Still and rotating myosin clusters determine cytokinetic ring constriction

The cytokinetic ring is essential for separating daughter cells during division. It consists of actin filaments and myosin motors that are generally assumed to organize as sarcomeres similar to skeletal muscles. However, direct evidence is lacking. Here we show that the internal organization and dynamics of rings are different from sarcomeres and distinct in different cell types. Using micro-cavities to orient rings in single focal planes, we find in mammalian cells a transition from a homogeneous distribution to a periodic pattern of myosin clusters at the onset of constriction. In contrast, in fission yeast, myosin clusters rotate prior to and during constriction. Theoretical analysis indicates that both patterns result from acto-myosin self-organization and reveals differences in the respective stresses. These findings suggest distinct functional roles for rings: contraction in mammalian cells and transport in fission yeast. Thus self-organization under different conditions may be a generic feature for regulating morphogenesis in vivo.

The rod is of length (the filament, red) that is drawn by a motor force !"# at !"# into the 116 direction of the arrow. The filament velocity is , its mobility . The black line indicates the 117 stress profile that results from the applied force and the filament friction with the environment.
The total filament number was 2 !" In the following, we will give the mathematical details of the physical model used in the main 143 text. 144

Model definition 145
With our theory, we try to capture essential features of the ring dynamics, such as, filament 146 polarity, rules of interaction between filaments through molecular motors. Consequently, the 147 final equations of motion describe the behavior of contractile rings independently of many 148 details of the molecular interaction rules. Still, in the following, we will evoke a specific image 149 to introduce the dynamic equations. 150 Consider a ring of perimeter L of actin filaments such that the filaments align with the ring 151 perimeter. We denote the co-ordinate along the ring perimeter by and describe the distribution 152 of (polar) actin filaments along by the densities ! for filaments with their plus-end pointing 153 clockwise and ! for filaments of the opposite orientation. Two filaments of opposite orientation 154 can join their plus-ends forming a bipolar filament (Fig. 4a, (i)). Indirect evidence for such a 155 process is given by the fusion of nodes observed in fission yeast 2 . While we refrain from 156 suggesting an explicit molecular mechanism, such bipolar filaments may be formed by motor 157 clusters linking the filaments. Also actin nucleating proteins of the formin family could be 158 involved (Figs. 2c and 3b). The distribution of bipolar filaments is denoted by !" and gives the 159 density of their centers. Bipolar filaments form at rate ! ! ! , bipolar filaments can split into 160 two filaments of opposite orientations at rate ! (Fig. 4a, (i)). 161 Actin filaments continuously turn over. In general, they assemble at the barbed end by addition 162 of actin monomers and disassemble at the pointed end by actin monomer removal or severing. 163 Assembly and disassembly can be captured by effective rates. These rates depend on the state of 164 the ends: Capping proteins can inhibit or promote assembly and disassembly. In a minimal 165 model of the ring dynamics, we refrain from giving a detailed account of filament assembly and 166 disassembly. Instead we assume that all filaments have a fixed length that is equal to the average 167 filament length . Bipolar filaments thus have a length of 2 and the total actin density at a point 168 is ! ! + + !" + + ! − + !" − . As a crude account of filament 169 turnover, we will assume that filaments assemble and disassemble at the two ends at the same 170 rate. This leads to an apparent motion of the polar filaments at velocity ± !" . 171 Let us now turn to the filament dynamics induced by molecular motors. They can induce relative 172 sliding between actin filaments. The corresponding velocities are between filaments of the 173 same orientations (Fig. 4a, (ii) ) and for filaments of opposite orientations (Fig. 4a, (iii)). We 174 use these parameters to quantify the strength of the motor-mediated filament-filament 175 interactions. We assume that motors are located at the filaments' plus-ends, such that ! + ! + 176 !" is the distribution of motors. Finally, fluctuations are accounted for by diffusion terms with 177 an effective diffusion constant . The corresponding dynamic equations read: 178 To assist the reader, let us state explicitly the difference of this model to the one discussed in 179 Ref. 3 , where the framework used here was developed. The present model extends the former 180 work by including the presence of bipolar filaments and processes of their assembly and 181 disassembly. Furthermore, in the original formulation 3 , filament assembly and disassembly were 182 neglected. Here, we include it in an effective way, by adding the treadmilling currents. That 183 treadmilling is an important part of the actin assembly dynamics was shown, for example, in 184 Ref. 4

. 185
For numerical solution of the dynamic equations, we used a first-order upwind scheme with 186 adaptive time stepping. For the calculation, we have used dimensionless parameters, where time 187 has been scaled by ! ! !! , length by , and the filament densities by ! , where ! is the 188 number of plus-and of minus-filaments. Consequently, !" is scaled by ! ! , and by ! , 189 and by ! ! ! . 190

Mechanism of the instability 191
First note that the interaction of two bipolar filaments with each other tends to align their centers. 192 For a homogenous ring of bipolar filaments, the force on each bipolar filament cancels out. As 193 soon as there is a perturbation, locally imbalances are present that will lead to an accumulation of 194 bipolar filaments, possibly at different positions along the ring, unless diffusion is dominating 195 and smoothing the perturbations. What is the typical distance one can expect between two 196 clusters of bipolar filaments? A bipolar filament can interact with all bipolar filaments that are a 197 distance away. These bipolar filaments extend a distance 2 from the original filament's 198 center, which suggests that the typical distance between two clusters is about 4 . This is indeed 199 the typical distance we observe after clusters have developed starting from a random perturbation 200 of the homogenous state. The typical distance is also affected by the system size (the ring 201 perimeter). For = 10 , which we use in the main text, only two clusters are seen. Note that the 202 distances between clusters changes on long time scales, which presumably eventually leads to a 203 single remaining cluster for systems of any size. However, this coarsening process takes place on 204 such long time scales that it is irrelevant for the dynamics of contractile rings and not further 205 discussed here. 206

Calculation of the stress in the bundle 207
The stress in the bundle is defined as the sum of the stresses in the individual filaments 3 . Stresses 208 in a filament are generated by motors that pull on the filaments and by friction with the 209 surrounding medium. Explicitly, force balance on a single filament gives 210 In this expression, is the co-ordinate along the filament, the stress in the filament, a 211 mobility, the filament's velocity, and !"# the force density exerted by motors on the filament. 212 Only the effects of motors cross-linking two filaments are accounted for. The stress along a 213 filament is thus piece-wise linear in with slope , where = ± or = ± depending on 214 the orientation and the relative position of the partner filament, the motor is connected to. If there 215 is no motor at a filament end, then the stress vanishes at this point, and the stress jumps by an 216 amount at the positions !"# , where motors are bound to the filament, see Fig. S11. The 217 total stress profile along the bundle is then obtained by summing the stress profiles along all 218 filaments in the bundle. Since the expressions are quite involved, we refrain from giving them 219 here explicitly. 220

Behavior after addition of blebbistatin 221
To capture the effect of the myosin inhibitor blebbistatin in our model, we reduced the motor 222 activity , compared to a case leading to a stationary state that corresponds to the pattern in 223 mammalian rings. For stationary states, the myosin clusters subsequently broadened, see Fig.  224 S12. This qualitatively agrees with the behavior observed in mammalian rings, see For mammalian cells, we measured the total and mean fluorescence intensities by tracing the 236 ring contour with ImageJ. The total and mean intensity were normalized by the total and mean 237 intensity of the cells at the onset of division (t = 0 s). The normalized values were averaged and 238 the standard deviation is given by the error bars ( Fig. 1 f, g). With intensities extracted from live 239 samples, we measured the bleaching rate of the cytoplasmic pool of fluorescent proteins, as a 240 good indicator for bleaching since the recovery times for FRAP at the ring were within seconds. 241 The results yielded minor corrections for intensity measurements, within 10%. In addition, fixed 242 samples gave the same measures as live samples, showing that corrections for photobleaching 243 were not needed. 244 Fission yeast rings can be fitted by a circle. Therefore, we measured the intensities by measuring 245 the intensity in circles of the dimensions of the outer ring diameter and the inner ring diameter 246 with ImageJ. The subtraction of these two values gave the total fluorescence intensity of the ring. 247 By dividing the total intensity by the area of the ring (which is the area of the outer circle minus 248 the inner circle), we calculated the mean intensity. Since the variations in intensity measures 249 between individual cells are small in fission yeast cells, we took snapshots of individual cells and 250 assigned time points to the rings as a function of their diameter. The averaged intensity curve 251 was then normalized with respect to the intensity value of 3.1 µm. For intensity measurements of 252 rings before constriction where the diameter is constant we analyzed timelapse movies. We 253 normalized the intensity with respect to the intensity at a diameter of 3.1 µm. Measurements on 254 snapshots and timelapses are in agreement and they are plotted in Fig. 1 i, j (timelapse data until 255 250 s, then data from fixed cells).The standard deviation is given by the error bars ( Fig. 1 i, j). Petri dish and the sticky side of Scotch tape was applied to the top of the filter 1 . The tape with 263 the filter was attached to a double sided tape attached to a Petri dish with the filter side exposed 264 to the air, the PDMS poured onto the filters (non-shiny face up), and the mixture allowed to cure 265 overnight, followed by 4 h curing at 65°C before the stamp was peeled off. 266 Alternatively the stamp can be fabricated by means of microfabrication ( Supplementary Fig. 1) 267 We used regular arrays of microcavities surfaces prepared using standard lithographic methods 268 on silicon wafers 5 . Circular patterns on a mask can be transferred to a Si-Wafer. The surface will 269 contain holes of the size of the wells. PDMS is mixed with curing agent (10:1) and poured on the 270 wafer. Air bubbles are removed by degassing for 30 min. After 4 h at 65°C the PDMS will be 271 cured and the stamp can carefully be cut out and peeled of the wafer. 272 The stamp was exposed to a plasma cleaning for 1 min (Harrick Plasma, PDC-32G, high setting 273 power), followed by a 10 min exposure to Chlorotrimethylsilane 97% (Sigma-Aldrich, C72854, 274 TMCS) vapor or by the deposition of an anti-adhesive layer (Sigma-Aldrich, SL2 Sigmacote). 275 The liquid degassed PDMS mixture was spread on a glass coverslip #0 (25 mm in diameter, 276 Fisherbrand) with a Pasteur pipette 5,6 , after its cleaning with a 1 min exposure in the plasma 277 cleaner. The silanized stamp was then placed onto the PDMS coated coverslip, allowed to cure at 278 room temperature overnight, followed by four hours curing at 65°C. The stamp was separated 279 from the coverslips, generating the well pattern on the upper layer of the 30 µm thick elastomer, 280 using the coverslip as the sealed bottom of the chamber. The overall thickness of the sample 281 allowed objectives with small working distances and high numerical apertures to be used. 282 We modified the protocol for larger cavities for mammalian cells. The PDMS stamps were 283 activated with a plasma cleaner and silanized with TMCS as described above. Liquid PDMS was 284 spin-coated on the molds at 1500 rpm for 30 s. The PDMS was cured for at least 2 h at 65°C. For 285 plasma binding of the cured PDMS layer to a coverslip (#0), both -the PDMS stamp and the 286 coverslip -were plasma activated. The thin PDMS layer was then pressed on the coverslip. The 287 pressure was maintained for several seconds. After about 30 min the PDMS stamp was unpeeled 288 and the thin PDMS layer containing the microcavities was plasma bound to the coverslip. 289 290 291