Positive Feedback Between Contractile Ring Myosin and Ring-Directed Cortical Flow Drives Cytokinesis

During cytokinesis, an equatorial actomyosin contractile ring rapidly transforms cell shape by constricting at a relatively constant rate despite its progressively decreasing size. The closure rate per unit length of the ring must accelerate as the ring gets smaller to maintain the overall constant rate of closure. Here, we examine the mechanistic basis for this acceleration by generating a 4D map of cortical flow in conjunction with monitoring ring component dynamics during the first division of the C. elegans embryo. This analysis reveals that acceleration arises because ring myosin pulls on the adjacent cortex generating ring-directed cortical flow that, in turn, accelerates constriction by delivering cortical myosin into the ring. We derive an analytical mathematical formulation that captures the positive feedback-dependent evolution of the contractile ring and use this formulation to provide a non-intuitive explanation for why reducing myosin activation by rho kinase inhibition slows contractile ring closure. Impact Statement: During cytokinesis, positive feedback between myosin motors in the contractile ring and ring-directed cortical flow drives constriction rate acceleration to ensure timely cell separation. Major Subject Areas: Cell biology, Computational and Systems Biology


INTRODUCTION 35
During cytokinesis in animal cells, constriction of an equatorial actomyosin ring cinches the 36 mother cell surface to generate a dumbbell-shaped structure with an intercellular bridge that connects 37 the two daughter cells (Fededa & Gerlich, 2012 that rings close at a faster rate per unit length as they get smaller. Prior work postulated that this 52 acceleration arises because force generators, either myosin motors (Wu & Pollard, 2005) or actin 53 filament-based contractile units (Carvalho et al., 2009), are retained during constriction, leading to an 54 increase in their amount per unit length. This retention model presumes that acceleration arises from 55 processes intrinsic to the contractile ring, ignoring potential influence of interactions between the ring and 56 the adjacent cortex. 57 Here, we examine the role of interactions between the ring and surrounding cortex on contractile 58 ring dynamics in the C. elegans embryo. Through 4D analysis of cortical flow in conjunction with 59 monitoring of ring component dynamics during closure, we show that acceleration of the per unit length 60 constriction rate does not arise from ring-intrinsic component retention, but instead results from positive 61 feedback between ring myosin and ring-directed cortical flow. We derive an analytical mathematical 62 formulation that captures the positive feedback-dependent evolution of the contractile ring and employ it 63 to analyze experimental data to assess the effects of rho kinase inhibition, uncovering a new, non-64 intuitive explanation for why reducing myosin activation by rho kinase inhibition slows ring closure. 65 66 RESULTS 67

The cortex at the cell poles expands in response to tension generated by the constricting ring, 68 whereas the intervening cortex flows towards the ring without expansion 69
To assess the significance of interactions between the contractile ring and surrounding cortex on 70 contractile ring dynamics, we generated a 4D map of cortical flow to determine how the cortex responds 71 to ring pulling. We monitored cortical movement at high time resolution (Figure 1A, Video 1) in embryos 72 expressing a GFP fusion with the heavy chain of non-muscle myosin II (NMY-2; hereafter myosin::GFP; 73   Our results indicate that the per unit length amount of contractile ring components increases 174 exponentially, and suggest that this increase is due to delivery by cortical flow along the direction 175 perpendicular to the ring. In this model, constriction in the around-the-ring direction does not alter the per 176 unit length amount of ring components, but instead drives ring disassembly that reduces the total amount 177 of ring components in proportion to the reduction in ring length ( Figure 4A, left panel). An alternative 178 model for the increase in the per unit length amount of ring components, proposed based on work in 179 fission yeast (Wu & Pollard, 2005), is that myosin and anillin could be retained within the ring rather than 180 lost as ring perimeter decreases during constriction ( Figure 4A, middle panel). In the retention model, 181 the total amounts of both components remain constant as the ring closes resulting in an increase in their 182 per unit length amount that is inversely proportional to the reduction in ring size. Comparison with the 183 total amounts of ring myosin and anillin suggested that, whereas the retention model fits the data well for 184 / !" between 0.2 and 0.6, there was significant deviation for timepoints outside of this range. In contrast, 185 the ring-directed cortical flow model fit the data for the entire measured interval ( / !" = 0.0 to 0.8; Figure  186  To distinguish between the retention and ring-directed flow models using an independent 188 approach, we photobleached myosin in the entire division plane at ~30% closure, and monitored its 189 subsequent recovery in the ring ( Figure 4C). The ring-directed cortical flow model predicts that the per 190 unit length amount of bleached myosin should be constant and, since cortical myosin turns over faster 191 than myosin in the ring (t 1/2 of ~30s (Mayer, Depken, Bois, Julicher, & Grill, 2010; Salbreux,Charras,& 192 Paluch, 2012)), cortical flow should rapidly deliver unbleached fluorescent myosin to the ring, leading to 193 an exponential increase comparable to that in controls. In contrast, the retention model predicts that the 194 per unit length amount of bleached myosin and any residual fluorescent myosin that is retained in the 195 ring will increase in proportion to the decrease in the ring size (~1/R). We found that the per unit length 196 amount of fluorescent myosin in the ring increased exponentially following bleaching, and the difference 197 between the control and the bleached embryos, which reflects the amount of bleached myosin, remained 198 constant, both of which agree with the predictions of the ring-directed cortical flow model ( Figure 4C). 199 We note that this data also suggests that the recovery of myosin fluorescence in the ring in not due to 200 exchange with myosin in the cytoplasm. If ring myosin were turning over due to exchange with 201 cytoplasmic myosin, we would expect the FRAP curve to approach the control curve and the difference 202 between the FRAP and control curves to disappear. Instead, the two curves remained parallel and the 203 difference remained constant ( Figure 4C). This data suggest that rather than being due to exchange with 204 cytoplasmic myosin, the recovery of ring fluorescence is due to a mechanism in which myosin on the 205 cortex adjacent to the ring turns over, allowing resumption of delivery of myosin to the ring by cortical 206 flow. 207 The conclusion that the per unit length amount of contractile ring components increases 208 exponentially during constriction is in apparent contradiction to analysis in 4-cell stage C. elegans 209 embryos, where we had previously reported an ~1.3-fold increase in myosin, anillin and septins as the 210 ring perimeter decreased 2-fold (from 50 to 25 µm). However, this is in fact consistent with the prediction 211 of the ring-directed cortical flow model (see The exponential accumulation of contractile ring components during constriction due to positive 220 feedback means that the properties of the ring (component levels and constriction rate) are continuously 221 changing. Thus, analysis of perturbations requires fitting temporal profiles of ring size or component 222 levels and deriving meaningful quantitative parameters from these fits. In order to assess the 223 consequences of molecular perturbations, we therefore translated our experimental findings 224 (summarized in Figure 5A) into an analytical mathematical framework (see Methods for detailed 225 derivation), consisting of three equations and three model parameters, that we named the Cortical Flow 226 Feedback (CoFFee) model ( Figure 5B). Based on our photobleaching data, we assume that: (1) 227 constriction in the around-the-ring direction does not alter the per unit length amount of ring components, 228 but leads to ring disassembly that reduces the total amount of ring components in proportion to the 229 reduction in ring length, and (2) myosin in the contractile ring does not turn over by exchange with 230 myosin in the cytoplasm. Thus, increases in the per unit length amount of ring myosin are solely due to 231 delivery by cortical flow along the direction perpendicular to the ring. We posit that myosin increases 232 exponentially during constriction due to positive feedback between the per unit length amount of ring . This time reference also 248 avoids the difficulty of assessing cytokinesis onset. In this time reference, the equation for ring size is: 249 where ≔ / and !"! is the dimensionless characteristic ring size (held fixed at a value of 1.1; see 250 Methods). Any component that localizes to the cell cortex will be delivered to the contractile ring via the 251 same process as myosin, so contractile ring components all accumulate in a similar fashion, with 252 !"#$,!"#$ : = !,!"#$ − ln 2 !!" where !,!"#$ is the per unit length amount of the component at the half-way point of ring closure, 253 !"#$,!"#$ is the baseline amount of the ring component that does not increase exponentially, and !"#$ 254 ( !"#$ for myosin) is the concentration of the component on the cortex that is delivered to the ring. The 255 velocity of cortical flow and the constriction rate are 256 properties of the ring and cortex encoded in our three model parameters ( , , and !"#$ ), we directly 270 measured !"#$ and fit experimental measurements of ring size and ring myosin versus time to 271 equations (4) and (5) to determine the effects on and ( Figure 6A). Direct measurement revealed that 272 the amount of cortical myosin, !"#$ , was reduced by 20% in rho kinase depleted embryos compared to 273 Figure 6B). Next, we fit traces of ring size versus time using the ring size 274 equation (4) to determine characteristic times, ( = 1/ !"#$ ), for each embryo. This analysis revealed 275 that was 1.3-fold higher in rho kinase-depleted embryos compared to controls (120 ± 20 s versus 276 90 ± 10 s in controls; Figure 6C, middle row) indicating that !"#$% !"! , we conclude that !"#$% = !" ; thus, the ability of the cortex to be compressed by 278 ring myosin is not affected by rho kinase depletion. To determine the effect on , we measured the mean 279 per unit length amount of myosin::GFP in the ring versus time in control and rho kinase depleted 280 embryos and fit the data to the equation for ring myosin (5). Interestingly, the per unit length amount of 281 myosin for a given ring size was the same in control and rho kinase depleted embryos, resulting in an 282 equivalent exponential prefactor for the two conditions ( !"#$% conclusion that reducing the concentration of cortical myosin makes it more difficult for rings of the same 312 size with the same amount of myosin to constrict. We suggest that this may be because the 313 compensatory increase in cortical flow that restores ring myosin to control levels leads to an 314 overabundance of other components (e.g. anillin) that increase resistance of the ring to constriction. 315 More broadly, the analysis of rho kinase inhibition, employing straightforward-to-measure experimental 316 parameters, highlights the utility of the mathematical formulation we present to explain the complex and 317 non-intuitive effects of molecular perturbations on cytokinesis. 318 319 DISCUSSION 320 Despite the physical connection between the contractile ring and adjacent cortex, how these 321 interconnected regions function together to change cell shape during cytokinesis has not been clear. 322 Here, we explore this question during the first division of the C. elegans embryo by generating a that feedback between contractile ring myosin and ring-directed cortical flow will be a broadly conserved 335 property of contractile rings in animal cells. The feedback-based mechanism we describe here, in which 336 the increase in myosin levels in the ring is due to cortical flow along the direction perpendicular to the 337 ring contrasts with prior models, including a model previously proposed by our group, that constriction 338 rate acceleration arises from the ring-intrinsic retention of force generating units (Carvalho et al., 2009;339 Wu & Pollard, 2005). 340 In addition to ensuring timely cell content partitioning, the feedback-based mechanism that we 341 describe renders the ring robust to defects in the cytokinesis machinery that increase the difficulty of ring 342 constriction, such as in the inhibition of rho kinase that we investigate here, and/or to internal or external 343 mechanical challenges, such as cell-cell contacts or obstacles in the crowded cell interior. In all of these 344 cases, the feedback loop between ring myosin and cortical flow would lead to the progressive build up of 345 contractile ring components until constriction proceeded. An interesting caveat, suggested by modeling 346 (Figure 7-figure supplement 2) is that molecular perturbations that reduce the ability of the ring to be 347 constricted by ring myosin (reduce in the mathematical formulation) do not alter the kinetics of 348 contractile ring closure. Instead, they introduce a time delay that allows the ring to accumulate enough 349 myosin to overcome the reduced . After this delay, constriction proceeds with kinetics identical to 350 controls, but with higher component levels and flow velocities throughout closure. Experimentally, this 351 means that perturbations that make ring constriction more difficult will not be detected by monitoring 352 constriction kinetics in the absence of a reliable time reference for cytokinesis onset, since the introduced 353 delay may be relatively small. The second signature feature of these perturbations, higher component 354 levels throughout closure, would likely be easier to measure (e.g., by quantifying ring component levels 355 at the closure halfpoint). 356 We note that ring-directed flows have also been observed in the context of wound healing 357 The cortical flow map and laser ablation analysis indicate that recruitment of myosin to the 366 equatorial cortex leads to local compression that places the adjacent cortex under tension. In response 367 to this tension, the polar cortex expands; in contrast, the cortex between the poles and the equator flows 368 towards the ring without expanding. These observations suggest that the polar cortex has distinct 369 mechanical properties. These distinct properties could arise from different, non-exclusive mechanisms. 370 The polar cortex may be less stiff than the rest of the cortex, causing it to stretch and thin in response to 371 ring constriction-induced tension. Consistent with this idea, a reduction in f-actin intensity at the cell poles 372 has been reported during cytokinesis in Drosophila cells due to delivery of a phosphatase by segregating 373 chromosomes (Rodrigues et al., 2015). Alternatively, the polar cortex may turnover more rapidly, leading 374 to a higher rate of surface renewal after stretching. A third possibility is that the polar cortex is more 375 prone to rupture, repair of which would locally increase cortical surface. Consistent with this last idea, 376 blebs have been reported at the cell poles in cultured vertebrate and Drosophila cells, where they have 377 been proposed allow cells to elongate in anaphase and release tension at the poles ( Early conceptual models of cytokinesis hypothesized that polar relaxation coupled to a global 382 upregulation of surface tension could trigger a flow of tension-generating elements towards the equator 383 that would compress into a circular band and initiate a feedback loop similar to the one we describe here 384 (Greenspan, 1978;Swann & Mitchison, 1958;Taber, 1995;White & Borisy, 1983;Wolpert, 1960;385 Zinemanas & Nir, 1987, 1988. Although polar relaxation could drive cytokinesis on its own, the 386 compressed band of cortex would be sensitive to the mechanical properties of the cortex and the amount 387 and timing of relaxation at each pole. Any non-uniformity, for example due to cell-cell contacts, could 388 lead to unstable positioning or collapse of the ring to one side (Greenspan, 1978). Similarly, mechanisms 389 that promote cortical contractility at the cell equator could initiate ingression; however, in the absence of 390 cortical relaxation, cytokinesis would stall due to progressively increasing cortical tension. Coupling 391 equatorial contractility to polar relaxation, as we observe in the C. elegans embryo, has two beneficial 392 effects: (1) it releases the isotropic tension produced by compression of the equatorial cortex along the 393 direction perpendicular to the ring, leading to filament alignment and ring narrowing that reduces 394 resistance from cytoplasmic pressure and, (2) it allows the ring to establish a pattern of ring-directed 395 cortical flow to generate a feedback loop that provides components to the ring in proportion to the 396 velocity of cortical flow rather than the rate of network turnover. 397 Information on constriction kinetics and patterning of cortical compression/expansion suggests 398 that a similar coupling may also support ring constriction in sea urchin embryos. Cleaving sea urchin 399 embryos from a variety of species exhibit constriction kinetics essentially identical to those during the first 400 division of the C. elegans embryo (Mabuchi, 1994). Pioneering work by Katsuma Dan monitoring surface 401 expansion and compression by measuring the distance between surface-adhered particles and the 402 distribution of pigmented cortex-associated granules (Dan, 1954 The ability to analyze the effects of mutations and other molecular perturbations is essential to 418 defining molecular mechanisms. The exponential accumulation of contractile ring components during 419 constriction due to positive feedback means that the properties of the ring (component levels and 420 constriction rate) are continuously changing. The existence of the feedback loop can also to somewhat 421 counterintuitive results-for example, perturbations that increase the difficulty of ring constriction delay 422 constriction onset rather than slowing constriction kinetics. Deconvolving the phenotypes observed 423 following specific perturbations therefore poses a significant challenge. To address this challenge, we 424 generated a straightforward analytical mathematical formulation (the CoFFee model) consisting of three 425 differential equations and three parameters that reflect the empirical properties of the ring and cortex. In 426 addition to describing the processes underlying the evolution of the contractile ring, the CoFFee model 427 provides a simple framework for analyzing experimental data. As we demonstrate here for rho kinase 428 depletion, assessing the effects of a perturbation on model parameters provides insights into the 429 underlying mechanistic effects of the perturbation. For example, the analysis of rho kinase depleted 430 embryos suggests that reducing the concentration of cortical myosin leads to a compensatory increase in 431 cortical flow that restores ring myosin to control levels-we note that the reason for this compensation is 432 a fascinating topic for future work. Since the CoFFee model encapsulates experimental data to 433 accurately describe the dynamics of the contractile ring and associated cortical network, an additional 434 interesting future direction will be to use parameter changes derived from the CoFFee model as input for 435 a finite-element model (similar to (Turlier et al., 2014)) in order to predict the evolution of cell shape given 436  inserted either just before (nmy-2) or after (arx-7 and ani-1) the start codon. The single copy nmy-2 448 transgene was generated by injecting a mixture of repairing plasmid (pOD1997, 50ng/µL), transposase 449 plasmid (pJL43.1, Pglh-2::Mos2 transposase, 50ng/µL), and fluorescence selection markers (pGH8, 450 Prab-3::mCherry neuronal, 10ng/µL; pCFJ90, Pmyo-2::mCherry pharyngeal, 2.5ng/µL; pCFJ104, Pmyo-451 3::mCherry body wall, 5ng/µL) into EG6429 (ttTi5605, Chr II). Single copy ani-1 and arx-7 transgenes 452 were generated by injecting a mixture of repairing plasmid (pSG017 (ani-1) or pOD1998 (arx-7), 453 50ng/µL), transposase plasmid (CFJ601, Peft-3::Mos1 transposase, 50ng/µL), selection markers (same 454 as for nmy-2 strain) and an additional negative selection marker (pMA122; Phsp-16.41::peel-1, 10ng/µL) 455 into EG6429 (ttTi5605, Chr II). After one week, progeny of injected worms were heat-shocked at 34°C for 456 2-4 hours to induce PEEL-1 expression and kill extra chromosomal array containing worms (Seidel et al., 457 2011 Cortical flow was monitored in images of the cortical surface in embryos expressing myosin::GFP 472 obtained from adult hermaphrodites by dissection. Embryos were mounted followed by sealing with a 473 coverslip on double thick (1 mm) low percentage agarose (0.5%) pads to prevent compression that 474 biases the initial angle of furrow ingression (Figure 1 -Figure Supplement  Orienting embryos with their anterior end to the top: Acquired z-plane images were convolved with a 500 10-pixel Gaussian kernel to reduce noise. An optimal signal threshold that partitioned the embryo interior 501 from exterior was identified by finding a local minimum in the intensity histogram that produced a binary 502 mask with expected area (~120000±50000 pixel 2 ). The orientation of the AP axis was identified by fitting 503 an ellipse to the thresholded area in the middle plane of the z stack. The anterior side was identified by linear interpolation. In further iterations, ! and !" were refined for every embryo by minimizing the 544 residuals between its normalized ring size, , and the average dimensionless ring size, < > , 545 throughout the entire timecourse of cytokinesis, thus increasing the number of time points available for 546 fitting ! and !" (6-10 values per embryo). After refining time alignment and normalization for each 547 embryo, average dimensionless ring size was re-calculated and ! and !" were refined for each embryo 548 again. The refinement process was repeated until changes in average dimensionless ring size, < > 549 , were smaller than 0.001 on average (achieved within a few iterations). The collective fitting of all ! 550 and !" at every iteration was performed under restriction that the line fit through < > between 0.8 551 and 0.3 intercepted 0 at = 0 and 1 at = 1. This restriction ensured that ! and !" determined from fits 552 of individual embryos to the average ring size would be consistent with their original definition. The 553 dimensional ring kinetics, < > , can be recovered using the following equation 554 where < !"# >= 14.7 ± 0.7 and < !" >= 200 ± 30 are average embryo radius and time of 555 cytokinesis accordingly. , , and averaged according to its position and time 578 579

Calculation of expected cortical surface flow profiles 580
To aid in the interpretation of experimental results, expected profiles for cortical surface movement were 581 calculated for defined patterns of cortical surface increase and plotted ( Figure 1B and Figure 1 -Figure  582 Supplement 5). The general form of surface movement velocity is given by the following equation 583 where is the amount surface gain and is the velocity of asymmetric ring movement, which could 584 be positive or negative, depending on whether the ring is moving towards or away from the surface. 585 From equation (12)  Cortical laser ablations, presented in Figure 2, were performed using a robotic laser microscope 594 system (RoboLase) (Botvinick & Berns, 2005). Embryos expressing myosin::GFP were mounted using 595 standard procedures. A cortical cut, approximately 10 µm long, was made on the anterior side of the 596 embryo when the ring was at ~50% closure (7µm radius). The cut was confirmed by comparison of 597 cortical fluorescence images before and after the cut and was considered successful if the foci moved 598 away from the cut area (~3.5µm distance), indicating cortical tension release. Contractile ring closure 599 rate was calculated by measuring the difference in ring sizes before and after the cut, assessed from two 600 4x2µm z-stacks acquired immediately before the cut and 13s later. Errors in measuring the radius at the 601 two timepoints were determined from the procedure used to fit the data to a circle and were propagated 602 to determine the errors in the constriction rate measurements for individual embryos; mean errors are 603 S.E.M. The cortical opening after ablation was approximately 35µm 2 ; this translates into an additional 604 reduction in ring radius by ~0.8µm, if the cortical surface tension dominates the ring closure rate. This 605 additional decrease in ring size within 13s should correspond to increase of the control rate (0.22µm/s) 606 by ~30% (0.06µm/s). The experiment was repeated 19 times for no cut condition, 14 times for parallel 607 cut, and 15 times for perpendicular cut. All imaging was performed over the course of 5 days. The 608 number of embryos was chosen to achieve sufficient accuracy in the determination of mean ring closure 609 rates to assess whether it was altered by the cuts. 610 611

Calculation of the surface area flowing into the division plane 612
We calculated the amount of surface area flowing into the division plane from flow measurements 613 made 7 µm away from the position of the furrow on the anterior and posterior sides (as illustrated in 614 Figure 3A). The rate of the surface flow is 615 where ! is -7 µm and 7 µm for the rate of flow from the anterior or the posterior sides, respectively. The 616 total amount of surface area that entered the division plane from any time ! to is obtained by 617 integrating equation (13) over time 618 The increase in area of the division plane was calculated as following 619 In Figure 3A we used ! = −0.2. The extra cortex delivered into the ring can be inferred from the 620 difference between the surface area entering the division plane and the area of the division plane 621  Figure 3 -Figure Supplement 4). The intensity profiles in z from 13 embryos were fitted to an 651 exponential using the same characteristic attenuation depth for all embryos 652 which yielded a characteristic depth of attenuation, !"" , of 15 µm.

Derivation of the Cortical Flow Feedback (CoFFee) model for cytokinesis 677
The CoFFee model formalizes the following conceptual view of cytokinesis: Active RhoA recruits 678 contractile ring components to the equatorial cortex, where myosin engages with actin to exert an 679 isotropic force that compresses the underlying cortex. Polar relaxation releases tension in the direction 680 perpendicular to the ring, but not in the around-the-ring direction, generating anisotropic boundary 681 conditions that cause the system to exhibit distinct behavior in the two directions. Disassembly in the 682 around-the-ring direction reduces ring components in proportion to the reduction in length, and does not 683 alter the per unit length amount of myosin. Thus, changes in myosin levels are determined solely by ring-684 directed cortical flow along the direction perpendicular to the ring, which can be solved as a one-685 dimensional problem. We assume that the cortical compression rate (between and + ) is 686 proportional to local myosin concentration, ( , ), which exerts stress onto the actin network resulting in 687 where is the cortical strain (i.e. change in length of cortical surface per unit length) and is a 688 proportionality constant that reflects the ability of the cortex to be compressed by ring myosin. The 689 velocity of cortical surface movement is obtained from the following relationship (see also equation (12)). 690 The conservation of mass for myosin flow results in the following 691 If we integrate equation (20) over x on (-w, w) domain we obtain 692 where !"#$ ≔ , ! !! is the total per unit length amount of engaged ring myosin, 2 is the 693 width of the contractile ring/active zone where myosin is engaged and compressing cortex and !"#$ ≔ 694 ( , ) is the concentration of myosin on the cortex delivered into the contractile ring. The velocity of 695 ring-directed cortical flow is 696 The one half is included to account for the fact that flow comes in from both sides. The solution of 697 equation (21) is 698 where we define the characteristic time of myosin accumulation, , as ! !! !"#$ . Note that the total amount 699 of myosin in the ring will be the amount of engaged myosin plus an added baseline that would include 700 any myosin not involved in compression (see equation (5)). We assume the rate of ring shrinkage is 701 proportional to the amount of ring myosin, as observed in our data, 702 where is a proportionality coefficient that reflects the ability of the ring to be constricted by ring myosin. 703 Using equations (23) and (24), we obtain the dynamics of contractile ring size over time 704 where !"! is the dimensionless characteristic size of the ring; essentially the radius at minus infinity if the 705 same exponential process controlling contractile ring assembly extended back in time infinitely. Instead, 706 in vivo cytokinesis initiates when spindle-based signaling activates RhoA on the equatorial cortex leading 707 to the abrupt recruitment of contractile ring components. If the time frame of reference is chosen so that 708 = 0 is cytokinesis onset immediately following the initial patterning of the cortex by RhoA, ! !"#$ is the 709 amount of ring myosin immediately following this event and the initial size of the ring is 710 To compare our model with data we use the time frame of reference where = 0 is the point of 50% 711 closure (i.e. = 0 = ! ! ). In this reference, ! !"#$ = !" ! !"! !" , and by defining dimensionless velocity as 712 : = , we obtain equations (4-8). Note that equation (4) can be rewritten in the following way 713