Physical mechanisms of ESCRT-III–driven cell division

Significance Cell division is an essential requirement for life. Division requires mechanical forces, often exerted by protein assemblies from the cell interior, that split a single cell into two. Using coarse-grained computer simulations and live cell imaging we define a distinct cell division mechanism—based on the forces generated by the supercoiling of an elastic filament as it disassembles. Our analysis suggests that such a mechanism could explain ESCRT-III–dependent division in Sulfolobus cells, based on the similarity of the dynamics of division obtained in simulations to those observed using live cell imaging. In this way our study furthers our understanding of the physical mechanisms used to reshape cells across evolution and identifies additional design principles for a minimal division machinery.

on its own, and then the filament in contact with the vesicle, where the target radius of the filament equals that of the cell. At setup are available on the following link * . 55 Fig. S2. Filament coiled according to randomised protocol displays more perversions (blue curve) than the one coiled according to the sequential protocol (green curve) with the same rate. Instantaneous protocol displays the largest number of perversions. This data has been collected on a filament attached to a membrane without disassembly, as this scenario is more relevant for the division mechanism than the filament in solution (which displays the same trend). The amount of the curvature change is Rtarget/R cell = 5.0%. 56 To understand how the rate of curvature change vcurv influences the division probability, we systematically study its influence 57 on the filament geometry. To this end, the simulations for a constriction to 5% of the original cell radius were repeated 58 without disassembling the filament for the instantaneous, sequential, and randomised curvature change protocol. The number 59 of perversions (4, 5) that occurred in the equilibrated filament were measured as a function of vcurv. As shown in Fig. S2, for 60 sequential constriction, a faster constriction leads to the formation of more perversions, whereas for randomised constriction 61 the number of perversions generated seems to be rate-independent. Instantaneous constriction generates the most perversions.

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The randomised protocol distributes the perversions more evenly along the filament, hence it also leads to a higher 63 constriction of the filament overall. As visible from Fig S3a, the filament's overall radius of gyration is smaller when the 64 filament is constricted to the same degree in the randomised than in the sequential protocol. This higher overall supercoiling of Fig. S3. The filament coiled according to randomised protocol (blue curves) has a smaller radius of gyration (top panel), higher tension (middle panel), and creates higher membrane curvature (bottom panel) than the one coiled according to the sequential protocol (green curves). In all the panels, the membrane was present and the filament was disassembled following the constriction. Orange and purple vertical lines indicate the start and the end of the filament disassembly, respectively. Speed of constriction: 10 2 vcurv = 6.7/τ ; speed of disassembly: 10 2 v dis = 2.7/τ ; and curvature reduction: Rtarget/R cell = 5.5%. Fig. S4. Midcell diameter evolution after constricting the filament instantaneously by Rtarget/R cell = 5.5% and varying the disassembly rate v dis . Each curve is the mean of 10 random seeds. Note that disassembly must be slow (10 2 v dis < 6.67τ ) for divisions to complete.

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In the manuscript we measure how the division probability and evenness depend on how much we change the filament curvature 80 Rtarget/R cell for the instantaneous protocol (Fig.2, Fig.3). These measurements explore the role of the force that the filament 81 exerts -showing that division fails if this force is too large or too small. To test if these conclusions depend on the exact 82 protocol of force application, we have repeated the same measurements for the sequential and randomised protocol. As shown 83 in Fig. S5 (continuous lines), division fails for too high or too low filament curvature changes. Interestingly, the region for 84 the successful division is broader for the randomised curvature change. Please note that, because of this, to see the full 85 non-monotonic behaviour for the randomised curvature we would need to go to very low Rtarget values, which is prohibitive in 86 our model due to filament volume exclusion.

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Cells appear to divide less evenly when using the sequential protocol than the randomised or instantaneous protocol, which 88 both achieve similar amounts of division evenness (dashed lines). However, the randomised protocol achieves this high symmetry 89 of division for a broader range of curvature changes.  Fig. 2. The dashed lines compare the division evenness between the three protocols. The rate of curvature change was set to the value that matched best with the normalized cell diameter evolution curve (Fig.6): 10 2 vcurv = 5/τ for the sequential and 10 2 vcurv = 16/τ for the randomized protocol; the rate of disassembly is 10 2 v dis = 2.7/τ .

The role of filament tension in division 97
For simplicity, in the manuscript we kept the filament bond strength at k = 600kBT . Here we varied the filament bond strength 98 to probe its effect on division probability when using the randomised protocol. As shown in Fig. S7, our simulations predict 99 that if the tension in the filament is decreased, division becomes less reliable. In experiments this can be tested by deleting one 100 of the two proteins that make up the contractile CdvB1/2 filament. Deleting either one of the proteins would yield a filament 101 that has lower tension. Indeed, in experiments we find that the deletion of either B1 or B2 proteins produces less reliable or 102 slower division (6), in agreement with the predictions of our model. 103 Fig. S7. The average division probability vs the amount of tension in the filament, showing that below a certain tension threshold division fails. The simulations were carried out using the randomised curvature change protocol with a rate of curvature change: 10 2 vcurv = 16/τ ; rate of disassembly: 10 2 v dis = 2.7/τ ; and curvature reduction: Rtarget/R cell = 5.0%.

Thermodynamic work for different curvature change protocols
We find that consistently around 75% of the supplied energy to the filament is dissipated, but we do not observe a significant 118 difference in dissipation between different protocols (Fig. S8). However, there is a substantial difference in how the non-dissipated 119 portion of the energy is spent in different protocols. In the randomised curvature change protocol the highest proportion of the 120 non-dissipated energy supplied to the filament is transferred into membrane deformation, which is the productive work in our 121 case, while in the sequential protocol the highest proportion is transferred into filament detachment. This is consistent with 122 the resulting higher fidelity of division in the randomised protocol measured in our simulations.
123 Fig. S8. The average partitioning of the invested energy in the sequential and randomised protocol when constricting by the same amount and using the same constriction and disassembly rate. On average ∼ 75% of the energy invested in the curvature change is dissipated. In the randomised protocol more of the non-dissipated potion of energy is spent on productive work (membrane deformation) than in the sequential protocol. All the energies were computed directly from the relative potential energies by taking the difference between the final and initial energies. Rate of curvature change: 10 2 vcurv = 20/τ ; rate of disassembly: 10 2 v dis = 2.7/τ ; and curvature change Rtarget/R cell = 5.5%. 124 The simulation setup in the main paper does not account for the fact that cells are filled with cytoplasmic content. 20-30% of a 125 cell's volume is occupied by proteins (7). In order to investigate the presence of non-compressible cytoplasmic content on the 126 dynamics of cell division, we place volume-excluded particles inside the simulated cells. The volume exclusion is implemented 127 via a Lennard-Jones potential of =2 k B T , cut and shifted at the potential minimum.   Not enough tension: The filament target curvature radius is around 33% of the initial cell radius, and it does not have enough tension to deform the membrane. Blue curve -Successul constriction: With target curvature radius around 9.5% of the initial one, the filament here has the right amount of tension to drive considerable neck constriction. Orange curve -Too much tension: For target curvature radius around 7.5% of the initial one, the filament has too much tension and detaches from the membrane. b) Snapshot of the constriction scenario after relaxation. It is interesting to note the numerous perversions formed along the filament, also observed in the molecular dynamics simulations (see main text).

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The microscopy field of view covered an area of ∼ 150µm × 150µm, displaying multiple cells at once. We used a machine 175 learning algorithm to detect any dividing cells and cropped them from the master image stack (using ImageJ) to a more 176 appropriate scale as can be seen in Fig. S11. can be used to describe the measurement of the width of an imaged object when the edges of the image are not sharp (9). The 181 intensity profile of the midcell diameter can be extracted using ImageJ's line tool as shown in Fig. S12a. 182 We then fitted the intensity profiles via a Gaussian, and calculated the FWHM of the fit via where σG is the standard deviation of the fitted Gaussian. If two peaks were present, two Gaussians were fitted to the profile instead of one. These fits were background-subtracted 188 such that their offsets were zero (Fig. S12b, middle panel), and then combined to create a unified curve (Fig. S12b, right   189 panel). This process was repeated for each frame of the data where the profile was poorly represented by a single Gaussian.

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The combined double-Gaussian fit does not have a single well-defined σG. In this case, the FWHM was measured as the width 191 of the combined fit profile at half the intensity of the higher-amplitude peak.
192 11. Midcell diameter evolution rescaling 193 The shape of the curves describing the midcell diameter of the dividing cells over time resemble a sigmoidal function, but can 194 be quite asymmetric if the filament curvature changes quickly. We hence fitted them using this generalised logistic function 195 that allows the curves to be both asymmetric and symmetric: 196 since we know they all start at the same original diameter. 200 We then fit the diameter over time (for all random seeds) using this function and extract the fitting parameters to 201 scale the curve in the x and y direction. Scaling in the y-direction is simple, as we only need to subtract the minimum 202 asymptote value from the data points and then divide them by the difference between the upper and lower asymptote: Movie S1. Example of a successful cell division simulation following the instantaneous constriction protocol.