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Cross-linkers both drive and brake cytoskeletal remodeling and furrowing in cytokinesis

Cell shape changes such as cytokinesis are driven by the actomyosin contractile cytoskeleton. The molecular rearrangements that bring about contractility in nonmuscle cells are currently debated. Specifically, both filament sliding by myosin motors, as well as cytoskeletal cross-linking by myosins and nonmotor cross-linkers, are thought to promote contractility. Here we examined how the abundance of motor and nonmotor cross-linkers affects the speed of cytokinetic furrowing. We built a minimal model to simulate contractile dynamics in the Caenorhabditis elegans zygote cytokinetic ring. This model predicted that intermediate levels of nonmotor cross-linkers are ideal for contractility; in vivo, intermediate levels of the scaffold protein anillin allowed maximal contraction speed. Our model also demonstrated a nonlinear relationship between the abundance of motor ensembles and contraction speed. In vivo, thorough depletion of nonmuscle myosin II delayed furrow initiation, slowed F-actin alignment, and reduced maximum contraction speed, but partial depletion allowed faster-than-expected kinetics. Thus, cytokinetic ring closure is promoted by moderate levels of both motor and nonmotor cross-linkers but attenuated by an over-abundance of motor and nonmotor cross-linkers. Together, our findings extend the growing appreciation for the roles of cross-linkers in cytokinesis and reveal that they not only drive but also brake cytoskeletal remodeling.


Description of modeled components
Each fiber is modeled as a set of equidistant points, subjected to bending elasticity determined by the persistence length of F-actin (~15 µm), resulting in a bending rigidity of 0.06 pN· µm 2 . Fibers are semi-flexible, with degrees of freedom representing position, orientation, and bending. Twist is ignored in our 2D simulation space, and fibers are non-extensible and noncompressible. The number of model-points used to describe a fiber is determined by the segmentation parameter and the length of the fiber. It is automatically adjusted to ensure that the actual segmentation is as close as possible to the segmentation parameter. The simulation is twodimensional and steric interactions between fibers are ignored. Fibers grow during the first ~30% of closure (up to about 1.4 µm), and then shrink to ~1.3 µm. While contraction could create high density within the ring, many fibers become disconnected from the ring as it constricts. Fiber density only increases 20% between when the ring is 0% closed to when it is ~90% closed.
Simulations are modeled to occur in a low Reynold's Number environment, such that inertia can be ignored. While Brownian motion contributes to diffusion, the evolution of the system is mostly determined by the interactions between fibers created by crosslinkers and motors.
Motors and crosslinkers are modeled as containing two functional units, as was done previously (Ennomani et al., 2016). The motor complex, which represents N individual non-muscle myosin heads and tails, has a stiffness of 100xN pN/µm, resulting for N=14 in a total stiffness of 1400 pN/ µm. With only two fiber-binding subunits, motors and crosslinkers can bridge at most two fibers. When crosslinkers bridge fibers, they are represented by a Hookean spring of stiffness of 100 pN/µm; motors are represented by a spring with a stiffness of 1400 pN/µm. Motors are modeled to end-track a shrinking fiber, as done previously (Mendes Pinto et al., 2012;Oelz et al., 2015). At the microscopic scale, end-tracking is determined by motor (NMM-II) binding-and unbinding and fiber disassembly, but the associated rates are not known for the cytokinetic ring.
We thus used a simpler model of end-tracking by assuming that motors do not detach from shrinking fiber ends. This corresponds to a regime in which re-binding is always productive, possibly because it is much faster than disassembly.
Motor and crosslinker turnover: Motors and crosslinkers, as modeled are stably bound with ~10% of their respective populations becoming unbound at one end at any time point, but none being fully unbound. Crosslinkers are removed from the simulation at a variable rate such that roughly 88% of crosslinkers remain at 25% simulation time, and ~55% remain by the completion of the simulation. The slower initial loss of crosslinkers, and quicker secondary loss are slightly slower than published measurements (Wu and Pollard, 2005), in order to maintain ring integrity.

CytoSim configuration file
The following is the complete configuration file sufficient to generate our simulations in CytoSim (version PI). On the left are CytoSim commands; on the right are comments and annotations (inclusion of "%" enables the program to ignore text immediately following this notation).  (Thoresen et al., 2011;Oelz et al., 2015) hold_growing_end = 0 % whether motor hands persistently associate with the newest region of growing fibers; % binary 1 = true, 0 = false hold_shrinking_end = 1 % (Mendes Pinto et al., 2012;Oelz et al., 2015); binary 1 = true, 0 = false display = (size=4; color=0xFF000005;) % display specifications } set couple myosin % defines bipolar motor ensembles { hand1 = myoII % subunit attached on first end of ensemble hand2 = myoII % subunit attached on second end of ensemble stiffness = 1400 % stiffness of spring when linking two fialments in pN/microns; % Empirical and high to reflect presumed high stiffness of many bundled NMM-II tails fast_diffusion = 1 % we assume that diffusion of unbound molecules is so fast that the % distribution of unbound motors remains uniform activity = crosslink % Denotes that couples cannot bind the same fiber with both hands. length = 0.32 % resting length of spring, in microns, when linking two fibers; (Shutova et al., 2014) } set hand statlinkh % defines non-motor crosslinker "hands" (Ennomani et al., 2016) {