve occurs gives the dropletspacing measured for this micro-tubule, according to λ = 1 /f .19 ig. S2 Supplementary Fig. 4: The Rayleigh-Plateau instability.
The viscosity of thecondensed film is µ , γ is the surface tension of the interface, and p ∞ is the far fieldpressure provided by the solvent. The microtubule has radius r i . Initially, the interface isflat at ξ ( z, t = 0) = r o , but this scenario is unstable against the capillary pressure γ/r o ,so ξ ( z, t ) will evolve to a lower energy state. The unit normal n and unit tangent t trackthe geometry of the interface during its evolution.20 -2 Inverse wavelength 1/ (nm -1 ) -4 -2 S pe c t r a l po w e r ( n m ) UncoatedInitially coatedAfter droplet formation max = 268 nm -2 Inverse wavelength 1/ (nm -1 ) -4 -2 S pe c t r a l po w e r ( n m ) Initially coatedAfter droplet formation max = 338 nm -2 Inverse wavelength 1/ (nm -1 ) -4 -2 S pe c t r a l po w e r ( n m ) UncoatedInitially coatedAfter droplet formation max = 111 nm
Length along microtubule (nm) H e i gh t ( n m ) Length along microtubule (nm) H e i gh t ( n m ) Length along microtubule (nm) H e i gh t ( n m ) ± μ ± μ ± μ Length along microtubule (nm) H e i gh t ( n m ) UncoatedInitially coatedAfter droplet formation0 200 400 600 800 1000 1200
Length along microtubule (nm) H e i gh t ( n m ) UncoatedInitially coatedAfter droplet formation0 200 400 600 800 1000 1200
Length along microtubule (nm) H e i gh t ( n m ) UncoatedInitially coatedAfter droplet formation
Fig. S6
Supplementary Fig. 5: AFM height profiles and power spectra at additionalTPX2 concentrations.
For 0 . ± . µ M, the power spectrum is averaged over N = 5topographies after the droplet pattern had formed. For 0 . ± . µ M, N = 3. For0 . ± . µ M, N = 4; the uncoated height profile for this specific microtubule is unavailablebecause the sample moved after TPX2 addition. Height profiles were smoothed using amoving-average window of 40 nm; power spectra were taken from raw height profiles. Allpower spectra after droplet formation show mean ± standard error of the mean.21 Time nlDt I n t e r f a c e po s i t i on / r i Concentration c ( M) F il m t h i ck ne ss h / r i Fig. S3c ab increasing c Supplementary Fig. 6:Growth of the condensedfilm. (a) Schematic of the modelfor growth of the condensed pro-tein film. Microtubules of radius r i are spaced periodically by a dis-tance 2 ¯ R , where ¯ R = 1 / √ πnl where l is the typical microtubulelength and n is the number den-sity of microtubules. Solubleprotein with diffusivity D phaseseparates from solution and nu-cleates a spatially uniform con-densed film on the microtubulesurface, whose interfacial positionwe denote by r = ξ ( t ). (b) Fi-nal film thickness h versus ini-tial concentration c as measuredby atomic force microscopy (blue)and as predicted by equation (19)(black), using 1 /c R (cid:0) ¯ R /r − (cid:1) as a least-squares fit parameter.(c) Evolution of the interfacial po-sition of the film ξ/r i over time πnlDt for S = r i /r o ∈ [0 . , . S . Solid linesare the exact solution and dashedlines are the asymptotic formula(34b).22
50 100 150
Time t (min) -1 S pe c t r a l a m p li t ude ( n m ) Mean peak power exp( max t ) -3 -2 Inverse wavelength 1/ (nm -1 ) -2 -1 S pe c t r a l po w e r ( n m ) ab -2 Inverse wavelength 1/ (nm -1 ) -2 -1 S pe c t r a l po w e r ( n m ) Uncoated MTTPX2 initial coating1 0.5 M Kinesin-10.6 0.3 M C-terminal TPX2
Supplementary Fig. 7: Average power spectra from AFM data for all con-centrations of TPX2 and for uncoated, initially TPX2-coated, C-terminal-TPX2-bound, and kinesin-1-bound microtubules; the growth of the instabil-ity for early times is exponential. (a) Peaks indicate characteristic wavelengthsthat correspond to a typical droplet spacing (Supplementary Table 3, N = 18, 17, 14,and 8 microtubules, respectively, for increasing TPX2 concentration). Also includedare average power spectra for uncoated microtubules ( N = 29 microtubules), micro-tubules initially coated uniformly with TPX2 ( N = 25 microtubules), kinesin-boundmicrotubules ( N = 19 microtubules), and C-terminal-TPX2-bound microtubules ( N = 4microtubules)—none of which show any characteristic spatial features. For kinesin-boundmicrotubules, h = 2 . ± . h = 3 . ± . ± standard deviation. Shaded regions represent 95% bootstrap confi-dence intervals. (b) The average spectral amplitude (black line, N = 21 microtubules)at the most unstable frequency grows exponentially for early times. Spectral amplitude= √ spectral power. Individual measurements are black dots. The shaded region repre-sents 95% bootstrap confidence intervals. At later times, the spectral amplitude levelsoff due to nonlinear forces as the pattern sets in. For the exponential fit (red line), σ max = 0 .
03 min − , with R = 0 .
75. 23 ig. S8
Supplementary Fig. 8: γ -TuRC localization onbare microtubules. Typicalelectron microscopy experimentwith just γ -TuRC and micro-tubules. No TPX2 or augminis present. The localizationof γ -TuRC to the microtubulewithout TPX2 and augmin isnegligible (Supplementary Table4). Scale bar is 100 nm.24 Colocalization time k on l P r obab ili t y P DropletsFilm Total number of each factor N -1 C o l o c a li z a t i on t i m e k on l Film: s =100Film: s =10Film: s =1Droplets: s =100Droplets: s =10Droplets: s =1 Fig. S5ab
Supplementary Fig. 9:Parametric study of MonteCarlo simulations. (a) Time τ to colocalize two distinct factors,and hence the minimum time toform a branch, as a function ofthe total number of each factor N and s = k on l/k off for a uni-form and periodic protein coat-ing. For a given s , the peri-odic coating is always more effi-cient at colocalizing well-mixedfactors. Each data point is theaverage of 10 independent sim-ulations. (b) Typical histogramof 10 independent simulationsfor two factors with N = 50 and s = 10.25 Theory and simulation
Assuming conditions are met for the nucleation and growth of the condensed film to betime scale separated from the subsequent Rayleigh-Plateau instability (SupplementaryInformation 1.2), we proceed to analyze the stability of a condensed film of thickness h ona microtubule of radius r i (Supplementary Fig. 4). The nanometric length scales of thesystem are such that inertial effects are completely negligible, and so our starting pointis the Stokes equations ∇ · u = 0 (1a) µ ∇ u = ∇ p (1b)for the velocity u and pressure p fields in the condensate, where µ is the viscosity of thecondensate. This simplified starting point is the only difference between our derivationand that in [22].For the boundary conditions at r = r i we require no-slip on the microtubule surface u ( r = r i , z ) = . (2)At the interface r = ξ ( z, t ) between the film and the surrounding fluid, we assume thatthe viscosity of the solvent µ s is much less than µ , which is an excellent assumption forcondensed proteins [36]. Hence, we can write for the tangential stress condition t · τ · n = 0 at r = ξ ( z, t ) , (3)where τ = µ (cid:0) ∇ u + ∇ u T (cid:1) is the viscous stress tensor. n = ( e r − ξ (cid:48) e z ) / (cid:112) ξ (cid:48) is theunit normal and t = ( ξ (cid:48) e r + e z ) / (cid:112) ξ (cid:48) is the unit tangent expressed in local cylindrical26oordinates with e α · e β = δ αβ , where ξ (cid:48) = ∂ z ξ . When the interface is crossed, there is aYoung-Laplace jump in the normal stress due to the surface tension γ of the form − p ∞ − n · σ · n = γ ∇ · n at r = ξ ( z, t ) , (4)where σ = − p I + τ is the total stress. Lastly, we account for the kinematic conditionthat the interface advects with the flow( ∂ t + u · ∇ ) ( r − ξ ) = 0 at r = ξ ( z, t ) , (5)which closes the problem.To make progress, we assume that all motions are axisymmetric by the symmetry ofthe cylindrical geometry, so that u φ = 0 and ∂ φ (cid:55)→
0. In this case we can introduce thestream function ψ by u r = (1 /r ) ∂ z ψ and u z = − (1 /r ) ∂ r ψ , which automatically satisfiesequation (1a). Equations (1b) simplify to ∂ r p = µ (cid:18) ∂ r + 1 r ∂ r − r + ∂ z (cid:19) r ∂ z ψ (6a) ∂ z p = − µ (cid:18) ∂ r + 1 r ∂ r + ∂ z (cid:19) r ∂ r ψ. (6b)Taking ∂ z of equation (6a) and equating it to ∂ r of equation (6b) furnishes, after simpli-fication, the fourth-order equation D ψ = 0 , (7)where D = ∂ r − (1 /r ) ∂ r + ∂ z .The equilibrium solutions that satisfy equation (7) and boundary conditions (2-5) are ψ = 0, p = p ∞ + γ/r o , and ξ = r o . We now perform a linear stability analysis about theseequilibrium fields by positing the expansions( ψ, p, ξ ) = (0 , p ∞ + γ/r o , r o ) + (cid:88) k (cid:16) ˆ ψ k ( r ) , ˆ p k ( r ) , ˆ ξ k (cid:17) e σt + ikz , (8)27here we interpret the hatted quantities as small perturbations to the equilibrium solu-tions, and hence we will retain at most up to linear order in hatted quantities. Therefore,by linearity, we can conduct the analysis for each mode k separately, with the end goalbeing to arrive at the dispersion relation σ = σ ( k ).Substituting equation (8) into equation (7) results in D k ˆ ψ k = 0 , (9)where D k = d / d r − (1 /r ) d / d r − k . To solve equation (9), we construct the solutionˆ ψ k = ˆ ψ (1) k + ˆ ψ (2) k , where D k ˆ ψ (1) k = 0 and D k ˆ ψ (2) k = ˆ ψ (1) k . Making the substitution ˆ ψ (1) k = rζ ( r ) transforms D k ˆ ψ (1) k = 0 into a modified Bessel equation of order 1 for ζ ( r ), so thesolution is ˆ ψ (1) k = ArI ( kr ) + BrK ( kr ), where I p and K p are modified Bessel functions ofthe first and second kind of order p , respectively. We can now write D k ˆ ψ (2) k = rI ( qr ) + rK ( qr ), where we will let q → k . The particular solution is found by inspection tobe ˆ ψ (2) k = [ rI ( qr ) + rK ( qr )] / ( q − k ). We perform the necessary limit by letting q = k + (cid:15) and taking (cid:15) →
0. Extracting the only surviving linearly independent terms,while regularizing integration constants where necessary, furnishes the particular solutionˆ ψ (2) k = r I ( kr ) + r K ( kr ). Henceˆ ψ k = ArI ( kr ) + BrK ( kr ) + Cr I ( kr ) + Dr K ( kr ) , (10)where the integration constants A, B, C and D are to be determined by the boundaryconditions (2-5).It is now convenient to define the dimensionless variables K = kr o , Σ = µr o σ/γ , and S = r i /r o . We substitute the stream function (10) into the no-slip condition (2), resultingin AI ( KS ) + BK ( KS ) + ˜ CI ( KS ) + ˜ DK ( KS ) = 0 (11a) AI ( KS ) − BK ( KS ) + ˜ CI ( KS ) − ˜ DK ( KS ) = 0 , (11b)28here ˜ C = Cr o and ˜ D = Dr o . Substituting the expansion (8) into the tangential stresscondition (3) while retaining only the leading-order terms, then using the stream function(10) gives2 AI ( K ) + 2 BK ( K ) + ˜ C [ I ( K ) + I ( K )] + ˜ D [ K ( K ) + K ( K )] = 0 . (12)Next, we substitute the expansion (8) into the normal stress condition (4), replacing ˆ p k with ˆ ψ k by taking ∂ z of equation (4) and using equation (6b). We then replace ˆ ξ k by ˆ ψ k using the kinematic condition (5). The leading order result is AF A + BF B + ˜ CF ˜ C + ˜ DF ˜ D = 0 , (13)where F A = − I ( K ) / /K − K/ Σ + 1 / Σ K ] I ( K ) − I ( K ) (14a) F B = 3 K ( K ) / /K − K/ Σ + 1 / Σ K ] K ( K ) + K ( K ) (14b) F ˜ C = 2 I ( K ) /K − I ( K ) + [2 /K − K/ Σ + 1 / Σ K ] I ( K ) (14c) F ˜ D = 2 K ( K ) /K + 2 K ( K ) + [2 /K − K/ Σ + 1 / Σ K ] K ( K ) . (14d)Equations (11-13) are a set of four linear equations for the constants A , B , ˜ C , and ˜ D .For there to be non-trivial solutions, we require that the matrix of the linear system hasvanishing determinant. This condition establishes the dispersion relation between Σ and K , Σ ( K ; S ) = ( K − G ( K, S )2 K G ( K, S ) , (15)29here G = 1 + KS (( − KSI ( KS ) + 2 I ( KS ) I ( KS ) + KSI ( KS )) K ( K )+ I ( K )( − KSK ( KS ) − K ( KS ) K ( KS ) + KSK ( KS ))+ I ( K ) K ( K )( − KS + I ( KS )((2 + K S ) K ( KS ) − KSK ( KS ))+ I ( KS )( − KSK ( KS ) + ( − K S ) K ( KS )))) (16)and G = 1 + 2 /K + S + KSI ( K )( KSK ( KS ) + 2 K ( KS ) K ( KS ) − KSK ( KS ))+ S/K ( I ( KS )( K SK ( K ) − KS (1 + K ) K ( K ))+ I ( KS )( − K SK ( K ) + KS (1 + K ) K ( K ))+ I ( K ) I ( KS ) K ( K )((2 + K (2 + S )) K ( KS ) − K ) KSK ( KS ))+ I ( KS )( I ( KS )( − K K ( K ) + 2(1 + K ) K ( K ))+ I ( K ) K ( K )( − K ) KSK ( KS ) + ( − K ( − S )) K ( KS )))+ I ( K )( − KSK ( K )+(1+ K ) I ( K )( − KSK ( KS ) − K ( KS ) K ( KS )+ KSK ( KS )))) − I ( K ) K ( K )( − K S ( I ( KS ) K ( KS ) + I ( KS ) K ( KS ))) (17)The observed dimensionless wavelength Λ max corresponds to the maximal growth rate,i.e., Λ max ( S ) = 2 π/ argmax K Σ ( K ; S ), and it depends only on the geometric parameter S (Fig. 3b).One might suspect that the rheology of protein condensates is not truly Newtonian,and that viscoelastic effects might play a role [37]. However, it has been shown that sucheffects are subdominant on wavelength selection in this instability, even at high Weis-senberg number, which measures the magnitude of elastic effects. Finite viscoelasticityonly “swells” the droplets without affecting their peak-to-peak spacing, as well as slowsdown the instability time scale [23]. 30n order to contextualize our experimental measurements, we can estimate the capillaryvelocity γ/µ by noting that the typical time scale for droplet formation T ∼ r o µ/γ . Using r o = O(10) nm as the typical length scale (from Fig. 2b) and using T between O(10) s(TIRF conditions; Fig. 1a and Supplementary Movie 1) and O(10 ) s (AFM conditions;Supplementary Fig. 7b), we estimate a capillary velocity γ/µ between O(1) nm / s andO(10 − ) nm / s. In Supplementary Information 1.1, we assumed that the nucleation and growth of thecondensed film on the microtubule occurred much faster than the subsequent Rayleigh-Plateau instability, and so we neglected it in our hydrodynamic model. This is indeedwhat we observe experimentally. Here, we explicitly consider the nucleation and growthdynamics, thereby elucidating the precise conditions where we expect the observed timescale separation to hold.We consider a suspension of microtubules in a pool of soluble protein, whose concentra-tion field we denote c = c ( x , t ). In our experiments, we spike in protein at a concentration c ( x , t = 0) = c , where c is large enough such that there is a thermodynamic drivingforce for the soluble protein to phase separate into a protein-rich condensed phase of con-centration c R and a protein-poor “gas” phase of concentration c P , where c P < c < c R . c P and c R are constrained by the lever rule and are in general functions of c [38]. Forsimplicity we take each microtubule to be periodically spaced by a distance 2 ¯ R , so thatthe number density of microtubules in solution is n = 1 /π ¯ R l (Supplementary Fig. 6a).Given this geometry we can also assume radial symmetry throughout: c = c ( r, t ) only.At t = 0, phase separation begins, and the microtubules offer sites for heterogeneousnucleation of the condensed phase. For simplicity we neglect any homogeneous nucleation31f condensed protein droplets in the bulk. We assume that all of the initial soluble proteinpool eventually contributes to the condensed film on the microtubules, effectively reaching c P = 0, which is a nonessential but mathematically simplifying assumption in polymerphase separation [38, 39]. As a check, we compute the ratio of fluorescence intensitiesbetween condensed TPX2 on the microtubule versus background soluble TPX2 (Supple-mentary Fig. 2a), finding c P /c R between 10 − and 10 − over all tested concentrations,consistent with our assumption.We consider the transport of soluble protein towards the condensed film to be diffusive,so that ∂ t c = D ∇ c in the bulk, where D is the diffusivity constant of the soluble protein.We denote the position of the film interface as r = ξ ( t ), which will grow in time as solubleprotein from the bulk diffuses onto the film. Hence we can write the global mass balancedd t (cid:0) c R (cid:2) πξ l − πr l (cid:3)(cid:1) = − dd t (cid:90) ¯ Rξ πlr d r c ( r, t ) (18a)= ⇒ ξ ( t ) = c c R ¯ R + r (cid:18) − c c R (cid:19) − (cid:90) ¯ Rξ r d r c ( r, t ) c R , (18b)where in going from equation (18a) to equation (18b) we integrate in time and apply theinitial conditions c ( r, t = 0) = c and ξ ( t = 0) = r i . By just considering equation (18b)we see that the final film thickness h = ξ ( t → ∞ ) − r i is reached after depleting thecondensable solute c ( r, t → ∞ ) →
0. Therefore r o := r i + h = (cid:115) c c R ¯ R + r (cid:18) − c c R (cid:19) , (19)where we have defined the final position of the film interface as r o for later convenience.The main feature of (19) is that it predicts the film thickness h to be an increasingfunction of protein concentration c and a decreasing function of microtubule density n .To check if the relationship h = h ( c ) as specified by equation (19) is consistent with ourexperimental AFM observations, we plot measured values of h against c and fit the data32o equation (19) (Supplementary Fig. 6b). In doing so we neglect the dependence of c R on c . We observe qualitative agreement, suggesting that the final film thickness is indeedset by depletion of locally condensable solute.At r = ξ ( t ) there is a sink of soluble protein as it is adsorbed onto the condensedfilm, and so we demand that c ( r = ξ, t ) = 0, which enforces the driving force for thesoluble protein to diffuse towards the film. Due to our assumption of a periodic arrayof microtubules, we can additionally enforce the symmetry condition ∂ r c (cid:0) r = ¯ R, t (cid:1) = 0.This closes the problem statement for the desired fields c and ξ , with the goal being todetermine the dynamics of the film thickness h ( t ) = ξ ( t ) − r i .To solve for the fields, we work with the rescaled variables C = c/c , R = r/ ¯ R , I = ξ/r o , and T = t/ (cid:0) ¯ R /D (cid:1) . The rescaling has been carefully chosen such that C ∈ [0 ,
1] and I ∈ [ S,
1] for all parameter values. This choice dramatically speeds up the subsequent rootfinding calculations. Under these transformations the boundary value problem becomes ∂ T C = 1 /R ∂ R ( R ∂ R C ) (20a) C ( R, T = 0) = 1 (20b) C ( R = M I ( T ) , T ) = 0 (20c) ∂ R C ( R = 1 , T ) = 0 (20d) I (0) = S (20e) I ( T ) = 1 − βM (cid:90) MI ( T ) R d R C ( R, T ) , (20f)where β = c /c R is a thermodynamic parameter, S = r i /r o is a geometric ratio, and M = r o / ¯ R is a measure of the microtubule density. For the remainder of this sectionwe take M = 1 /
10. However, equation (19) relates these parameters, so only two areindependent. Squaring equation (19), dividing through by ¯ R , and using the definitions33or β , S , and M gives, after rearrangement, β = M (1 − S )1 − M S . (21)In our AFM experiments described in the main text, we directly measure S and fix M , andso the only value of β self-consistent with our theory is given by equation (21). For ourexperimentally observed range of S ∈ [0 . , . β = O (10 − ), which is consistentwith c R > c .Equations (20) would be a standard boundary value problem were it not for theintegral constraint (20f). Nevertheless, we can proceed with the separation anzats C = f T ( T ) φ ( R ; I ) where the notation φ ( R ; I ) signifies that we treat I as a fixed parameterwhen treating φ . This construction is valid so long as the dynamics of I ( T ) quasi-statically adjusts to the dynamics of C ( R, T ). Proceeding with the usual steps [40], thetwo boundary conditions (20c,d) can be satisfied by the eigenfunctions φ = φ n ( R ) = J ( λ n R ) − [ J ( λ n ) /Y ( λ n )] Y ( λ n R ), where the eigenvalues satisfy the time-dependent relation J ( λ n M I ) Y ( λ n ) − Y ( λ n M I ) J ( λ n ) = 0 . (22) J p and Y p are Bessel functions of the first and second kind of order p , respectively. Thisis to say that for every I = I ( T ) there is a corresponding set of eigenvalues λ n . We alsonote that, for any choice of I , λ < λ < . . . . We find that f T = e − λ n T , so that thesolution may be represented by C ( R, T ) = ∞ (cid:88) n =1 a n e − λ n T φ n ( R ) , (23)where the a n are chosen to satisfy (20b) and are calculated using orthogonality of the eigen-functions under the inner product (cid:104) f, g (cid:105) = (cid:82) MI R d R f g . We find a n = (cid:104) φ n , (cid:105) / (cid:104) φ n , φ n (cid:105) ,34here (cid:104) φ n , (cid:105) = { J ( λ n ) − M IJ ( λ n M I ) + J ( λ n ) /Y ( λ n ) [ M IY ( λ n M I ) − Y ( λ n )] } /λ n (24) (cid:104) φ n , φ n (cid:105) = { J ( λ n ) + J ( λ n ) − ( M I ) [ J ( λ n M I ) + J ( λ n M I )]+ 2 J ( λ n ) /Y ( λ n )[( M I ) [ J ( λ n M I ) Y ( λ n M I ) + J ( λ n M I ) Y ( λ n M I )] − J ( λ n ) Y ( λ n ) − J ( λ n ) Y ( λ n )] + J ( λ n ) /Y ( λ n )[ Y ( λ n ) + Y ( λ n ) − ( M I ) [ Y ( λ n M I ) + Y ( λ n M I )]] } / . (25)Upon substitution of the expansion (23), the integral constraint (20f) becomes F ( I ) := I ( T ) + 2 βM ∞ (cid:88) n =1 a n e − λ n T (cid:104) φ n , (cid:105) − . (26)The solutions C ( R, T ) and I ( T ) are constructed in the following self-consistent way.For every time T we numerically compute the zero of F ( I ) defined by equation (26). Thisgives the value of I that satisfies the integral constraint, which then fixes the values of λ n by equation (22). Hence, the solution C ( R, T ) is fully determined by the expansion (23).We plot the position of the film interface I ( T ) for various S within our experimentallyprobed range (Supplementary Fig. 6c).Equation (26) is a nonlinear, implicit relation for I ( T ), and therefore admits no suc-cinct closed form solution. However, we can derive explicit asymptotic representations for I ( T ) working in the early and late time limits.For early times t →
0, the soluble bulk has hardly been depleted, so it is as if thefilm grows in an unbounded medium of solute. Therefore we may replace the integralconstraint (20f) with dd t (cid:0) c R (cid:2) πξ l − πr l (cid:3)(cid:1) = (cid:32) D ∂c∂r (cid:12)(cid:12)(cid:12)(cid:12) r = ξ (cid:33) πξl (27a)= ⇒ d ξ d t = Dc R ∂c∂r (cid:12)(cid:12)(cid:12)(cid:12) r = ξ , (27b)35hich is the statement that the film grows due to the diffusive flux of solute at r = ξ ( t ).Moreover, the film interface is far from reaching its final value r o , so to analyze this limitwe choose the different rescaling X = r/r i − δ = ξ/r i −
1, and ˜ T = t/ ( r /D ). Thisrescaling measures distances from the surface of the microtuble, so that δ (cid:28) T → ∂ ˜ T C = ∂ X C + ∂ X C/ (1 + X ) (28a) C (cid:16) X, ˜ T = 0 (cid:17) = 1 (28b) C (cid:16) X = δ (cid:16) ˜ T (cid:17) , ˜ T (cid:17) = 0 (28c) C (cid:16) X → ∞ , ˜ T (cid:17) → δ (0) = 0 (28e)d δ d ˜ T = β ∂C∂X (cid:12)(cid:12)(cid:12)(cid:12) X = δ . (28f)We now introduce the similarity transformation η = X/δ and demand that C = C ( η )only. Under these conditions equation (28a) becomes − η δ d δ d ˜ T C (cid:48) = C (cid:48)(cid:48) δ + C (cid:48) δ (1 + ηδ ) , (29)where ( . ) (cid:48) = d( . ) / d η . For ˜ T → η = O (1) near the film and δ (cid:28)
1, hence we can neglect ηδ compared to 1. Doing so and rearranging gives − (cid:18) ηδ d δ d ˜ T + δ (cid:19) C (cid:48) = C (cid:48)(cid:48) . (30)On physical grounds we assume the ordering O (cid:16) d δ/ d ˜ T (cid:17) > O ( δ ), so we can again ignorethe isolated δ to find − η (cid:18) δ d δ d ˜ T (cid:19) C (cid:48) = C (cid:48)(cid:48) . (31)To satisfy C = C ( η ) only, we are forced to choose δ d δ/ d ˜ T = b for some constant b to be determined self-consistently later. Using the initial condition (28e) this equation36ntegrates to δ = (cid:112) b ˜ T , which allows for the three conditions (28b-d) to collapse to onlytwo conditions C ( η = 1) = 0 and C ( η → ∞ ) →
1. Using these conditions the solution toequation (31) is C = 1 − erfc (cid:0) η/ √ (cid:1) erfc (cid:0) / √ (cid:1) , (32)where erfc ( z ) = 2 / √ π (cid:82) ∞ z d x e − x is the complementary error function. Substitutingequation (32) into equation (28f) determines b = δ d δ/ d ˜ T = 2 β/ (cid:0) √ πe erfc (cid:0) / √ (cid:1)(cid:1) . Thefinal asymptotic formula for the film thickness is therefore ξr i ∼ (cid:115) β √ πe erfc (cid:0) / √ (cid:1) ˜ T for ˜ T → , (33)which displays the typical ∼ (cid:112) ˜ T behavior expected for an interface that grows by diffusiveprocesses.For late times t → ∞ , the leading term n = 1 in the expansion (23) surely dominates.Moreover, I ( T → ∞ ) ∼
1, so we can simply evaluate the first eigenvalue λ using equation(22) taking I = 1. We find λ (cid:39) . C ( R, T ) ∼ ( a φ ( R )) | I =1 e − λ T (34a) I ( T ) ∼ (cid:113) − β/M ( a (cid:104) φ , (cid:105) ) | I =1 e − λ T for T → ∞ . (34b)Fortuitously, equation (34b) gives a uniformly excellent approximation to the exact resultfor all times (Supplementary Fig. 6c).In this section, we have established the dynamics for the nucleation and growth ofa condensed film on a microtubule. As long as these dynamics are fast compared tothe subsequent Rayleigh-Plateau instability, i.e. ¯ R /D (cid:28) r o µ/γ , we may neglect themin the hydrodynamic analysis. Substituting the estimates r o µ/γ (cid:39) / .
03 min (fromSupplementary Fig. 7b), ¯ R (cid:39) µ m, and D (cid:39) µ m / s we find that ¯ R /D = O(10 − ) sand r o µ/γ = O(10 ) s. Hence time scale separation is well satisfied.37 .3 Kinetic Monte Carlo simulations We now consider the stochastic chemical kinetics involved in F branching factors bindingto and unbinding from a microtubule lattice of length l coated with condensed protein.We wish to determine the first time τ at which F factors find each other on the proteincoated lattice for two different coating geometries: a uniform coating and a periodiccoating of droplets with spacing λ . This serves as a model for how multiple necessaryfactors may find each other on a microtubule and subsequently nucleate a branch, and weseek to answer the question of whether a periodic coating of protein droplets acceleratesthis process.For concreteness, we choose l = 5 µ m and λ = 0 . µ m, although these choices donot affect the results. We fix d = 25 nm to be the distance below which we considerthe F factors to have colocalized, which is the typical size of a γ -TuRC. Let N ( u ) α ( t ) and N ( b ) α ( t ) be the random variables specifying the number of well-mixed unbound and boundfactors of species α , respectively, where α = 1 , , . . . , F . We start the dynamics with N ( u ) α (0) = N and N ( b ) α (0) = 0. Each distinct factor can bind to the coated lattice withrate constant k on and unbind with rate constant k off . Factors can only bind to the coatedlattice (Fig. 4a), so the total length of available binding sites is L = l for a uniform coatingand L = l/ s = /k off /k on l .We set k on = 1 . · − µ m sec − [4] and study the dynamics with respect to s , keepingin mind that microtubule associated proteins bind tightly so that s >
1. We assumebinding and unbinding events are Poisson processes with rates r α, on ( t ) = k on LN ( u ) α ( t )and r α, off ( t ) = k off N ( b ) α ( t ), respectively.The dynamics are updated using a standard Gillespie algorithm [29]. In this schemetime is updated by the rule t i +1 = t i − log (1 − u ) /m , where u is a uniform random38ariable on [0 ,
1] and m = (cid:80) α r α, on ( t i ) + r α, off ( t i ). One event is then realized at time t i +1 , a binding event of a factor of species α with probability P ( b ) α = r α, on ( t i ) /m or anunbinding event of a factor of species α with probability P ( u ) α = r α, off ( t i ) /m . If a bindingevent occurs, the factor may bind anywhere on the coated lattice with uniform probability.The dynamics are stopped once any F distinct factors come within a distance d of eachother, and the colocalization time τ is reported.Considering F = 2, which is the minimum number of distinct factors needed forbranching nucleation, we find that a periodic coating is more efficient at colocalizingfactors than a uniform coating for all relevant values of s and N (Supplementary Fig. 9a).Of course, as N increases the time τ to colocalize decreases, as there are simply morefactors that can sample the lattice. As s decreases, the factors are more likely to unbindand hence can more efficiently sample the lattice, and so τ correspondingly decreases. Wealso provide a typical histogram that shows a long time tail, characteristic of Poissonianchemical kinetics (Supplementary Fig. 9b). Recombinant GFP-TPX2 was purified as previously described [4, 41]. Full-length TPX2or C-terminal TPX2 (amino acids 377 to 716) tagged on the N-terminus with Strep-6xHis-GFP-TEV was cloned into the pST50 expression vector. TPX2 was expressed in E. coli(strain Rosetta 2) for 7 hours at 25 ◦ C and the cells lysed using EmulsiFlex (Avestin) inlysis buffer (50 mM tris-HCl, pH 8.0, 150mM imidazole, 750 mM NaCl, 2.5 mM PMSF,6 mM BME, 1 cOmplete TM EDTA-free Protease Inhibitor tablet (Sigma), and 1000 UDNase I (Sigma)). Then, the protein was affinity purified using Ni-NTA beads in bindingbuffer (50 mM tris-HCl pH 8.0, 750 mM NaCl, 15 mM imidazole, 2.5 mM PMSF, 6 mM39ME). Next, the protein was eluted with 200 mM imidazole, and the protein was pooledand diluted 4-fold to a final NaCl concentration of 200 mM. After, nucleotides wereremoved from the eluted protein with a HiTrap Heparin HP column (GE Healthcare).Protein was bound to the column in 250 mM NaCl and then isocratic elution was done in750 mM NaCl. Solutions were prepared in Heparin buffer (50 mM tris-HCl, pH 8.0, 2.5mM PMSF, 6 mM BME). Finally, peak fractions were pooled and loaded onto a Superdex200 pg 16/600 gel filtration column. Gel filtration was performed in CSF-XB buffer (10mM K-HEPES, pH 7.7, 100 mM KCl, 1 mM MgCl , 5 mM EGTA) with 10% (w/v)sucrose. TPX2 was flash frozen in liquid nitrogen and stored at -80 ◦ C until use. ThisTPX2 was used for all fluorescence, electron, and atomic force microscopy experiments.Recombinant GFP-tagged augmin holocomplex was purified as previously described [18,42]. Briefly, Sf9 cells were coinfected with several baculoviruses, each carrying a subunitof the augmin complex, at multiplicity of infection (MOI) values of 1–3. Cells were col-lected and lysed 72 hours after infection. Augmin subunit HAUS6 had an N-terminalZZ-tag, subunit HAUS2 had a C-terminal GFP-6xHis, and the other six subunits wereuntagged. The complex was affinity purified using IgG-Sepharose (GE Healthcare) andeluted via cleavage with 100200 µ g of GST-HRV3C protease. The HRV3C protease wassubsequently removed using a GSTrap 5 mL column (GE Healthcare). The sample wasfurther purified and concentrated using Ni-NTA agarose beads. The protein complex wasdialyzed overnight into CSF-XB with 10% (w/v) sucrose. Augmin was flash frozen inliquid nitrogen and stored at -80 ◦ C until use. γ -TuRC was purified from X. laevis meiotic cytoplasm as done previously [18]. First,5-10 mL of cytoplasm was prepared from
X. leavis eggs using standard procedures [43,44]and 10-fold diluted in CSF-XB with 10% sucrose (w/v), 10 µ g each of leupeptin, pepstatin,and chymostatin, 1 mM GTP, and 1 mM DTT. The larger particles in the cytosol were40emoved by low-speed centrifugation at 3000 g for 10 minutes at 4 ◦ C. The supernatantwas 2-fold diluted with buffer and passed through 1.2 µ m, 0.8 µ m and 0.22 µ m poresize filters. γ -TuRC was precipitated by adding a solution of 6.5% (w/v) PEG 8000 tothe filtrate on ice for 30 minutes. After centrifuging at 17000 g for 20 minutes at 4 ◦ C,the pellet was resuspended in 15 mL of CSF-XB with 10% sucrose (w/v), 10 µ g each ofleupeptin, pepstatin, and chymostatin, 1 mM GTP, 1 mM DTT, and 0.05% (v/v) NP-40detergent. The resuspension was centrifuged at 136000 g for 7 minutes at 4 ◦ C. Then,protein A sepharose beads (GE Healthcare) were used to pre-clear the supernatant for 20minutes at 4 ◦ C. The beads were removed via centrifugation. 2-4 mL of γ -tubulin antibody(Sigma) at 1 mg/mL was added and the mixture was rotated for 2 hours at 4 ◦ C. Afterthis, 1 mL of washed protein A sepharose beads was incubated with the sample on arotator for 2 hours at 4 ◦ C. The beads were collected by spinning and then transferredto a column with CSF-XB with 10% sucrose (w/v), 10 µ g each of leupeptin, pepstatin,and chymostatin, 1 mM GTP, 1 mM DTT, and 0.05% (v/v) NP-40 detergent. The beadswere washed with CSF-XB with 10% sucrose (w/v), 10 µ g each of leupeptin, pepstatin,and chymostatin, 1 mM GTP, 1 mM DTT, and 150 mM additional KCl, then with CSF-XB with 1 mM ATP, and finally with CSF-XB to remove the ATP. For biotinylation of γ -TuRC, done for visualizing the reconstitution using fluorescence microscopy, 25 µ M ofNHS-PEG4-biotin (Thermo Scientific) was incubated with the beads in CSF-XB for 1hour at 4 ◦ C, and unreacted reagent was washed away with CSF-XB before elution. Toelute γ -TuRC from the beads, 2 mL of a γ -tubulin peptide (amino acids 412451) at 0.5mg/mL in CSF-XB was added to the column and incubated overnight. The eluted samplewas collected from the column the following day. γ -TuRC was then concentrated usinga 100 kDa centrifuge filter (Amicon). This concentrated sample was loaded onto a 10-50% (w/w) continuous sucrose gradient in CSF-XB buffer with 10 µ g each of leupeptin,41epstatin, and chymostatin, 1 mM GTP, and 1 mM DTT. The gradient was centrifugedfor 3 hours at 4 ◦ C at 200000 g in a TLS55 rotor (Beckman). Fractions of the sucrosegradient were collected manually from the top of the gradient. The fractions with thehighest γ -tubulin signal as determined by western blot were combined and concentratedas above using a 100 kDa centrifuge filter. Purified γ -TuRC was used within two dayswithout freezing and kept on ice.The kinesin-1 fragment K560-GFP was purified as previously described [45]. Aminoacids 1-560 of human conventional kinesin tagged on the C-terminus with GFP and 6xHiswas cloned into pET17B vector [45]. K560-GFP was expressed in E. coli for 4 hours at37 ◦ C and the cells lysed as above using lysis buffer (50 mM NaPO , pH 8.0, 250 mMNaCl, 1 mM MgCl , 20 mM imidazole, 10 mM BME, 0.5 mM PMSF, 0.5 mM ATP, 1cOmplete TM EDTA-free Protease Inhibitor tablet, 1000 U DNase I). Then, K560-GFPwas affinity purified using a HisTrap HP column (GE Healthcare) using binding buffer(50 mM NaPO , pH 8.0, 250 mM NaCl, 1 mM MgCl , 20 mM imidazole, 10 mM BME,0.5 mM PMSF, 0.5 mM ATP) and eluted using binding buffer at 500 mM imidazole. Peakfractions were pooled and dialyzed overnight at 4 ◦ C into CSF-XB buffer.Bovine brain tubulin was labelled with Cy5 or Alexa568 NHS ester (GE Healthcare)or biotin-PEG4-NHS (Thermo Scientific) using methods previously described [46].Protein concentrations were determined by SDS-PAGE and then Coomassie stainingof a concentration series of the protein of interest alongside a BSA standard, or usingBradford dye (Bio-Rad). 42 .2 Expression and purification of RanQ69L and EB1-mCherryfor
Xenopus laevis meiotic cytosol experiments
Proteins used to visualize branching microtubule nucleation in
X. laevis meiotic cytosol,RanQ69L and EB1-mCherry, were purified as previously described [47]. RanQ69L, whichwas N-terminally BFP-tagged to improve solubility, was expressed and lysed as abovein lysis buffer (100 mM tris-HCl, pH 8.0, 450 mM NaCl, 1 mM MgCl , 1 mM EDTA,0.5 mM PMSF, 6 mM BME, 200 µ M GTP, 1 cOmplete TM EDTA-free Protease Inhibitor,1000 U DNAse I). RanQ69L was affinity purified using a StrepTrap HP 5 mL column (GEHealthcare) in binding buffer (100 mM tris-HCl, pH 8.0, 450 mM NaCl, 1 mM MgCl , 1mM EDTA, 0.5 mM PMSF, 6 mM BME, 200 µ M GTP) and eluted using binding bufferwith 2.5 mM D-desthiobiotin. Eluted protein was dialyzed overnight into CSF-XB with10% (w/v) sucrose. EB1-mCherry was expressed and lysed as above in lysis buffer (50mM NaPO , pH 7.4, 500 mM NaCl, 20 mM imidazole, 2.5 mM PMSF, 6 mM BME, 1cOmplete TM EDTA-free Protease Inhibitor, 1000 U DNAse I), affinity purified using aHisTrap HP 5 mL column (GE Healthcare) in binding buffer (50 mM NaPO , pH 7.4,500 mM NaCl, 20 mM imidazole, 2.5 mM PMSF, 6 mM BME) and eluted using bindingbuffer with 500 mM imidazole. Finally, peak fractions were pooled and loaded onto aSuperdex 200 pg 16/600 gel filtration column. Gel filtration was performed in CSF-XBwith 10% (w/v) sucrose. Xenopus laevis meiotic cytosol preparation and immunode-pletion
Meiotic cytosol, also known as CSF extract or egg extract, was prepared from
Xenopuslaevis eggs using standard protocols [43, 44]. Cytosol was diluted up to 75% for TIRFmicroscopy experiments. For depletion of TPX2 from cytosol, 30 µ g of purified anti-TPX243ntibody was coupled to 200 µ L protein A magnetic beads (Dynabeads, ThermoFisher)overnight. Then, 200 µ L of fresh meiotic cytosol was depleted in two rounds by incubatingwith 100 µ L beads for 30 minutes in each round. Control depletion was done using rabbitIgG antibody (Sigma). The efficiency of depletion was determined by western blot andconfirmed using functional tests.
Xenopus laevis frogs were cared for in accordance withthe recommendations in the Guide for the Care and Use of Laboratory Animals of theNational Institutes of Health. All animals were cared for according to the approvedInstitutional Animal Care and Use Committee (IACUC) protocol 1941-16 of PrincetonUniversity.
For TIRF microscopy experiments, double-cycled GMPCPP-stabilized microtubules wereprepared [46]. Bovine tubulin at 20 µ M was polymerized in BRB80 (80 mM K-PIPES,pH 6.8, 1 mM EGTA, 1 mM MgCl ), with 10 mM GMPCPP (Jena Bioscience), 2 mMAlexa568-labelled tubulin, and 2 mM biotin-labelled tubulin, at 37 ◦ C for 30 minutes. Mi-crotubules were pelleted via centrifugation at 126,000 g for 8 minutes, and depolymerizedby resuspending and incubating in ice cold BRB80 totalling 80% of original volume for 20minutes. GMPCPP was then added up to 10 mM, and microtubules were again polymer-ized and pelleted as in the first cycle. Microtubules were flash frozen in liquid nitrogenand quickly thawed prior to use. Microtubules were used at a final 1000-fold to 2000-folddilution for TIRF experiments.For electron and atomic force microscopy, bovine tubulin at 20 µ M was incubated inBRB80 with 10 mM GMPCPP (Jena Bioscience) on ice for 5 minutes and then centrifuged90 kRPM in a TLA100 rotor (Beckman) at 4 ◦ C for 15 minutes. The supernatant was thenincubated at 37 ◦ C for 30 to 60 minutes to polymerize microtubules. Finally, the reaction44ixture was 5-fold diluted and microtubules were pelleted via centrifugation as aboveat 27 ◦ C for 5 minutes. The supernatant was discarded and microtubules resuspended inthe original volume of BRB80. Microtubules were made fresh for each imaging day anddiluted 500-fold for electron microscopy and 10-fold for atomic force microscopy. in vitro
Branching microtubule nucleation was reconstituted in vitro using purified proteins as wasdone recently [18]. Briefly, microtubules were attached to a flow chamber made with PEG-passivated and biotinylated glass coverslips. A mixture of TPX2 (0.05 µ M), augmin (0.05 µ M), and γ -TuRC was incubated on ice for 5 minutes and then flowed into the chamber.Proteins were allowed to bind microtubules and excess protein was flowed out. Then,BRB80 with 30 mM KCl, 1 mM GTP, 5 mM BME, 0.075% (w/v) methylcellulose (4000cP), 1% (w/v) glucose, 0.02% (v/v) Brij-35, 250 nM glucose oxidase, 64 nM catalase, 1mg/mL BSA, 19 µ M unlabeled bovine tubulin and 1 µ M Cy5-labeled bovine tubulin wasflowed into the chamber to initiate the nucleation and growth of branched microtubules,which was visualized via TIRF microscopy.
Coverslip-bottomed culture well plates (Grace BioLabs) were functionalized with anti-Biotin antibody (Abcam) at 0.1 mg/mL concentration at room temperature for 10 min-utes. Excess antibody was washed out by serially diluting in buffer (CSF-XB at 50-60 mMKCl or BRB80) three times, and then the coverslip blocked with κ -casein at 1 mg/mL.Excess κ -casein was washed out by serially diluting in buffer as above. Then, microtubuleswere allowed to settle and bind the coverslip surface for 10 minutes at room temperatureand then unbound microtubules were removed via serial dilution as above. TPX2 was45hen pipetted into the wells during image acquisition, to visualize TPX2 coating andforming droplets on microtubules.Total internal reflection fluorescence (TIRF) microscopy was done using a Nikon TiEmicroscope with a 100x magnification, 1.49 NA objective. An Andor Zyla sCMOS camerawas used for acquisition. Images were acquired using NIS-Elements software (Nikon).For imaging of the reconstitution of branching microtubule nucleation, the objective washeated to 33 ◦ C using an objective heater (Bioptechs).
Droplet sizes and spacings used to create Supplementary Fig. 1a were calculated manuallyusing FIJI (ImageJ) [48]. To calculate the average power spectrum in SupplementaryFig. 1b, line scans of droplet patterned microtubules in the raw TIRF images were takenin FIJI and Fourier transformed in MATLAB. The power spectrum is given by the formula P ( f ) = | ˆ I ( f ) | = (cid:12)(cid:12)(cid:12)(cid:12) L (cid:90) L I ( z ) e − πifz dz (cid:12)(cid:12)(cid:12)(cid:12) , (35)where I is the intensity, z is the position along the microtubule, L is the length of themicrotubule, and f is a spatial frequency. These power spectra were then averaged. Thefrequency at which the average power spectrum is maximal corresponds to the averageobserved droplet spacing, as is also the case for the AFM data (Methods). X. laevis meiotic cytosol
Visualization of branching microtubule nucleation from TPX2-coated microtubules wasdone as previously described [4]. Briefly, coverslips were etched and cleaned using KOHand then silanized, following a previously described protocol [46], and used to make amicroscope flow chamber. Then, a TPX2 fragment (amino acids 320631) that does not46romote or inhibit branching microtubule nucleation [41] was flowed into the chamber at4 µ M. The protein was allowed to bind the coverslip at room temperature for 10 minutes.Excess protein was washed out using BRB80. Then, κ -casein at 1 mg/mL was flowed inand incubated at room temperature for 10 minutes to passivate unfunctionalized regionsof the coverslip, and washed out as above. Next, TPX2-depleted meiotic cytosol with 0.9 µ M Cy5-labeled tubulin and 0.5 mM vanadate was flowed into the chamber. Microtubulesgenerated in cytosol were allowed to bind the coverslip. The cytosol was washed out usingCSF-XB with 10% sucrose. 0.1 µ M GFP-TPX2 was incubated with meiotic cytosol for 20minutes, and then this cytosol supplemented with 0.1 µ M GFP-TPX2, 10 µ M ranQ69L,0.9 µ M cy5-labeled tubulin, 0.2 µ M EB1-mCherry and 0.5 mM vanadate was flowed intothe chamber. Initial TPX2 coating of the pre-attached microtubules and subsequentbranching nucleation was visualized using TIRF microscopy.
For experiments with TPX2 alone, microtubules were incubated for 5 minutes at roomtemperature on plasma-treated carbon grids (Electron Microscopy Sciences). Excess mi-crotubules were wicked away, and then TPX2 diluted in BRB80 was pipetted onto thegrids. TPX2 was incubated with microtubules on the grids for 1 hour to let droplets form,and kept in a humidity chamber to prevent evaporation.For experiments to see colocalization of γ -TuRC with TPX2 droplets, microtubuleswere first incubated with a mixture of 50 nM TPX2, 50 nM augmin, and γ -TuRC. Thesample was diluted 10-fold with BRB80 and 5 µ L of the diluted sample was immediatelyapplied onto plasma-treated carbon grids. Then, for both experiments, all liquid waswicked away and samples were stained by blotting the grids three times with 2% uranylacetate, after which the grids were left to dry. Images were collected with a CM100 TEM47Philips) operated at 80 keV with 64000x magnification and recorded using an ORCAcamera.
Microtubules were bound to mica disks using divalent cations, as previously described [49],as follows. Mica disks were mounted on metal disks using double-sided sticky tape (sup-plies all from Ted Pella). A fresh mica layer was exposed by peeling off the previous layerusing tape, and 1 M MgCl was pipetted onto the mica disk and incubated at room tem-perature for 10 minutes. MgCl was blown off using nitrogen gas and microtubules wereimmediately deposited onto the mica disk and allowed to bind for 10 minutes. Unboundmicrotubules were washed off using BRB80. The mica disk was mounted onto the AFMstage using a magnet.AFM was done using the Bruker BioScope Resolve operated in peak force fluid tappingmode using PEAKFORCE-HIRS-F-A or -B probes (Bruker), with peak force set to 25-40pN. These probes have a nominal tip radius of 1 nm and a stiffness of 0.35 N/m or 0.12N/m, respectively. A 1 Hz scan rate per pixel row was used to sample a 2 µ m × µ marea every (cid:39) ×
256 pixels, yielding alateral pixel size of about 8 nm. After uncoated microtubules were imaged, TPX2 orkinesin-1 was gently pipetted into the fluid meniscus above the mica disk and below theAFM scan head so that protein can bind microtubules while the AFM was scanning thesample. The AFM continued scanning the sample for 40 to 180 minutes as TPX2 dropletsformed. We waited a similar amount of time to probe kinesin-bound microtubules. Formost experiments, once TPX2 droplets had formed, after about 40 minutes, we movedthe stage to a different area of the sample to collect more data from microtubules thatwere not probed as the droplets formed. This confirms that the act of poking TPX2-48oated microtubules repetitively with the AFM probe does not generate the patterns.For these microtubules, for which the initial film thickness was not measured, the filmthickness used for Fig. 3c was the average of all film thicknesses measured in that sample,which was always within (cid:39) . ± . µ M, larger topographies (4 µ m × µ m, sampled 512 ×
512 at a 0.5 Hz scanrate per pixel row so that the speed of the AFM probe was the same across the sampleas in the smaller images) were taken once the patterns formed to enable sampling largerdistances between droplets.
Raw AFM topographies were processed using Gwyddion data visualization and analysissoftware [50] to remove horizontal scarring and linear and polynomial background, and toalign rows along the scan axis. Topographical line scans along bare, uncoated microtubulesand along microtubules within one frame ( (cid:39) P ( f ) were calculated by squaring the absolute value of the normalized finite Fouriertransform ˆ h ( f ) of the height profiles h ( z ), where z is spatial position and f is spatialfrequency: P ( f ) = | ˆ h ( f ) | = (cid:12)(cid:12)(cid:12)(cid:12) L (cid:90) L h ( z ) e − πifz dz (cid:12)(cid:12)(cid:12)(cid:12) . (36)49 is the length of the height profile. For each microtubule, multiple (at least 3) spectrawere averaged. The inverse of the frequency at which the peak power in the averagedspectrum occurs corresponds to the spacing between TPX2 droplets: λ = 1 /f , as is alsothe case for the fluorescence data (Methods). Due to the noise in the AFM data, thesize of the droplets were not measured, as they were manually measured for EM andTIRF microscopy data. Microtubules that broke apart or were too short to sample lowspatial frequencies, and sections of microtubules that curved or were under crisscrossingmicrotubules, were excluded from analysis. Some line scans were high-pass filtered viaa large smoothing window ( (cid:39) In our AFM experiments, we have as the volume of the entire reaction chamber, V chamber ≈ µ L. At a bulk TPX2 concentration c bulk TPX2 = 100 nM, we have N TPX2 = c bulk TPX2 · . · · V chamber ≈ · total molecules of TPX2 in the reaction chamber.Based on the lengths of the microtubules measured in five AFM fields, we determine themicrotubule length density ρ MT length , or microtubule length per area of reaction chamber,to be ρ MT length = 1 . · − ± . · − nm − (mean ± standard deviation). The reactionchamber rests on a mica desk with a diameter d = 12 mm. Thus, the total length ofmicrotubules in the reaction chamber is l total MT = ρ MT length · A mica disk ≈ · µ m. Wemeasured the initial thickness of the TPX2 film on the microtubule to be h TPX2 film =503 . ± . ± standard error of the mean). Assuming an inner radius that isthe radius of a microtubule, r i = 12 . r o = r i + h TPX2 film ,this gives the following for the total volume of the TPX2 film on all microtubules in thereaction chamber, V total TPX2 film = π ( r o − r i ) l total MT ≈ · µ m .Finally, assuming that all TPX2 molecules condense, we find the density of TPX2 inthe condensed film to be ρ TPX2 in film = N TPX2 /V total TPX2 film ≈ µ m − or ≈ µ M.This estimate is consistent with equation (21) for S = 1 / M = 1 /
10. This puts theconcentration of condensed TPX2 within the range of recent concentration measurementsof condensed proteins in vivo [52]. 51
Supplementary tables
Microtubule ≥ ≥ ≥ ≥ ± standarddeviation 6 . ± . . ± . . ± . ± ± Table 1:
Analysis of branching microtubule nucleation efficiency from TPX2 droplets invitro . Data from [18] was included in this analysis.52xperimentalmethod Buffer and reactioncomponents TPX2 concentra-tion Microtubule dilu-tionTIRFM (TPX2alone) CSF-XB with 50-60 mM KCl, 1 mMGTP, and equimo-lar bovine tubulin 0.1, 1 µ M 1:1000-1:2000TIRFM (Branchingreconstitution) BRB80 with 30 mMKCl, 1 mM GTP, 5mM BME, 0.075%(w/v) methylcel-lulose (4000 cP),1% (w/v) glu-cose, 0.02% (v/v)Brij-35, 250 nMglucose oxidase,64 nM catalase,1 mg/mL BSA,19 µ M unlabeledbovine tubulin and1 µ M Cy5-labeledbovine tubulin 0.05 µ M 1:2000EM (TPX2 alone) BRB80 0.1 µ M 1:500EM (TPX2, aug-min, γ -TuRC) BRB80 with 50nM augmin and γ -TuRC 0.05 µ M 1:500AFM BRB80 0.05–1.2 µ M 1:10
Table 2:
A summary of the in vitro conditions used in this study. In one set of conditionsused for TIRFM (1 µ M TPX2 and CSF-XB with 50-60 mM KCl, 1 mM GTP, and 1 µ Mbovine tubulin), we observed droplets forming within a few minutes (Fig. 1a). In anotherset of conditions used for AFM and EM ( (cid:39) µ M TPX2 and BRB80 with no GTPor bovine tubulin), we observed droplets forming over tens of minutes (Fig. 2). Therefore,different physicochemical conditions affect the timescale of the capillary physics, which isconsistent with the hydrodynamic theory. In addition, the film thickness—and thereforethe spacing between droplets—is sensitive to both the TPX2 concentration in bulk andthe amount of microtubules in the sample (Fig. S6b, Supplement 1.2). Thus, the dropletspacing may vary between experimental methods.53PX2 concentration ± estimated range ( µ M) Average peak wave-lengths ± standarddeviation (nm)0.1 ± ± ± ± ± ± ± ± Table 3:
The peak wavelength for each concentration tested. Peak wavelengths arereported as the peak from bootstrapped average spectra ± standard deviation of the peakacross the bootstrapped distributions. Note that for the highest concentration tested, 0.8 ± µ M, longer line scans were taken using a 4 × µm window. A longer line scan samplesa greater number of low frequencies that have high spectral power. This increases theerror on our wavelength measurement, which is taken from bootstrapped distributionsover all individual spectra. 54M Grid γ -TuRCson grid surface γ -TuRCson bare MTs Total γ -TuRCs γ -TuRCson bare MTs/ γ -TuRCson grid surface1 40 0 40 0.002 36 3 39 0.083 51 4 55 0.08Mean ± stan-dard deviation 42 ± ± ± . ± . Table 4:
Analysis of γ -TuRC localization to microtubules (MTs) without TPX2 andaugmin using electron microscopy (EM). Results are normalized accounting for total MTlength per grid. Data from [18] was included in this analysis.EM Grid γ -TuRCson gridsurface γ -TuRCs onbare MTs γ -TuRCs indropletson MTs Total γ -TuRCs γ -TuRCs indropletson MTs/ γ -TuRCs onbare MTs γ -TuRCs onMTs/ γ -TuRCson gridsurface1 30 4 10 44 2.5 0.472 34 2 13 49 6.5 0.443 24 4 21 49 5.3 0.52Mean ± standarddeviation 29 ± ± ± ± . ± . . ± . Table 5:
Analysis of γ -TuRC localization to microtubules (MTs) in the presence of TPX2and augmin using electron microscopy (EM). Results are normalized accounting for totalMT length per grid. Data from [18] was included in this analysis.55 Supplementary movie captions
Supplementary Movie 1.
Rayleigh-Plateau instability of TPX2 on microtubules visual-ized using TIRF microscopy. 1 µ M GFP-TPX2 was spiked onto a passivated glass surfacecovered with Alexa568-labeled microtubules at t = 0 s. TPX2 coats the microtubule andthen proceeds to break up into droplets. Scale bar is 1 µ m. Supplementary Movie 2.
Microtubule branches nucleating from TPX2 droplets on apreexisting microtubule. During acquisition, only the soluble tubulin channel was im-aged to enable capturing nucleation and polymerization of branched microtubules at hightemporal resolution. Scale bar is 5 µ m. Supplementary Movie 3.
Branching microtubule nucleation visualized using TIRFmicroscopy. 0 . µ M TPX2 is added to
X. laevis meiotic cytosol purified from eggs. TPX2coats the mother microtubule from which daughter microtubules then nucleate, leadingto an autocatalytic branched network. Frame dimensions are 16 µ m × µ m. Supplementary Movie 4.
Rayleigh-Plateau instability of TPX2 on microtubules probedusing AFM. 0 . ± . µ M GFP-TPX2 was spiked onto a mica surface covered with mi-crotubules during acquisition. The uniform film of TPX2 is established at t = 0 s, afterwhich the film breaks up into droplets. Frame dimensions are 2 µ m × µ m.56 eferences [1] Shin, Y. & Brangwynne, C. P. Liquid phase condensation in cell physiology anddisease. Science (2017).[2] Alberti, S., Gladfelter, A. & Mittag, T. Considerations and challenges in study-ing liquid-liquid phase separation and biomolecular condensates.
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