Crowder and Surface Effects on Self-organization of Microtubules
CCrowder and Surface Effects on Self-organization of Microtubules
Sumon Sahu, Lena Herbst, Ryan Quinn, and Jennifer L. Ross ∗ Department of Physics, Syracuse University Department of Microbiology, UMass Amherst Department of Biochemistry and Molecular Biology, UMass Amherst
Microtubules are an essential physical building block of cellular systems. They are organizedusing specific crosslinkers, motors, and influencers of nucleation and growth. With the addition ofanti-parallel crosslinkers, microtubule pattern goes through transition from fan-like structures tohomogeneous tactoid condensates in vitro. Tactoids are reminiscent of biological mitotic spindles,the cell division machinery. To accomplish these organizations, we use polymer crowding agents.Here we study how altering the properties of the crowders, such as size, concentration, and molecularweight, affect microtubule organization. Using simulations with experiments, we observe a scalinglaw associated with the fan-like patterns in the absence of crosslinkers. Tactoids formed in thepresence of crosslinkers show variable length, depending on the crowders. The subtle differencescorrelate to individual filament contour length changes, likely due to effects on nucleation andgrowth of the microtubules. Using quantitative image analysis, we deduce that the tactoids differfrom traditional liquid crystal organization, as they are limited in width irrespective of crowdersand surfaces, and behave as solid-like condensates.
I. INTRODUCTION
Cytoskeletal systems inside a cell, composed of actin,microtubules, and their associated proteins, are dynamicand complex in nature. The organization and rapid reor-ganization of cytoskeletal fibers are responsible for var-ious crucial jobs in cells such as cell division, morpho-genesis, and cell motility. One essential microtubule-based organization is the mitotic spindle. This dynamic,self-organized machine spontaneously forms around con-densed and duplicated chromosomes in order to align andeventually separate the DNA into two daughter cells dur-ing cell division. The physics behind spindle formation,steady-state behavior, and transition into later stages ofcell division is of primary interest to cell biology, bio-physics, and could reveal important information to activematter physics.The spindle-shape and fluidity of the mitotic spindleis reminiscent of the tactoid liquid crystalline phase [1].Since the spindle is made from microtubule filaments,which have a high aspect ratio, it is reasonable to ap-ply liquid crystal models and theories. Tactoids are con-densed phases of aligned liquid crystal molecules in abackground of isotropic molecules. They can be bipolaror homogeneous, depending on the director field withinthe condensate. Bipolar tactoids have two point defectson the surface (boojums) at opposite ends, while homoge-neous tactoids’ defects exist at infinity, so that the direc-tor vector field is constant within the condensate. Liquidcrystal theory has been used previously to explain thatthe shape and director field inside tactoids [2, 3]. Theorganization is a result of the interplay between the de-formation of bulk elastic energy and anisotropic surfaceenergy. ∗ Corresponding authorEmail address: [email protected]
Tactoid organization has been observed in a variety ofbiological systems composed of elements with high aspectratio, including actin [4–6], amyloid fibrils [7, 8], tobaccomosaic viruses (TMV) [9, 10], and fd viruses [11, 12].Condensation of biological tactoids can be driven by spe-cific crosslinking [4], macromolecular crowding [13], andhigh density of particles [5, 14], which are passive, en-tropic drivers of self-assembly.We have previously shown that microtubules canalso condense into tactoids with the addition of a pas-sive microtubule crosslinker, MAP65, a member of theMAP65/Ase1/PRC1 family used in spindle organiza-tion in plants, mammalian, and yeast cells [15]. Thisfamily of crosslinkers has been found in the mitoticspindle midzone in metaphase, telophase, cytokinesis[16, 17]. MAP65 binds specifically to anti-parallel mi-crotubule overlaps via a single microtubule-binding do-main. MAP65 has a self-associating “crossbridge region”that is flexible, about 25 nm in length [18–20]. MAP65has comparatively low affinity K D ∼ . µM (low dwelltime on microtubule) with contact angle dependent bind-ing [21]. Interestingly, it has been shown MAP65 pro-motes tubulin assembly, influencing nucleation by reduc-ing critical concentration of tubulin required for spon-taneous filament nucleation [22, 23]. In our prior work,we found that microtubule tactoids were homogeneous,nematic tactoids [15]. Unlike other biological tactoids orthe mitotic spindle itself, our homogeneous microtubuletactoids were not able to coalesce or internally rearrange,as would be expected for a liquid [15], leading us to con-clude that they were jammed. Another interesting ob-servation was that these microtubule tactoids displayeda limited width.Our prior work was performed in specific conditions,leading us to speculate that the jammed nature of thetactoids could be due to the experimental methods weused. Specifically, we used polymers in solution to aidin condensation and polymers on the surface of the ex- a r X i v : . [ phy s i c s . b i o - ph ] S e p perimental chamber to eliminate protein adhesion. Wesupposed that these large, complex polymers could alterthe filament interactions to reduce mobility of the micro-tubule organizations.In order to explore these possibilities, we quantifymicrotubule organization in the presence and absence ofMAP65 with altered macromolecular crowding agentsand surface treatments. We perform new experiments aswell as simulations to understand the mechanism of mi-crotubule pattern formation and scaling associated withthese patterns. Overall, we find that the microtubulecontour length and the presence of the MAP65 have thelargest effect on the organizations observe. The crowd-ing agent could affect filament nucleation and growth,thus affecting the filament length, but it is a minorperturbation. The surface interaction is important inthe absence of MAP65, when microtubules self-organizeonto the surface, driven there by depletion interactions.These new studies demonstrate the reproducibility ofmicrotubule tactoids, which continue to be homogeneousand jammed with fixed width. II. METHODS AND TECHNIQUES
Tubulin Preparation : Unlabeled and fluorescentlylabeled (Dylight 488) lyophilized tubulin from porcinebrain is purchased from Cytoskeleton. Tubulins areresuspended to 5 mg/ml in PEM-80 buffer (80 mMPIPES, pH 6.8, 1 mM MgSO , 1 mM EGTA). Fluo-rescent and unlabeled tubulin are combined to a finallabeling ratio of ∼
4% fluorescently labeled tubulin.Tubulin is aliquoted, drop-frozen, and stored in − o C for later use. Aliquots are thawed on ice prior to use. MAP65 : The microtubule crosslinker, MAP65-1(MAP65) plasmid was a gift from Ram Dixit (Washing-ton University, St. Louis). The full protein purificationprotocol is detailed in [24]. Briefly, protein is expressedusing
Escherichia coli
BL21(DE3) cells, grown to anOD of 1, lysed, and clarified. MAP65 protein isrecovered from the lysate through affinity between the6 × histidine tag and Ni-binding substrate [24]. Purifiedprotein is checked on an SDS-PAGE gel, aliquotted, andstored at − o C . Crowding Agents : Stock solutions of macromolecu-lar crowders used for these experiments are dissolved inPEM-80 buffer or dd H O . The polymers we used are: 8kDa Polyethylene glycol (8 kDa PEG), 14 kDa Methyl-cellulose (14 kDa MC), 88 kDa Methylcellulose (88 kDaMC), and 100 kDa Polyethylene glycol (100 kDa PEG).Stock solutions of each polymer are as follows: 20% (w/v)8 kDa PEG in water, 2.4% (w/v) 14 kDa MC in PEM-80,3% (w/v) 88 kDa MC in water, and 5% (w/v) 100 kDaPEG in PEM-80. For experiments, outlined below, eachof these stock solutions are diluted in the experimental mix to 1% (w/v) for 8 kDa PEG, 0.12% (w/v) for 14 kDaMC, 0.15% (w/v) for 88 kDa MC, and 0.25% (w/v) for100 kDa PEG.Kinematic viscosity of the polymer solutions as well asdd H O are measured using an Ubbelohde glass capillaryviscometer (Cannon Instrument) at 37 o C using a hotwater bath. Kinematic viscosity (cSt) is multiplied bythe density of the solutions (g/ml) to produce dynamicviscosity (cP) (viscosity used in this paper). The dd H O viscosity we measure at 37 o C was 0.77 cP. Silanized coverslip:
Coverslips are treated toinhibit protein binding to the surface using a block co-polymer brush of Pluronic-F127, as previously described[15]. Cover slips are cleaned with ethanol, acetone,potassium hydroxide (KOH) from Sigma, and treatedwith 2% (w/v) dimethyldichlorosilane solution (GEHealthcare) to make the surface hydrophobic. Slides arecleaned with ethanol and water, but are not treated withsilane. The full silanization protocol can be found in [25].
Lipid surface preparation:
For samples with lipidbilayer surface coatings, we create small unilamellarvesicles (SUVs) to coat the surface. SUVs <
100 nm indiameter are made of phospholipid 1-palmitoyl-2-oleoyl-glycero-3-phosphocholine (POPC) (Avanti) suspendedin PEM-80 buffer. First, 40 µ l of 10 mg/ml POPC inchloroform is well mixed with and additional 70 µ l ofchloroform. The lipid-chloroform mixture is dried under N gas and further desiccated inside a vacuum desiccatorfor >
15 min. Dried lipid is resuspensed with PEM-80buffer to form giant unilamellar vesicles (GUV), whichare sonicated for three minutes to form SUVs. The SUVsolution is kept sealed with parafilm in 4 o C for use overone week.
Flow Chamber preparation:
Experimental flowchambers with an approximate volume of 10 µ l aremade from a cleaned glass slide (Fisher), glass cover slip(Fisher) and double-sided tape (3M). Glass slides arecleaned with 70% ethanol and water (milliQ), and driedusing Kimwipes.For samples with polymer surface coatings, silanizedcover slips are coated with 5% Pluronic F127 by flowinginto the chamber and incubating for 7-10 minutes in ahumid environment to avoid evaporation. For sampleswith lipid bilayer coated surfaces, cover slips and slidesare cleaned with UVO for 10 minutes prior to flowingin small unilamellar vesicles (SUVs). Flow chambers areincubated in humid chambers to allow SUVs to adhere tothe surface for ∼
10 minutes. Chambers are washed withPEM-80 buffer to remove excess SUVs from solution.After surface treatments are performed, the experi-mental assay mix (see below) is added, and chambersare sealed immediately afterwards with epoxy to preventevaporation and flow in the sample. One chamber is usedto inspect the experiment using fluorescence imaging(see below) while additional chambers are incubated at37 o C for 1 ∼ Experimental Assay Mix:
The experimental assaymix contained tubulin, crowding agents, and crosslink-ers as specified for each experiment. Each experimentalassay mix contained 13.6 µM tubulin fluorescently la-beled (4% labeling ratio), drop frozen, and thawed onice. To enhance microtubule nucleation and stabilize mi-crotubules, 1 mM GMPcPP (Jena Bioscience) is added.For experiments in polymer coated chambers, 0.5 % F127is added to maintain polymer surface coating. To limitphotobleaching and photodamage during the experiment,25 mM DTT is added with 0.25 mg/ml glucose oxidase,0.075 mg/ml catalase from bovine liver, and 7.5 mg/mlglucose (all reagents from Sigma) as an oxygen scaveng-ing system. Oxygen scavengers are added last, just priorto mixing and flowing into the chamber.Specific crowders are added to the experimental mixas specified above. MAP65 crosslinkers are added to thesolution, according to the percentage bound to tubulin:0%, 3%, and 10 % which corresponds to 0, 0.44 µM , and1.49 µM final concentration [15].We have found that the results can be sensitive tothe batches of chemicals and protein reagents used.For all the data presented here, we use the exact samechemicals and protein preparations to ensure quantita-tive reproducibility. Qualitatively, the experiments arehighly reproducible. Imaging:
Imaging of microtubule organizations isperformed using total internal reflection fluorescence(TIRF) microscopy on an inverted Nikon Ti-E micro-scope with Perfect Focus. The TIRF is illuminated usinga 488 nm diode laser aligned into the epi illuminationpath [25]. Image data is taken using a 60x oil-immersion(1.49 NA) objective, expanded with a 2.5x lens ontoan Andor Ixon EM-CCD camera. The image scale is108 nm per pixel. Images are displayed and recordedusing the Nikon Elements software. Time series imagesare taken every 2 seconds for 1 hour. Still images ofchambers are imaged within 1 ∼ Data Analysis:
Self-organization of microtubuleswithout crosslinkers are studied by orientation domainanalysis using a FIJI plugin called OrientationJ andfurther analyzed by custom MATLAB scripts, similarto prior methods [15]. First, raw gray scale imagesare smoothed two times in FIJI to reduce noise andpixelated regions. Next, we employ OrientationJ tothe image to create a colored image map, where thecolor indicated the directionality of microtubules ormicrotubule bundles within the image. The settingsused to create color maps is a 5 × π ]. In order to determine the size of domainspointing at similar angles, we use the color threshold command in FIJI to select regions in the following fourcolor ranges: [0, 63], [64, 127], [128, 191], [192, 255].The image map is turned into four individual orientationregion maps by binarizing the thresholded image maps.For each pixel (x,y) in the original image, the locationis assigned to one of the four thresholded image mapsand assigned a directionality index, m = 1, 2, 3, 4corresponding to the four different directionality/colorbins ([0, 63], [64, 127], [128, 191], [192, 255]).The four m-type binary images are analyzed using mo-ment analysis and spatial auto correlation function anal-ysis to determine the area parameters of the domainspointing in each direction. We use MATLAB’s built-in analysis tool, region properties (regionprops) that re-turns the number and size of domains present in a binaryimage. We use definitions from percolation theory to de-scribe our domains or clusters [26]. We define the domainsize distribution, as: n s ( L ) = Λ L (1)where L is the size of our system or window size, and Λ isthe number of domains with area, s, within a window ofsize, L . The average of n s ( L ) over all binary images pro-duce n s ( L ), which is used to compare between differentdata sets.The 0 th moment of the n s ( L ) distribution is the num-ber of m-type domains in an binary image, defined as: N ( L ) = s max (cid:88) s =1 n s ( L ) (2)The 1 st moment of the n s ( L ) distribution defines theprobability of m-type domains appearing in an image as: p ( L ) = s max (cid:88) s =1 sn s ( L ) (3)Since we quantize our continuous rotation symmetry tofour values, we should expect p ( L ) ∼ . (cid:104) s ( L ) (cid:105) in an image of windowsize L is defined by (cid:104) s ( L ) (cid:105) = (cid:80) s max s =1 s n s ( L ) p ( L ) (4)We can also compute the correlation length for the do-main images using the binarized orientation images. Thecorrelation length is found using the radially averagedpair auto-correlation function. Spatial auto-correlationscan be calculated using the Fast Fourier Transform(FFT) [27] according to Weiner-Khinchin theorem: G ( r ) = F − ( | F ( I ) | ) ρ M ( r ) (5)where F ( I ) is the FFT of a binarized image, I , | F ( I ) | isthe power spectrum of the FFT of I , and F − ( I ) is theinverse FFT of I . To avoid artifacts due to the periodicnature of the FFT, the images I are increased in sizearound the boundary by adding 200 additional pixels thatare all set to zero (black).In the denominator, ρ is the intensity per unit area inthe image. Since the image is binarized, ρ exists between0 and 1. The form of M ( r ) = F − ( | F ( W ) | ), whereW is the window function used for normalization as de-scribed in [28]. Specifically, W is an image of the samesize as I (with padding) where all pixels are set to white(intensity equals 1). G ( r ) is averaged in the azimuthaldirection to give ˜ g ( r ) = 1 + Ag ( r ), where A is the am-plitude. The function g ( r ) is fitted and determine thecorrelation length, ξ . For each experimental parameter, g ( r ) is averaged over several independent images to give g ( r ), which is used to compare between data sets.Tactoid growth dynamics are analyzed using the MAT-LAB function regionprops on cropped binary images cre-ated using the Otsu global thresholding algorithm [29].The area, major and minor axis lengths are determinedfor each frame and plotted over time. Intensity profilesalong the major axis length are plotted to visualize thechange over time. Each video is one hour long and datawas averaged over three tactoids.Microtubule tactoids made in the presence of 10%MAP65 are imaged as still frames and analyzed in FIJIby hand to find the length and width. For comparison,we also use homemade MATLAB scripts using binarizedimages and the regionprops functions. Cytosim:
Agent based modeling is performed usingan open source cytoskeletal fiber simulating package, Cy-tosim [30]. Parameters used for the simulation are givenin TABLE I. Fibers are allowed to grow from an initiallength of 0.02 µm with an initial growth rate of 30 nm/s [31]. Filaments grew until the tubulin concentration isexhausted and they reached an average contour length.Thus, the final contour length is well defined by the totaltubulin concentration and the number of initial micro-tubule seeds. We specifically chose the total polymermass to represent the experimental concentration, whichis controlled by the packing fraction φ [32]. The packingfraction is, φ = 2 R s L N l c (6)where L is the window size in the simulation, R s is thesteric radius of the fiber, N is the total number of fibers,and l c is the average steady-state contour length of thefibers.Depletion forces are mimicked by an effective attractivespring force, F ( r ) = kr , where r is the distance from thecenter of the filament. The spring constant depended on r as: k = k s , if 0 ≤ r ≤ R s k d , if R s ≤ r ≤ R d , otherwise (7) where k s is purely replusive within the steric radius, R s and k d is attractive inside the depletion radius, R d .Simulations are run long enough for the fibers to reachtheir equilibrium length and thermalize as observed inexperiments. Simulated filaments did not perform dy-namic instability because we set the catastrophe rate tozero. This is done because experiments used GMPcPP,the slowly hydrolyzable analog of GTP, that virtuallyeliminates the catastrophe rate [33]. Since dynamic in-stability is blocked in our simulation, microtubules onlygrew and the growth curve flattened as they exhaustedthe total dimer concentration in the system. All simula-tions are run for 1600 sec to allow the system to reachsteady state. The last frame is rendered as an imagefor analysis identical to that performed for experimentalimages. TABLE I. Cytosim parametersSimulation parameters ValuesTime step 0.1 sSimulation time 1600 sViscosity 0.025-2.5 Ns/ m Geometry (Periodic) 50 × µm (2D)Packing fraction 0.5-1Rigidity 20 pN µm Growing speed 0.03 µm/s
Catastrophe rate 0 /sGrowing Force 1.7 pNTotal Polymer 12500, 18750, 25000Mean MT length 2, 4, 6, 8 µm MT segmentation 1 µm Initial MT length 0.02 µm Steric radius 0.05 µm Steric force constant 50 pN/ µm Depletion radius 0.03 µm III. RESULTS AND DISCUSSION
We seek to understand how the self-organization of mi-crotubules depends on non-specific forces from depletionagents and specific interactions via diffusible crosslink-ers. Both mechanisms are responsible for minimizing themicrotubule overlap area by contributing to the lateralinteraction between them. As we demonstrated previ-ously [15], nucleating and polymerizing microtubules re-sulted in microtubule self-organizations that showed atransition from fan-like domain patterns to elongated ho-mogeneous tactoids as a function of MAP65 crosslinkerconcentration. These prior experiments used methylcel-lulose polymer as a crowder and a polymer-coated sur-face, which could have affected the organized patternswe observed. The tactoid phase was different from otherbiological tactoids in two ways: the width of the tactoidswas constant and the microtubules inside the tactoidswere jammed. (B)(A)
88 kDa MC14 kDa MC 100 kDa PEGControl8 kDa PEG M AP C on c en t r a t i on μ m FIG. 1. (A) Schematic of experimental system: Tubulindimers (green) polymerize into microtubles in the presence ofcrowders (brown polymer balls) and crosslinkers (orange Z)on polymer-coated surfaces imaged with TIRF illumination.(B) Images showing microtubule patterns on polymer brushcoated surfaces in the presence of crowders with differentmolecular weights (8 kDa PEG, 14 kDa MC, 88 kDa MC, 100kDa PEG) with tubulin concentration 13.6 µ M and MAP65at 0% (top), 3% (middle), 10% (bottom). Fan-like patternswere observed without MAP65 (top row). High MAP65, 10%,induced tactoid like condensates (bottom row). The red boxindicates 88 kDa MC, the condition used in our prior work,and serves as our reproducibility control state [15]. Scale baris 10 µ m for all images. In order to determine the mechanism driving the pat-tern formation we previously observed, we have alteredthe crowding agent type, size, concentration, and viscos-ity. Qualitatively, for polymer-coated surfaces, all crow-ders show a similar phase transition, from fan-like struc-tures to tactoids (Fig. 1).To test the effect of the surface on microtubule pat-terns, we repeat the experiments for 0% and 10% MAP65in the presence of 88 kDa MC on a lipid-coated surface.The lipid surface is a fluid bilayer with a slip condition.We observe that the lipid surface retains the fanlike pat-tern with increased disorder in the absence of MAP65.Tactoids form robustly on lipid surfaces (Fig. 2).In an attempt to understand the mechanisms drivingthe patterns we observe, we simulate the microtubulegrowth and self-organization using the Cytosim softwarepackage (Fig. 3). Simulations allow us to alter the viscos- ity, depletion interactions, filament length, and growthrates independently and examine the dynamics to steady-state. (A) lipid bilayer M AP pe r c en t age % % μ m (B) LipidPolymer Brush
Polymer brush
FIG. 2. Surface effects on patterns. (A) Schematic diagramof F127 polymer brush surface (top) and lipid bilayer sur-face (bottom). (B) Images of microtubule organizations onpolymer-brush surfaces (left column) and lipid surfaces (rightcolumn) in the presence of 0% MAP65 (top row) and 10%MAP65 (bottom row). Scale bar is 10 µ m for all images. A. Microtubule organization without crosslinkers
Polymer surface with different crowders:
Qual-itatively, fan-like surface patterns in the absence ofcrosslinkers are similar for different crowding agents(Fig. 1). To quantify the patterns of microtubule self-organization, we perform a domain analysis similar toour prior publication [15], where the direction of locally-oriented microtubule bundles in the pattern were iden-tified automatically using the OrientationJ plugin inFIJI/ImageJ (Fig. 4A(i)). The image is colored accord-ing to the local orientation (Fig. 4B(ii)) and that colormap (Fig. 4A(iii)) is used to identify domains where theangles are similar (Fig. 4A(iv)). We threshold the col-ors to a specific range of colors divided into four differentbins that we identify with an index, m. Fig. 4B(v) showshow the angle thresholding is used to create a binary im-age where the selected angle range is portrayed in white,and all other angle orientations are in black. For eachoriginal image of microtubule patterns, we create fourbinary images highlighting one of the four angle ranges(m = 1,2,3,4). We use the binary images to quantify thedomain parameters of the images. Domain parameters ofinterest included: average domain size (s) distributions n s ( L ) (Fig. 4B) used to determine the average domainsize (cid:104) s ( L ) (cid:105) and the radially averaged spatial mean au-tocorrelation function, g ( r ) (Fig. 4C) that allows us todetermine the mean correlation length ξ of the images(Fig. 4D).Similar to our prior study, we first perform microtubuleself-organization experiments using polymer-coated sur-faces with four different crowding agents. We find that t=20 s t=200 s t=400 s t=600 s t=800 s t=1000 s t=1200 s t=1400 s(i)(ii) l c = 2 μm l c = 4 μm l c = 6 μm l c = 8 μm (iii)η = 0.0025 η = 0.025 η = 0.25 η = 2.5 FIG. 3. Cytosim simulations of microtubule organization without crosslinkers. (i) Time series of fan-like microtubule patterngrowth for parameters, φ = 0 . η = 2 . l c = 6 µm with steric interaction and depletion forces present. Fibersgrow from initial small seeds until they exhaust the dimer concentration. (ii) The effect of l c on patterns from left to right: l c = 2 , , , µm with viscosity η = 2 . φ = 0 .
75. (iii) The effect of viscosity on steady state patterns from left toright: η = 0 . , . , . , . φ = 0 . l c = 6 µm . Patterns obtained at higher viscosities match experimentalpatterns qualitatively. the average distributions of the domain areas (dividedby the window size, L in order to compare with simu-lations) are all power-law (Fig. 4B(i)). We test the dis-tributions using the Kolmogorov-Smirnov (KS-test) sta-tistical test and found that distribution for 14 kDa MCis different from rest of the distributions. When we com-pare the average domain sizes for these images, the aver-ages are different from one another, although they havelarge error bars due to ’critical-domain’ like behaviour(TABLE II).In our prior work, we measured the area distributionsfor one crowder type, 88 kDa MC, but changed the con-tour length of the filaments by altering the tubulin con-centration. Microtubule contour length varied with tubu-lin concentration, [ T U B ] in a linear fashion, as described[15]. For, [
T U B ] = 13 . µM, . µM, µM MT me-dian lengths are 6 . µm, . µm, . µm respectively.To determine the effects of the crowding agents, we needto compare our new results with different crowders tothe former results using the same moment analysis (Fig.4B(ii)). For 88 kD MC with increasing tubulin, we findthat the KS-test shows that the distributions for 70 µM tubulin is distinct from 13.6 µM tubulin distribution.Further, the average domain sizes show a trend depend-ing on the tubulin concentration (TABLE II), specifically,the average domain size decreases with increasing tubulinconcentration (decreasing contour length).Our data imply that the contour length of the mi-crotubule filaments is the control parameter for the do-main sizes we observe for microtubule self-organizationon polymer coated surfaces without crosslinkers. In orderto directly test this mechanism, we use Cytosim to poly-merize microtubules to final microtubule contour lengthsthat we could control and perform the same analysis onthe final, equilibrium images. When we plot the his- tograms of the domain areas, we observe a transition fromnon-power law to power-law distribution (Fig. 4B(iii)).Examining the average domain size, we recapitulate thedirect dependence on contour length. Namely, the longerthe contour length, the larger the average domain size(TABLE II). Thus, our simulation verifies that the fil-ament contour length is the control parameter for thedomain area for this polymer-coated surface.Due to the power-law nature of the distributions, KS-tests on these domain distributions show that all thesedistributions are statistically the same. Comparison tocritical phenomena theory indicates there is a regime shiftfor smaller contour length configurations. In this case,the distribution starts with a power-law, then falls off ex-ponentially indicating a cut-off cluster size present in thesystem. Whereas, configurations which contained largedomain areas hold the power law behavior, a signatureof critial phenomena, because of the finite size of the sys-tem.In order to further characterize the self-organizationpattern for microtubules in the absence of crosslinker,we characterize another intrinsic length parameter, thecorrelation length ξ . Using the same binarized imagesfor each orientation domain, we perform the spatial au-tocorrelation function, averaged over the multiple imagesto find g ( r ), and plotted as a function of the radius fromthe center of the image (Fig. 4C). For each binary image, g ( r ) is fit with a sum of two exponentials. In order tofind a single correlation length, we determine the valueof the radius where g ( r ) = 0 .
37 from the fit equations.This value is averaged over all binary images to definethe mean correlation length ξ for each experimental pa-rameter.When we perform the spatial auto-correlation analysisusing data with different crowders on polymer surfaces,the radially averaged mean auto-correlation functions g ( r ) did not show significant changes overall for thesecrowders (Fig. 4C(i)). The correlation lengths computedfrom the auto-correlation function are not identical for allcrowding agents, although the error bars are large (TA-BLE II).In order to compare to our prior results, we determinethe mean correlation length ξ from our previous data ondifferent tubulin concentrations. This data shows an up-ward trend as tubulin concentration decreases and con-tour length increases. The g ( r ) profiles obviously becameslower decaying in radius as the tubulin concentration isreduced (Fig. 4C(ii)).Using the simulated organizations, we find that theauto-correlation profiles, g ( r ) and the mean correlationlength ξ , depend on the filament contour length (TA-BLE II) and (Fig. 4C(iii)). The simulation shows thesame trend observed in experiments. Since longer con-tour lengths form from larger domain areas, this is rea-sonable and corroborates the fact that contour length offibers are the driving factor of length scales in the pat-terns that we observe in experiments. TABLE II. Fan-like pattern data characteristics.Configuration ξ ( µm ) (cid:104) s ( L ) (cid:105) ( µm )mean ± SD mean ± SD8 kDa PEG 4.2 ± ± T UB ] 14 kDa MC 4.3 ± ± . µM
88 kDa MC 5.0 ± ± ± ± µM ± ± µM ± ± µM ± ± µm ± ± φ = 0 .
75 4 µm ± ± µm ± ± µm ± ± We quantify the first three moments of n s ( L ) for theexperimental and simulated data and averaged over mul-tiple binary images. The zeroth moment is the averagenumber of m-type domains N ( L ), the first moment isthe mean probability that a position is in one of the fourm-type domains p ( L ), and the second moment is the av-erage domain size (cid:104) s ( L ) (cid:105) . We plot each as a function ofmean correlation length, ξ (Fig. 4D).The number of domains (zeroth moment) shows apower law dependence on the mean correlation lengthwith an exponent − . ± . ξ . As we showin Fig. 4C, the correlation length depends on the contourlength. The the probability of an m-type domain appearingin an image (first moment) is constant, p ( L ) ∼ .
25, nomatter that contour length or correlation length (Fig.4D(ii)). As described in the methods, this is expected,since the images were binned into four orientation direc-tions for analysis. If the local orientation of the domainis randomly determined and independent from its neigh-boring domain, we would expect the probability for thefour orientation bins to be ∼ .
25. This is a good checkon our moments analysis.The average domain size (second moment) (cid:104) s ( L ) (cid:105) ,scales with the mean correlation length ξ with exponent1 . ± . N andcontour length l c . This last mechanism is likely the causefor the subtle differences we observe for the average do-main size (cid:104) s ( L ) (cid:105) (Fig. 4, TABLE II).The different types of crowders we use, MCs and PEGs,are different in their biochemical properties. We cannotdirectly determine the contour lengths for microtubulesin the presence of these crowders because they also causebundling. We notice that the correlation lengths andthe average domain sizes appear to be higher for MCmolecules than for PEG molecules (Fig. 4, TABLE II).This would imply that PEG molecules could be better atenhancing nucleation and growth of microtubules thanMC polymers at the concentrations we use in our study.Tubulin nucleation and growth in the presence of PEGusing a turbidity method showed that the formation ofmicrotubules was proportional to the PEG concentra-tion [34]. Higher concentration of PEG increased themicrotubule association constant K a , a determinant ofmicrotubule growth. Further, PEG reduced the lag timefor nucleation. Even more interestingly, K a increased asthe PEG size increased [35]. The same K a enhancementwas achieved by lower concentrations of larger PEGmolecules. This implies that higher molecular weight (A)(B)(C)(D) (i) (ii) (iii)(i) (ii)(i) (ii)(i) (ii) (iii)(iv) (v) s (μm ) s (μm ) s (μm ) (iii)(iii) -5 -4 -3 -2 l = 2 μm c l = 6 μm c l = 4 μm c l = 8 μm c n S ( L ) r (μm) l = 2 μm c l = 6 μm c l = 4 μm c l = 8 μm c g (r) Crowder dataControl dataCytosim pf = 0.5Cytosim pf = 0.75Cytosim pf = 1 -5 -4 -3 n s ( L ) -5 -4 -3 μM μM μM μM μM μM n s ( L ) -0.200.20.40.60.810 5 10 15 20 25 r (μm) g (r) -0.200.20.40.60.810 5 10 15 20 25 r (μm) g (r) -2 -1 ξ (μm) Crowder dataControl dataCytosim pf = 0.5Cytosim pf = 0.75Cytosim pf = 1 slope = -1.6 ± slope= 1.5 ± color code Crowder dataControl dataCytosim pf = 0.5Cytosim pf = 0.75Cytosim pf = 1 ξ (μm) ξ (μm) p ( L ) s ( L ) N ( L ) FIG. 4. Orientation Analysis: (A)(i) Raw gray scale image with fire look up table (FIJI/ImageJ). (ii) Image(i) is color codedaccording to orientation of fibers shown. (iii) Color coded version of image(i) without fibers. (iv) Selected orientation domainis shown in black color. (v) A binary image is created from image(iv) by defining the black region as white and rest as black.(B) Average domain size distribution n s ( L ) for (i) different crowders with 13.6 µ M tubulin (n ≥
18) (ii) different concentrationof tubulin with MC 88 kDa crowder (n ≥
4) (iii) Cytosim simulation with four different length for φ = 0 .
75 (n=6). (C) Radiallyaveraged mean auto correlation function g ( r ) for (i) different crowders with 13.6 µ M tubulin (n ≥ ≥ φ = 0 .
75 (n=6). (D)(i)Average number of m-type (m=1,2,3,4) domains per image N ( L ) plotted over mean correlation length ξ in log-log scale. (ii)Mean probability of m-type cluster present in an image p ( L ) plotted as a function of ξ in semi-log scale. (iii) Average clustersize (cid:104) s ( L ) (cid:105) as a function of ξ in log-log scale. Color bar on the right corresponds to the median microtubule length expectedfor these data sets, as reported before in [15] PEG should be more efficient in enhancing microtubulenucleation and growth. For instance, we observe that 8kD PEG has similar correlation length and domainsizes compared to 100 kD PEG. In our study, we triedto keep the total polymer mass similar, so the effect wassubtle. For the MC polymers, we found that for similarconcentrations there is a difference in correlation lengthand domain sizes. Future experiments examining boththe molecular weight and the concentration of differentcrowders might be able to deduce the effects of thesedifferent polymers on microtubule nucleation and growth. μ m (i) (ii) (iii) (iv) (v)(i) (ii) (iii) (iv) (v) (A)(B) FIG. 5. (A) Timeseries of the organization on lipid surfacein presence of 8 kDa PEG, at (i) t = 25 sec (ii) t = 50 sec (iii)t = 100 sec (iv) t = 500 sec (v) t = 1000 sec. (B) Gallery ofmicrotubules on lipid surface: tubulin concentration 13.6 µM ,with 8 kDa PEG as a crowder. (i) Fan-like pattern withouta defect. (ii) Spindle-like gap/defect formed inside fan-likepattern. (iii) Triangle with round edge gap embedded insidefan-like pattern. (iv) Interconnected defects. (v) Jumbledpattern without any particular structure. Scale bar is 10 µm for all images. Lipid surface with different crowders:
Becausecrowding agents cause the depletion of microtubules ontothe surface, we wanted to test the effects of the surfacecoating on microtubule self-organization in the absenceof crosslinkers. We replace the polymer brush surfacewith a lipid bilayer surface to test the surface effects. Wefind that the microtubule patterns are less reproducibleon the lipid bilayer surfaces. We are able to recapit-ulate fanlike structures on the lipid surface (Fig.2(B)),although the patterns are not as consistent as those ob-served on polymer surfaces. We often observe patternsdisplaying rounded gaps decorated inside fan-like pat-terns (Fig. 5(B)). These gaps occur in different shapesas round, spindle-like, and triangles with round edges(Fig. 5B(ii)-(iv)). In our samples we find isolated aswell as interconnected gaps present. In some parts ofthe chambers, microtubule bundles are formed but jum-bled upon each other, rather than well distributed intofan-like patterns.When we examine the dynamics of the microtubulesforming on the lipid bilayer surfaces, we often observejumbled, overlapping arrangements and not the fan-likearrangement. We observe microtubules nucleate on thesurface fail to reorient themselves due to immobility onthe surface. Microtubules nucleated in bulk sedimente onthe surface due to the depletion interactions caused bythe crowders. These bundles stack upon earlier bundles and are unable to form fan-like structures (Fig. 5(vi)).Jumbled arrangements do not depend on the crowdingagent used. So, the fluidity of the surface play a role indetermining whether fan-like structures could be formedand appeared to be more influential than the crowdingagent. We would expect that a fluid surface would allowmicrotubules to fluctuate in position and orientation toaccommodate sedimenting microtubules to reach the fan-like structure. It is surprising that we did not observedynamical filament rearrangement on the lipid bilayerconsistently. It could be possible that the lack of mobilitycould be due to incomplete lipid coating in some regions.To understand the mechanisms that control the pat-terns on lipid bilayer surfaces, we perform simulationsusing Cytosim. Our experimental system is quasi-2D dueto crowders, so we limit our Cytosim simulation space to2D plus steric interactions (2D+S) [32]. In our simula-tion, fibers falling from bulk of the solution is not imple-mented. As shown in TABLE IV the viscosity (dynamic)of the polymer solutions is ∼ B. Microtubule organization with MAP65
Polymer surface with different crowders:
In ourprior work, we were able to reproducibly form micro-tubule tactoids in the presence of MAP65 at a concen-tration of 10% bound [15]. The competition betweenelastic deformation energy of the director/microtubuleorientation field inside and aniosotropic surface energyat the interface defines the shape of the tactoid and di-rector field inside it. As we showed previously, the tac-toids are homogeneous droplets with a constant directorfield along the major axis of the tactoid. Boojums (vir-tual defects) are located at infinity, not at the pole as forthe bipolar case. We showed that microtubule tactoidsare jammed/frozen inside the droplet, demonstrated byimperceptible microtubule turnover when probed usingFluorescence recovery after photobleaching (FRAP)[15].0FRAP did show that MAP65 turnover was fast due to therelatively weak binding constant, 1.2 µ M. Here, we areinterested in examining the effect of different crowdingagents on the formation, shape, and dynamics of thesetactoids. For all experiments, 13.6 µ M tubulin was usedon polymer brush surfaces in the presence of each poly-mer type (Appendix A).In order to quantify the tactoid shape, we use a cus-tom MATLAB code and manual measurement by FIJI,to measure the maximum length, L = 2 l , where l is thehalf-max length, and width, W = 2 r , where r is the half-max radius of imaged tactoids. We calculate the aspectratio L/W (TABLE III). Tactoids that form on poly-mer surfaces in the presence of PEG or MC have subtledifference in their lengths (Fig. 6(i)). The 88 kD MCmade slightly longer tactoids than other crowders. Usingthe KS statistical tests, we find that the length measure-ments are all statistically different from one another. Aswe observed previously, the widths of the tactoids are ina small range from 500 ∼ Lipid surface with different crowders:
On lipidsurfaces, we are able to reproduce tactoids consistentlyin the presence of different crowders. Tactoids grown onlipid surface are more reproducible compared to the pat-terns we observe in the absence of MAP65 on lipid sur-faces. We find that the crowder effects on tactoid length,width, and aspect ratio observed on polymer surfaces arereproducible on lipid surfaces. We focus on 88 kDa MCand 100 kDa PEG, which were both large crowders, but
Leng t h ( μ m ) W i d t h ( μ m ) A s pe c t R a t i o L / W (i) (ii)(iii) FIG. 6. Box-whisker plots of tactoid characteristics with10% MAP65 on the polymer brush surface: (i) length of thetactoids, (ii) width of the tactoids, and (iii) aspect ratio ofthe tactoids. Tactoids are formed in presence of 8 kDa PEG(orange, n = 205), 14 kDa MC (green, n = 217), 88 kDa MC(blue, n = 207), 100 kDa PEG (yellow, n = 240). showed distinct differences on polymer surfaces (Fig. 7).We find that surface treatment does not affect tactoidlength, width, or aspect ratio (Fig. 7). The subtle dif-ferences observed due to the different crowders (MC vs.PEG) persist on the lipid surface (Fig. 7((i)-(iii))).As with the fan-like patterns, we expect that thedifferences in length are due to the crowder affectingmicrotubule nucleation. As discussed earlier, highermolecular weight crowders are more efficient in tubulinassembly but we do not know to what extent they areinfluencing the kinetics. Yet, the reproducibility of thetrends with and without crosslinkers suggests that thepolymers induce the length dependence.
Microtubule tactoid shapes cannot be ex-plained by liquid crystal theory:
To compare ourtactoid results to liquid crystal theory and elucidate thephysical mechanisms controlling tactoid formation, weuse the length, width, and aspect ratio measurementsfrom all the data to examine the shape parameters (Fig.8). We plot the aspect ratio
L/W as a function of L , andfind that the dependence appeared linear with a positiveslope, m = 0 . ± .
04 (Fig. 8(ii)). The slope representsthe constant width of the tactoids. The plot has a largescatter, which is due to the wide range of data for thelengths and widths (Fig. 8(ii)). Despite the scatter, thisresult suggests that the width is constant for all tactoids,regardless of the crowding agent and surface used (Fig.8(ii)). Prior work on other systems of homogeneous tac-toids have not observed any dependence of the aspectratio on the length of the tactoids. For instance, homo-geneous tactoids made from carbon nanotubes have beenshown experimentally to display a constant aspect ratioas a function of tactoid length, implying that the length1
TABLE III. Tactoid features: mean length, mean width, mean aspect ratio and dimensionless anchoring strength. First fourrows and last four rows were measured using FIJI (by hand) and MATLAB, respectively. As indicted, the first four columnsare for polymer brush surfaces and last two columns are for lipid surfaces.Polymer brush surface Lipid surfaceCrowders 8 kDa 14 kDa 88 kDa 100 kDa 88 kDa 100 kDaPEG MC MC PEG MC PEG(mean ± SD) (mean ± SD) (mean ± SD) (mean ± SD) (mean ± SD) (mean ± SD)L( µm ) 5.9 ± ± ± ± ± ± µm ) 0.8 ± ± ± ± ± ± ± ± ± ± ± ± ω ∼
16 21 28 17 27 20L( µm ) 5.6 ± ± ± ± ± ± µm ) 0.9 ± ± ± ± ± ± ± ± ± ± ± ± ω ∼
10 12 20 11 22 15 A s pe c t R a t i o L / W Leng t h ( μ m ) W i d t h ( μ m ) polymer brush 88 kD MCpolymer brush 100 kDa PEGlipid 88 kDa MClipid 100 kDa PEG (i) (ii)(iii) FIG. 7. Box-and-whisker plots of (i) length, (ii) width, and(iii) aspect ratio of tactoids in presence of two different crow-ders, 88 kDa MC and 100 kDa PEG, on polymer brush andlipid coated surfaces. Legend shows polymer brush surfacewith 88 kDa MC (light blue, n = 207), polymer brush surfacewith 100 kDa PEG (yellow, n = 240), lipid surface with 88kDa MC (purple, n = 374) and lipid surface with 100 kDaPEG (orange, n = 152). and width grew together [37]. The difference betweenprior systems and the microtubule system can be thatour system is driven to condense using specific crosslink-ers.We can examine tactoid shape in another way, usingtwo additional variables to characterize the tactoid: R and α . These parameters come from thinking of the tac-toid surface as an arc of a circle with radius R making anangle 2 α at the center (Fig. 8(i)). These shape parame-ters can be related to the semi-major length l = L/ r = W/
2, as so: R = l + r r (8) α = sin − (cid:18) lrl + r (cid:19) (9)We find that when we plot R as a function of α wesee an inverse dependence. This dependence is anotherindicator that the width is constant in our system. Whenwe fit the data for R vs α with a power law, it yields anexponent δ = − . ± .
04. This behavior is the oppo-site of what was seen previously in case of actin tactoidcondensate [5].In the prior work, the actin tactoids were bipolar, sothat the R dependence on α could be used to determinethe rescaled Frank free energy constants for splay κ andbending κ . Bipolar tactoids are described by the elasticenergy terms as in eq. (B2) (see Appendix B for details).Although our tactoids are not bipolar, if we were to usethe same assumptions to fit our results, we would end upwith a negative bending constant, which is nonphysical.For homogeneous tactoids, which our microtubule tac-toids appear to be, the surface energy term in free energyshould dominate the energy landscape (eq. (B3)). Fur-ther, elastic terms are negligible because there are nodistortions in the director field. In liquid crystal theoryusing scaling arguments and minimization of free energyvia variational theory, it has has been shown that ω , theratio of the anchoring strength and the bare surface ten-sion, is the sole parameter describing systems with ho-mogeneous tactoids [3]. We calculated ω values using L/W = 2 ω / for ω (cid:29)
1. In our case,
L/W is muchlarger than one, as seen in TABLE III. Using the aver-age value of
L/W for our systems, we find that ω rangesfrom 16 - 28. These ω values are large compared to othersystems [4, 8, 37].Our tactoids are different from other system in afew ways. First, being a homogeneous tactoid, theaspect ratio does not stay constant with the length ofthe tactoid as would be predicted from liquid crystaltheories and experiments. Second, the ω value is highalthough the tactoids are homogeneous. In our priorwork, we demonstrated that microtubule tactoids areimmobile and likely jammed within the tactoid [15].2 R ( μ m ) A s pe c t r a t i o L / W silane PEG 8 kDasilane MC 14 kDasilane MC 88 kDasilane PEG 100 kDalipid MC 88 kDalipid PEG 100 kDa (iii)(ii) Rr lα (i) FIG. 8. Tactoid shape characteristics for all presented data. (i) Diagram shows the relationship between quantitativeparameters of the tactoid r , l , R and α . (ii) Plot of tactoid aspect ratio vs tactoid length L = 2 l shows a slope equivalent to theconstant width of the tactoids. (iii) Plot of R vs α demonstrates a power law behavior that features constant tactoid width. Since the microtubules within the tactoids do not move,they do not behave like liquid droplets. Microtubuletactoid condensate might be more reminiscent of solidcondensates in a liquid background - like the precipita-tion of crystals.
Tactoid Growth Dynamics:
In order to help under-stand the mechanism of tactoid formation, we directlyobserve how a single tactoid nucleated and grew. Weplot the intensity profile of the major and minor axesover time (Fig. 9(i)). We plot the length of three tac-toids as a function of time and found that the tactoidgrowth displayed two distinct growth phases (Fig. 9(ii)).In early times, the tactoid nucleates and grows quickly.The tactoid is short, likely having formed from small,highly mobile microtubule filaments associating via thecrosslinking protein. The tactoid length increases rapidlyat a rate v t = 10 . ± . t = 0s until t = 200s. The time denoted t = 0sis the time when chamber is placed on the microscopewhich was ∼ one minute after the elongation mix is flowedthrough the chamber. At longer times, the tactoid lengtheither reaches a stable state or shows very little growthwith a slope 0 . ± .
02 nm/s (95 % confidence bound).Interestingly, the width of these initial assemblies isalready in the range of 500 - 1500 nm, the same rangewe observe for all tactoids (Fig. 6(ii), 7(ii)). This dataimplies that the width of the tactoids is set very early innucleation and growth, and changes little over time. Weobserve a small change in width over time, with a slopeof 0 . ± .
002 nm/s (95 % confidence bound).The aspect ratio trend follows the length growth trend.The long-time phase shows a constant aspect ratio overtime for all three tactoids. Of the three we measured,two continued a slow growth in both length and width,while the last one is constant in length and width. Theconstant shape parameters imply that the tactoids havereached equilibrium likely due to exhausting the tubulindimer supply. Similar to the length and width measurements, the as-pect ratio
L/W also has two phases as a function of time(Fig. 9(iv)). Since the width does not change much, evenin the initial fast-growing phase, the aspect ratio’s trendfollows that of the length, with an initial fast growingrate of r t = 0 . ± . s − (95 % confidence bound).At the later phase, the aspect ratio does not change andhas a constant aspect ratio of L/W = 7 . ± . First frameAfter one hour I n t e n s i t y v a l u e ( b i t ) L / W (i) (ii) (iii) (iv) W i d t h ( μ m ) Leng t h ( μ m ) FIG. 9. Growth dynamics of the tactoids: (i) Intensitydistribution along tactoid length for the first frame (orangeline), over time (gray lines) and after one hour (green line) isshown. (ii) Tactoid length L is plotted over time. (iii) Tactoidwidth W is plotted over time, (iv) Tactoid aspect ratio, L/W,is plotted over time. For (ii)-(iv), three tactoids from thesame video are averaged. Data is taken in presence of 88 kDaMC on lipid surface. κ from elastic terms and ω from surface terms. Thehomogeneous to bipolar transition was observed experi-mentally in the CNT system [38] and amyloid fibril sys-tems [8].The primary physical aspect of microtubule tactoidsthat make in inaccessible to liquid crystal theory is thatthe tactoid width is approximately constant in all condi-tions. Due to the limited width, the aspect ratio dependsentirely on tactoid length. Further, the width was unaf-fected by experimental parameters such as the crowdingagent or the surface of the chamber.There are several possible reasons why the width isconstant. First, there is a crosslinker causing condensa-tion of the tactoids instead of being entropically driven,like previous systems [5, 8, 38]. There was a prior reportof actin filament tactoid formation driven by an actin-associated protein crosslinker [4, 39]. These actin tac-toids did behave as liquids with fluid interiors, rearrange-ments, and coalescence. The differences between the mi-crotubule and actin tactoids condensed with crosslinkerscould be that the crosslinker affinity, flexibility, or fila-ment length control which were different for each of thesesystems.Since MAP65 is a known anti-parallel microtubulecrosslinker, we anticipate that the initial nuclei of thetactoids are antiparallel upon nucleation. Based on priorwork from our group and others [17, 18, 40, 41], we ex-pect that MAP65 diffuses along the overlap between an-tiparallel filaments and generates compaction forces asthe tactoid grows bidirectionally. We have observed thatMAP65 has a higher affinity for overlapping filamentscompared to single filaments [17]. Given this, we wouldexpect a higher local concentration of MAP65 inside thetactoid than outside, which might inhibit adding new fil-aments to the tactoid surface.Further, already nucleated microtubules would be slowto diffuse and would not be able to join tactoids eas-ily. New microtubules are unlikely to be nucleated oncesignificant growth has started, due to the depletion ofdimers from the environment. The width limit could bea natural consequence of the nucleation and growth kinet-ics of microtubules. In adddition, for our case κ > ω cancome from the small tactoid volume or the strong bindingof the crosslinker. The equilibrium dissociation constantfor MAP65 to bind to microtubules, K D ∼ . µM , in-dicating weak affinity and a fast turn-over rate. Despitethe relatively weak binding, MAP65 can immobilize themicrotubules within the tactoid [15]. For future models,adding a specific ’crosslinking’ term in the free energy expression could be used to describe this system moreaccurately.Limited width in assemblies has been explored theo-retically by kinetically arrested aggregation models [42],self-assembly models with long-range interaction schemes[43], as well as chirality and geometrical packing frustra-tion problems [44]. Microtubules have an intrinsic chiral-ity present in their lattice, most obviously shown at thelattice ‘seam’. If MAP65 binding templates off the mi-crotubule lattice, the tactoids should be chiral bundles aswell. This inherent chirality could result in packing frus-tration which limits the width in the assembly [44, 45].Prior work on the structure of mitotic spindles hasshown that the overlapping microtubules, which arecoated with MAP65-like proteins, display a helical twistof the filaments of the spindle [46]. Cross-sectionalstudies of microtubule bundles in the presence of crow-ders have shown that increasing depletion can alter thepacking from square to hexagonal packing [47, 48], butthey did not examine if there was chirality. Similarhigh resolution studies using cryo-electron microscopyon microtubule tactoids have yet to be performed,but they may show a chairality and square packingto accommodate the anti-parallel preference betweenfilaments. IV. CONCLUSIONS
Using bottom-up reconstitution experiments, weshowed that how passive entropic forces control patternformation in a microtubule system. Focusing on four dif-ferent crowding agents with different types, sizes, andconcentrations we demonstrated that self-organizationpatterns subtly depend on crowding agents and surfacecoatings with or without crosslinkers.For experiments in the absence of specific crosslinkers,microtubules robustly create fan-like patterns in the pres-ence of any crowder. Crowders had subtle effects on thefilament nucleation and growth, resulting in changes inthe contour length, and ultimately the size of the surfacepatterns. The fan-like organizations were reproduciblein simulations. We found a scaling relation between do-main size and correlation length to describe these surfacepatterns.We sought to test if the surface treatment affected thepattern without crosslinkers. We found that the patternwas less reproducible on lipid bilayer surfaces as defectsoccurred frequently in these patterns. When trying toreproduce this effect by lowering the viscosity in simula-tions, we did observe some similarities, but the final pat-terns and evolution of the system in time did not matchthe patterns observed on lipid surfaces.In the presence of antiparallel microtubule crosslinkers,all systems result in tactoid shapes. There was a subtledifference on the tactoid length, following the same trendwhat we observed for the fan-like patterns. This is fur-4ther evidence that the crowders are likely affecting mi-crotubule contour length through changes in nucleationand growth of the filaments. The tactoids were not com-parable to liquid crystal theory because the condensedmicrotubule phase was not liquid, but appeared solid. Inaddition, the width of the condensed homogeneous tac-toids were constant regardless of crowders.In this study, only dynamic component was micro-tubule growth, which was halted when the free tubulinwas exhausted. The mitotic spindle and other biologicalcondensed tactoid systems are fluid and display steadystate dynamics. In order to capture these excitingactivities, we propose future experiments should includeactive components, such as motor proteins. Indeed,such experiments have resulted in active nematics ofmicrotubule bundles, asters, and vortices [31, 49–54].Other possible directions to explore to increase thefluidity of these condensates include adding enzymes,such as depolymerizing kinesins or microtubule severingenzymes, to limit microtubule length. Moreover, ad-dition of nucleating centers to restrict nucleation andgrowth, or crosslinker affinity alteration, or addition ofdifferent ionic species are future directions to explore.
V. ACKNOWLEDGMENTS
Experiments were performed by SS, LH, and RQ. Ini-tial analysis was performed by LH and RQ. SS and JLRcomposed the original manuscript. SS, LH, RQ and JLRedited the manuscript. This work was funded by a grantfrom the National Science Foundation NSF BIO-1817926and partially funded from the Keck Foundation to Dr.R. Robertson-Anderson, M. Das, M. Rust, and JLR.
Appendix A: Crowder Characterization
In our experiments, two types of crowders (methylcel-lulose and poly-ethylene glycol) were added to the sys-tem, with different molecular weights, and %( w/v ) con-centrations to modulate the depletion forces. In our ex-periments, we only used dilute polymer solutions wherepolymers are hydrodynamically separated and cannot in-teract with each other. Once the concentration of poly-mer, c , crosses the critical overlap concentration c ∗ , poly-mers become entangled and overlapped. In the diluteregime c < c ∗ , these blob configurations are character-ized by the radius of gyration R g . A scaling relation, R g = K R M νW , connects R g with weight-average molecu-lar weight M W , where ν is the Flory exponent, and K R is an empirical constant of proportionality. Similarly, theintrinsic viscosity,[ η ] = lim c → η − η cη is related to M W byMark-Howink-Sakurada(MHS) equation, [ η ] = K η M αW ,where c is the solute concentration, α is MHS exponent, K η is the empirical constant of proportionality, and η, η are the solution, solvent viscosity respectively. Polyno-mial expression of osmotic pressure Π is obtained viavirial expansion in terms of molar concentration C [55],Π = RT [ C + aC + bC + ... ] (A1)where R is universal gas constant and T is temperaturein K, and a and b are virial coefficients. For low polymerconcentration like our experiment, lower bound can bewell estimated by setting Π ∼ CRT ignoring higherorder terms.
TABLE IV. Crowder characteristics.Crowders 8 kDa 14 kDa 88 kDa 100 kDaPEG MC MC PEG K R (nm) 0.021 c a a c ν c a a c K η (ml/g) 0.0488 b a a b α b a a b R g (nm) ∼ c a a c η ref (cP) 0.91 b a a b c ∗ %(w/v) 4.7 1.21 0.33 0.72c %(w/v) 1 0.12 0.15 0.25c in µM η exp (cP) 0.97 0.96 2.13 0.98Π ( Nm ) > a ref. [56] b ref. [57] c ref. [58, 59] For each polymer, the physical properties such asviscosity η , radius of gyration R g , the critical entan-glement concentration c ∗ are estimated and shown inTABLE IV. Polyethylene glycols (PEGs) are a highlysoluble, linear, inert polymers, whereas Methylcellulose(MCs) are negatively charged, branched, polymersthat strongly influence the macroscopic viscosity. ForMCs, our parameters have been estimated from ref.[56]. The R g and M W are insensitive to temperaturewithin the range 15 o C − o C whereas η dependson temperature, and were performed at 15 o C andMC concentrations of 0.03 %( w/v ) (dilute regime)in the referenced article [56]. Aqueous PEG solutionat 35 o C viscosity parameters were estimated fromdata in [57]. The R g value and c ∗ = M W πR g N A , where N A is Avogadro’s number, was extrapolated for our M W from [58, 59]. Since our experiments were per-formed at 37 o C to enhance microtubule growth, we alsomeasured the viscosity independently η exp in TABLE IV. Appendix B: Tactoid Scales
In this appendix we have discussed briefly the theo-retical aspect of the tactoid assembly that has already5studied in [2, 3, 60, 61]. Total free energy functional ofa nematic tactoid can be written in general form as, F = F E + F S (B1)where F E is the bulk elastic energy, accounts for di-rector field deformation inside tactoid, and F S is the in-terfacial energy term. The elastic term in Frank energy,which is integrated over volume V can be written downas, F E = (cid:90) V d r (cid:20) K ∇ · n ) + K n · ∇ × n ) + K n × ∇ × n ) − K ∇ · [ n ∇ · n + n × ( ∇ × n )] (cid:21) in terms of director field n ( r ). Here, K , K , K , K are splay, twist, bend and saddle-splay deformation moderespectively. Usually, twist term K is dropped consid-ering no-twist condition. The K term can be absorbedinto K as it renormalizes K . F E = (cid:90) V d r (cid:20) K ∇ · n ) + K n · ∇ × n ) (cid:21) (B2)Interfacial surface energy term, which is integrated overtactoid surface, has two components with accompanyingparameters. First component comes from isotropic baresurface energy which is driven by τ interfacial tension.Second term introduces anisotropy in the surface energyterm which is driven by ratio of anchoring strength andinterfacial tension ω , a measure how directors interactwith interface. F S = τ (cid:90) S d r { ω ( q · n ) } (B3) Here q is the unit normal to the interfacial surface. Asdescribed earlier, this system can be parametrized us-ing R and α . From simple scaling argument it can beshown that, the bulk elastic deformation cost should scaleas ∼ KR multiplied by R which accounts for squarederivatives, i.e. ∼ KR and surface anchoring energy willhave ∼ ωR scaling. Total free energy written in termsof these two independent variable takes the form, F = F E + F S = (cid:88) i =1 , K i Rf ( i ) E ( α ) + τ R ( f S ( α ) + ωf W ( α ))= ⇒ (cid:101) F = (cid:88) i =1 , κ i ψ ( i ) E ( α ) + ψ S ( α ) + ωψ W ( α ) (B4)where (cid:101) F = FτV , κ i = K i τV , ψ ( i ) E ( α ) = f ( i ) E ( α ) v ( α ) , ψ W,S ( α ) = f W,S ( α ) v ( α ) are the scaled variables. Surface area and volumeare defined as A = R f S ( α ) and V = R v ( α ) f S ( α ) = 4 π (sin α − α cos α ) v ( α ) = 2 π (sin α − α cos α − sin α n ( r ) = n . In small volume limit V →
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