How Selective Severing by Katanin Promotes Order in the Plant Cortical

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PLANT BIOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY cortical array (23, 24) as well as its reorientation upon exter- AB nal cues (25, 26). At first sight, these findings seem at odds with the survival-of-the-aligned mechanism. It has been reported that, after severing, the newly formed plus end of the lagging MT typically starts in the shrinking state (27, 28). Previous theoretical work on noninteracting MTs has shown that, under this condi- tion, severing uniformly along the full length of MTs does not affect the total number of MTs but does reduce average MT length (29). These shorter MTs would have fewer interactions per average MT lifetime and, consequently, a reduced potential C D for alignment. Experiments suggest, however, that katanin actually promotes alignment. Loss-of-function mutations of the katanin p60 subunit result in reduced MT alignment in a range of cell types. The cells in these mutants tend to be more bulged, and the plants exhibit an overall dwarfed phenotype (20, 22). Ref. 19 also reported delayed MT alignment. Additionally, overexpression of RIC1, a protein that stimulates katanin activity, results in highly aligned arrays in pavement cells, which normally do not have aligned E arrays (30). Interestingly, in plants, MT severing appears to preferentially occur at crossover sites (31), and it has recently been shown that this requires the activity of katanin (26, 24, 32). Strikingly, these recent experimental observations also show that it is the “crossing” MT which is preferentially severed (26, 24), rather than the “crossed-over” one. Here we dissect the influence of MT severing on the alignment process of the cortical array using both computer simulations and theoretical arguments. In the simulations, we systematically address the roles of crossover-specific versus random severing, the significance of severing only the crossing MT at crossovers, and the importance of bundling following shallow-angle encounters. Our results show that katanin-mediated severing can indeed promote array alignment, but only when severing is more likely for the crossing MT and the probability of severing at shallow- angle crossovers is low. Our work uncovers an additional requirement for severing-mediated alignment of the cortical array: a Fig. 1. Location of severing determines its effect on alignment. (A) Angle- dependent collision outcomes. Bundling (“B”) occurs for collision angles previously unknown role for MT bundling. These insights lead ∗ ◦ to the understanding of mechanisms by which tuning MT sever- θ < θ = 40 ; for larger angles, induced catastrophes (“C”) occur with prob- ing can be used to control the organizational state of the array. ability Pcat , and, otherwise, MTs cross over (“X”). (B–D) Alignment (S2) as a function of Gc.(B) Random severing (rates per micrometer per second as Results indicated). (C and D) Crossover severing (rates per crossover per second as indicated) for (C) aselective (either side of crossover) and (D) crossing only. Key Aspects of the Modeling Framework. We model cortical MTs In B–D, error bars show the SEM, with n = 100 independent simulations per as series of connected line segments, where one free end, the data point. (E) Representative snapshots (SI Appendix, Unbiased Selection plus end, exhibits dynamic instability, and the other free end, of Snapshots) for various induced catastrophe rates rc and severing rates the minus end, shows steady depolymerization (“treadmilling”), (rs and rx ) at T = 30 h. MTs are colored according to their orientation, to characterized by parameters (SI Appendix, Table S1) that are enhance the visibility of domains with different orientations. based on experimental observations (16). MT collisions with relative angles θ ≤ θ∗ = 40◦ result in bundling. Collisions at larger angles result in either an induced catastrophe (with probability (disordered) system, and 1, for a system in which all MTs are Pcat ) or a crossover (Fig. 1A). As default, we used Pcat = 0.5 (16, aligned exactly parallel or antiparallel. 33, 34). In between collisions, MTs grow in straight lines. By measuring the order parameter S2 for a range of values of In our simulations, we characterize the strength of the inter- the control parameter Gc , we can compare, in a straightforward actions between the MTs using a control parameter Gc . This manner, how phenomena that are not part of Gc , such as sever- parameter, originally derived from a theoretical model (Mate- ing, influence the system’s potential for alignment (e.g., see ref. rials and Methods), combines all dynamic instability parameters 35). Using this framework, we investigated several modes of MT and the nucleation rate in a single dimensionless number. Gener- severing. ically, alignment occurs when the control parameter Gc exceeds ∗ a critical value G (Materials and Methods). In most biologi- Localization and Selectivity of Severing Determines Effect on cally relevant cases, Gc < 0, and its value can be interpreted as Alignment. (minus) the ratio of the typical length scale between interactions Random severing reduces the potential to align. We first tested and the average MT length in the absence of interactions. This the theory-based (29) prediction that random severing would interpretation intuitively suggests ways in which alignment can be reduce the potential to align also in the full interacting system. enhanced by making Gc less negative (see also Table 1). The con- For this analysis, we introduced a uniform severing rate per unit trol parameter has proven very useful in investigating the effects of MT length rs (in micrometers per second) in our simulations. of various processes on MT alignment, even outside the scope in Increasing rs reduced the regime of control parameter values which it was derived (16, 33, 35). (Gc ) for which the system can align, to the point that alignment To quantify the degree of alignment of our simulated corti- became impossible with the highest reported severing rates (Fig. cal arrays, we use the order parameter S2, originally formulated 1 B and E). Moreover, in line with the theoretical predictions for to measure the degree of alignment in liquid crystals (Materials the noninteracting MTs (29), we found that the MT length distri- and Methods). S2 takes values between 0, for a perfectly isotropic butions were no longer exponential, as without severing (refs. 18

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Fig. Appendix, (SI MTs also see 36; and MTs. noninteracting of r v v Change dynamics MT in changes of Impact 1. Table .Errbr ersn E,with SEM, represent (C bars severing. Error from S5). 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Severing. random selec- discordant by of from punishment works Protected tive mechanism survival-of-the-aligned Be original Must The Interactions increasing Shallow-Angle with gradually (F MT increased crossing regime the for aligned preference the of size hsrvra fefcswt simulations (P crossovers with more pronounced more with even align- effects was alignment on of enhancement of maximum effect reversal positive a this to (at alignment ment on effect detrimental aligned 3). the affects Fig. rate severing , Appendix the (SI which MT in regime way of the and shapes S2) Appendix, the Fig. (SI by supported distributions is length approach This catastrophes. an theory as the 17 in ref. severing we by crossing-only included prediction, we are angle” this, this with crossovers occurs; test “protected To shallow-angle a severing. if introduced from bundling, protected without somehow also ment, stark in 2A), (Fig. observed 1D we Fig. detrimental what to be indeed, contrast is, would shallow-angle this which exposes ordering; severing, to however, crossover dynamics, to in (16, interactions crossovers change align- by replaced these This that is bundling 18). shows for when possible severing possible still is without ment is work of severing Previous angles interactions. through with crossovers punishment” no “delayed are there sequently, nspotn aai cin udigcnetae katanin concentrates Bundling action: katanin supporting in fti ehns stu,te rsigol eeigwl only will severing crossing-only then true, is mechanism this If hstertclcnieainpeitdatasto rma from transition a predicted consideration theoretical This align- promote should severing crossing-only vein, same the In hs eut ugs htbnln ly rtclrole critical a plays bundling that suggest results These cat θ = θ p p ≈ 9,teptnilipc fcosn-nysvrn seven is severing crossing-only of impact potential the 0.09), 20 = C orsodn oteclrdarw:dr ryrepresents gray dark arrows: colored the to corresponding 38 tmswae omo uihetta induced than punishment of form α-times-weaker ◦ ◦ 40 , ihincreasing with ) ihbundling. with ◦ nraigCosn-nySvrn Rates Severing Crossing-Only Increasing n 60 and , cat 0 = ◦ <40 r rtce rmsvrn n turns and severing from protected are ) . 09; ◦ θ t TLnt Distributions Length MT p i.S4). 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PLANT BIOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY + remaining parameters: rescue rate (rr ), plus end growth (v ) A − and shrinkage (v ) velocity, and minus end retraction velocity (v t ). In the absence of severing, such changes are accounted for in the survival-of-the-aligned theory [“primary effect” in Table 1; either a switch from disordered (D) to aligned (A) is possible, or the reverse]. That is, they affect Gc (MT length and array B density) in such a way that the critical value G∗ for the transition between disorder and alignment remains the same. The presence and magnitude of crossing-only severing did not change this finding (SI Appendix, Figs. S9 and S10 and Table 1). We noted, how- t ever, increased maximum S2 levels with increasing v and, very C + − mildly, also with increasing v , but not with rr or v . Increas- ing v t thus could have two effects: complete loss of alignment if Gc becomes too small, or increased alignment. Ref. 37 suggested that short treadmilling MTs (stMTs) could enhance alignment, so we quantified the MT length distributions of MTs that grow against the majority orientation (SI Appendix, Measuring Length Distributions of Discordant MT Populations, Figs. S11 and S12, Fig. 3. Crossing-only severing is essential for alignment if induced catas- and Table S2). We found that, even when we fixed the value t trophes are rare, but is insufficient if they are absent. Bundling occurs for of Gc (through rc ), the simulations with high v had more dis- θ < 40◦.(A) Without severing, the reduced induced catastrophe probabil- cordant short MT segments (< 1.6µm) and fewer long ones. + ity Pcat = 0.09 is insufficient to find an aligned regime. (B) With high sever- We also observed an increase in short MTs with decreasing v , −1 ing (rx = 0.1s ), some degree of alignment occurs over the same range of but this was accompanied by a small decrease in maximum S2. parameters for all three values of Pcat .(C) Using optical instead of normal S2 to Increasing rx also increased the abundance of the shortest MTs measure alignment, i.e., counting every stretch of MT bundle as a single MT, and decreased maximum S2 levels (Fig. 4 and SI Appendix, Fig. regardless of the actual number in the bundle, alignment disappears toward S11 E and F). Increases in stMT abundance occurred both cor- Gc = 0 in the absence of induced catastrophes. Also, for intermediate values related and anticorrelated with increases in maximum alignment, of Gc, the arrays would appear poorly aligned. Error bars represent SEM, with so this class of MTs is not a significant driver of array alignment n = 100 independent simulations per data point. For other values of rx and in our simulations. buildup of alignment over time, see SI Appendix, Figs. S6–S8. Discussion Supported by our simulations, we have extended the theory severing to large-angle interactions by preventing the occurrence underlying the survival-of-the-aligned mechanism for sponta- of shallow-angle crossovers. neous alignment of plant cortical MTs (17, 35), to explain two previously conflicting findings: (i) the theoretical finding Crossing-Only MT Severing Requires Subsequent Catastrophes to that katanin-mediated MT severing reduces average MT length, Promote Alignment. Treating crossing-only severing as a weak which should result in less alignment (29), and (ii) the exper- form of induced catastrophes raises two questions: (i) Does the imental observations that katanin-mediated severing critically effectiveness of crossing-only severing depend on induced catas- contributes to normal alignment in plants (19, 20, 22, 38). trophes? (ii) Can crossing-only severing also induce alignment The core of the survival-of-the-aligned mechanism is that independently, i.e., without induced catastrophes? Central to spontaneous alignment occurs if the amount of interactions these questions is the mechanism that removes the leading MT fragments after severing. What is the relative contribution in this removal of induced catastrophes, spontaneous catastrophes, and a combination of bundling (to adopt the majority orientation) and minus end treadmilling? We investigated the importance of catastrophes by repeating the crossing-only severing simulations (with bundling) of Fig. 1D with fewer (Pcat = 0.09) and no (Pcat = 0) induced catastrophes. With Pcat = 0.09, as with the default Pcat = 0.5, we found a well-aligned regime for the −1 −1 high severing rates (rx = 0.01 s and rx = 0.1 s ), albeit with a somewhat reduced maximum average degree of alignment (Fig. 3 and SI Appendix, Fig. S6). Without any induced catastrophes, the maximum degree of alignment further declined. Moreover, the “optical” version of the S2 alignment parameter—for which each bundle is counted as a single MT, regardless of how many MTs are part of it—was much lower still. Particularly the arrays with −1 rx = 0.01 s would visually appear disordered over the entire parameter range (SI Appendix, Figs. S7 and S8). We conclude that, in the absence of induced catastrophes, crossing-only severing can, at best, induce a low to moderate degree of alignment. Furthermore, such alignment could only be induced if Gc is not too high, that is, if sufficiently many Fig. 4. Opposite correlations between order and short MTs for crossing- spontaneous catastrophes occur. This result means that crossing- only severing and minus end treadmilling. Increasing vt resulted in increased only severing can only induce alignment with help of enough S2 and increased the fraction of short MTs (<1.6 µm) with and without catastrophes. crossing-only severing (three curves as indicated in legend). For any given t value of v , increasing rx resulted in an increase of the short MTs but a t Severing and MT Stability. To allow the reader to predict how decrease of S2. The v ranges from 0.01 to 0.06 µm/s, and identical values of various mutations and chemical treatments affecting MT stabil- vt are connected by gray lines (SI Appendix, Measuring Length Distributions ity interact with katanin-mediated severing, we varied each of the of Discordant MT Populations).

4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1702650114 Deinum et al. 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(24, MTs affects crossing predominantly and the 31) 26, (24, sites crossover MT at occurs In concor- observed: between exper- are specif- imentally conditions difference These have and enhanced. the is We MTs crossovers MTs, discordant 28). and at dant crossing (27, occurs the vitro severing targets in when ically observed only is that as found state, life- starts end and shrinking plus lengths the formed MT newly in the reduced because to happens only leads this i.e., times; punishment, a as time acts life MT alignment. unsta- spontaneous become average driving to state thus in disordered) ble, in (i.e., isotropic which difference the MTs, causes aligned) a turn (i.e., concordant in and MTs, discordant results between discordant “pun- sufficiently if this are only direction, ished”; majority happens the against this growing and i.e., threshold, a exceeds enme al. et Deinum (longitudi- as discordant the terms severing of In katanin-mediated repressor (26). than renders rather orientation this amplifier (longitudinal) cross- an framework, the new affected our the events of of severing many MTs growing as the ing times in 1.7 started and ends state, plus severing-created of population suggest for thus organization. targets MT results potent controlling potentially Our are ). S12 properties and treadmilling MT S11 that Figs. Appendix , (SI magnitude, ment observed with experimentally only the and, of abundance stMT them. in in incorporate changes to substantial that interesting found exper- be We firmly will been accounted have it effects not established, such are imentally Once and simulations. our how- reorientation in length-dependent MT organiza- for is, and/or MT affect age It MT in that as role alignment. properties such a factors of play of because decrease could tion stMTs a that abundance with conceivable stMT ever, correlated in increase the was small a explain and only in cannot produced this abundance from however, because sites the stMT simulations, crossover in protect Our increased to stMTs (23). proposed of three- is severing abundance to SPR2 twofold the cells. a tread- petiole in reported few 37 increase process, Ref. fold either expected. are Without MTs release (32). only their milling for sites not responsible katanin nucleation also that is from fact but crossovers, the at from MTs understood severs be can this (37); (40). environmen- organization to MT responding changing and for sever- stimuli target MT tal important tuning an that indeed suggest is findings ing these of in All perturbations (25). photore- mechanical mutants through to responds manner array less MT light-dependent far the Moreover, blue (26). PHOT1/PHOT1 a ceptors in p60 and katanin and experi- can (30), RIC1 activity between Other interactions severing through organization. regulated katanin-mediated katanin be MT that (p80) suggest normal data regulatory for mental crucial and is (p60) subunits catalytic between balance u iuain hwta Tsvrn,dpnigo the on depending severing, MT that show simulations Our hid- major a uncover to us allowed approach mechanistic Our severing that confirm severing random using simulations Our uigbu ih eretto ftehpctlary sub- a array, hypocotyl the of reorientation light blue During the in absent were stMTs The v t hscag orltdwt nrae align- increased with correlated change this , Tsvrn predominantly severing MT Arabidopsis, Arabidopsis katanin spr2 v t 3) ugsigaproper a suggesting (39), yincreasing by or katanin oso-ucinmutant loss-of-function v + rdcdachange a produced loss-of-function spr2 r x mutant alone, r 16. ref. of information For supporting density. the length see MT details, total technical varying, continuously the, on dependent length events unit per In rate constant treadmilling. shrinking a the starts r with MTs in immediately severs starts end katanin mode, end minus “random” plus formed the MT newly formed the newly and The state, MT. single a of ering Action. Katanin of Modes Source circumference. cell plant a (42). to available 80 is similar simu- periodic code dimensions a previous used with our have in i.e., We as domain, 34). same 33, for the (16, values are studies varying parameters lation with dynamic researchers, basic other These by 15). used also is proba- function with occur catastrophes induced bility angles, larger For angle occurs. relative bundling the on depends of collisions MT–MT rate of nucleation effect isotropic uniform a with nucleated with (treadmilling) retraction steady rate with severed. was bundle selected selected the was bundle 1C (Fig. either random which “ase- at in performed also simulations We severing” intersection. crossover the call lective of the We side to intersection. cytoplasmic corresponds the the usually on at bundle bundle second second The arrived severing.” that (possibly “crossing-only only) bundle this the MT from one severing of for selected consisting is MT single a of Typically, rate park. used highest our s that 41 of severing until can MTs with event-based MTs. ends fast, individual plus a all their v is includes at this straight explicitly (16); grow that 1.26.1 version platform CorticalSim simulation using out Parameters. ried and Methods Simulation Basic Methods and Materials organization MT show end that minus phenotypes. mutants of in measurements parameters vivo. encourage treadmilling in organization moreover, MT results, in is changes Our severing induce MT to why and target alignment, excellent MT an in severing MT of role the for topic a remains reorientation of investigation. type further particular are complete mechanisms a possible in a fix different involved of so and Which favor array, orientation. new to the the mechanism(s) to additional directionality how- requires option, final reorientation Neither a rate. imposes rescue and/or ever, velocity increas- growth and/or sponta- the velocity the ing shrinkage decreasing and/or by rate reached stubs catastrophe be MT neous could leading regime of This removal unbound 3). the the (Fig. hampers enter this temporarily as popula- would regime, cell MT growth the reorien- discordant if initiate occur the to could of tion needed enhancement is Additional what tation. exactly population, MT nal) ae h retto fteainetadtksvle from values takes 90°). and to alignment same (−90 the the of exactly orientation with the MTs angle cates corresponding (all A 1 polarity). to segment system) disregarding orientation, isotropic (perfectly 0 domain. simulation from the ically within segment the of tation planar Alignment: pro- Measuring a in results effectively 40 bundling of of angle tected presence and the 2 simulations, Fig. default in used only is modification x s − prmcoee e eod;ti ae h optto fsimulation of computation the makes this second); per micrometer (per (s ntecosvrmd,itretosaesvrdwt osatrate constant with severed are intersections mode, crossover the In eitoue rtce angle protected a introduced We ncnlso,orwr rvdsmcaitcudrtnigof understanding mechanistic provides work our conclusion, In = −1 r l c n P 0.16 e nescin hn ta.(4 eotda vrg atn time waiting average an reported (24) al. et Zhang intersection. per ) cat n ecerate rescue and S stelnt fidvda Tsget and segments MT individual of length the as 2 .Tesm interaction same The 1A). (Fig. another one over cross MTs the or , re aaee 1,33), (16, parameter order µm·s S 2 −1 = ◦ hycnsic ewe hs ttswt catastrophe with states these between switch can They . and . q ± r S r P ,uo hc igeM from MT single a which upon S3), Fig. Appendix, SI 2 4si rnvreary (mode: arrays transverse in s 14 = . n omauetedge fainet euethe use we alignment, of degree the measure To 0 s 0.007 nalcss eeigeetrslsi h sev- the in results event severing a cases, all In l n cos(2θ r x = −1 s 0.1 n θ epciey(1.Almnsed show ends minus All (41). respectively , ) P p v hra,i the in whereas, S5, Fig. Appendix, SI eo hc osvrn cus This occurs. severing no which below 2 −1 t n v + = l + n shg u tl naraitcball- realistic a in still but high is 0.01 = P l iuain aebe car- been have simulations All 0.08 n µm·s l NSEryEdition Early PNAS n sin(2θ µm·s θ n −1 S r steaslt orien- absolute the as 2 × n nrae monoton- increases θ = n −1 1) e T are MTs New (12). 80 ) For . 0s,indicating s), <40 0.001 2 rsrn with shrink or , µm , θ < θ to 2 −π/ 2 µm simulation ∗ θ P | −2 s cat = 2 f6 of 5 The . indi- 40 2 π/ (14, [1] ◦ ,

PLANT BIOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY Control Parameter Gc . Control parameter Gc, which combines all dynamic with instability parameters and the nucleation rate, was originally derived with- 4 Z π/2 out minus end treadmilling (17, 18) and has subsequently been extended to ˆcn = sin(θ) cos(nθ)Pcat (θ)dθ [4] π 0 include treadmilling in the absence of MT bundling (33), and θ as the (relative) collision angle (16). s + t 2 − t Due to the spatial correlations that typically occur in real (and simulated) 3 2(v − v ) (v + v ) rr rc Gc = − . [2] cells, the transition between isotropic and aligned states typically deviates r v+(v+ + v−) v− + vt v+ − vt n somewhat from the theoretical value, but G∗ remains a powerful concept to predict the direction of change (16). For Gc ≥ 0, there is no intrinsic bound on MT growth, which, in practice, To predict the effects of crossing-only severing in the absence of results in an absence of steady-state solutions with biologically relevant den- bundling, we assume that it effectively functions as an 0 < α ≤ 1 times- sities, so that natural systems can, at most, transiently exist with Gc ≥ 0. We −1 −1 as-effective induced catastrophe, based on the detrimental effects random use parameter rc ∈ [0.003 s , 0.02 s ] to vary Gc, which we emphasize severing on alignment. This results in using the subscript c. With our default parameters, Gc ≈ − 63.2s rc + 0.182. 4 Z π/2 ˆc = sin(θ) cos(nθ) {P (θ) + α [1 − P (θ)] q(θ>θ )} dθ, [5] ∗ n cat cat p Critical Value G of Gc . The mean field theory in Hawkins et al. (18) predicts π 0 that the system will align for > ∗, and not for smaller . The biolog- Gc G Gc with q(θ >θp) as a step function that is 1 for collision angles larger than ically relevant range for alignment is thus ∗ < < 0. In the absence of G Gc protected angle θp and zero otherwise. severing, the value of G∗ depends on the zeroth and second Fourier coeffi- cient of interaction function as ACKNOWLEDGMENTS. We thank Pier Siersma for his input during the initial stages of this work. The work of B.M.M. is part of the research program of ∗ p3 ˆc0 G = −2ˆc2 − 1 , [3] the Foundation for Fundamental Research on Matter, which is part of the −2ˆc2 Netherlands Organisation for Scientific Research.

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