Asymmetric Flows in the Intercellular Membrane during Cytokinesis Vidya V. Menon, S.S. Soumya, Amal Agarwal, Sundar R. Naganathan, Mandar M. Inamdar, Anirban Sain  Biophysical Journal  Volume 113, Issue 12, Pages 2787-2795 (December 2017) DOI: 10.1016/j.bpj.2017.10.011 Copyright © 2017 Terms and Conditions

Figure 1 Schematic diagram of the asymmetric septum closure based on (22,28). (a) Given here is the longitudinal view and (b) x-y view of the shrinking hole located off-center. The r0 is the initial furrow radius, and d is the distance between the initial center of the circular outline of furrow and center of the contractile ring. Biophysical Journal 2017 113, 2787-2795DOI: (10.1016/j.bpj.2017.10.011) Copyright © 2017 Terms and Conditions

Figure 2 The evolution of the hole radius and the corresponding velocity pattern for symmetric closure. (a) Here, we compare the analytic and numerical curves (both using the same parameters and v0 = 0.084 μm/s) with the experimental data taken from (25) (same as that in (22)). The initial ring radius in the simulation was set at r = 13 μm, whereas the outer radius was at r0 = 14 μm, same as in the experimental data. Therefore, the starting point of the simulation r/r0 = 0.93 was matched with the experiment by shifting the simulation data by 20 s (see Numerical Solution and Methods for details). Here, (b) and (c) shows the evolution of the inhomogeneous velocity pattern, where (b) is for intermediate time when the closure speed—slope of the curves in (a)—is high and (c) is for the terminal slow phase. Biophysical Journal 2017 113, 2787-2795DOI: (10.1016/j.bpj.2017.10.011) Copyright © 2017 Terms and Conditions

Figure 3 Various initial and boundary conditions are categorized according to the symmetry of the incoming flow at the outer boundary and the initial location of the hole. The resulting flow structure is azimuthally symmetric for the left panel (a–d), and asymmetric for the right panel (e–i). However, all the figures in the right panel, except (i), have one mirror symmetry axis, shown by the dotted line. Rotation of the ring can be observed for (c) and (d) and (i). In (c) and (d), it is trivially due to the nonzero vθ at the boundary, whereas in (i) it is due to lack of any mirror symmetry. Biophysical Journal 2017 113, 2787-2795DOI: (10.1016/j.bpj.2017.10.011) Copyright © 2017 Terms and Conditions

Figure 4 Off-center hole remains off-center for v0 = 0. A uniform radial inflow is imposed throughout the outer boundary of an off-center hole. Here, (a) shows the initial off-center hole and (d) is the final position of the closed hole, which is still off-center. Here, (b) and (c) are the velocity plots at intermediate times during the ring contraction. The resulting flow in the interface has only a radial (vρ) component. The hole moves off-center and closes itself off-center under the strong action of the contractile ring. Because of one mirror symmetry, the inner boundary does not have any net rotation. Biophysical Journal 2017 113, 2787-2795DOI: (10.1016/j.bpj.2017.10.011) Copyright © 2017 Terms and Conditions

Figure 5 Comparison of the numerical data with experimental results. (a) Fluorescence images of asymmetric septum closure in C. elegans embryo on the septum (see Movie S1). (b) Numerical solution of the Eq. 4 showing time course of asymmetric septum closure with asymmetric radially inward velocity: v0 = 0.2 μm/s on the lower half, and v0 = 0.02 μm/s on the upper. The initial hole was also displaced off-center along the positive y axis. As a result of the flow the center of the hole moves further up in the beginning and later the hole closes due to strong line tension in the ring (see Movie S2). (c) The curve shows the time evolution of the radius of the contractile ring. The circle and the line indicates the experimental data and numerical results, respectively. (d) The graph shows the extent of asymmetry in ring closure with time, where d is the distance between the initial center of the circular outline of the furrow and the instantaneous center of the contractile ring and r0 = 14 μm is the initial furrow radius (see Movie S1). Biophysical Journal 2017 113, 2787-2795DOI: (10.1016/j.bpj.2017.10.011) Copyright © 2017 Terms and Conditions

Figure 6 Rotation of the central hole. Here, (a–c) show evolution of a maximally asymmetric flow where all mirror symmetries are violated by (i) choosing a nonzero radial inflow on the left half, and (ii) displacing the initial hole along the y axis. Nonzero vθ is generated in the interior, although the input at the boundary has vθ = 0. Absence of any mirror symmetry leads to net rotation of the hole in the counterclockwise direction. We computed the sum of vθ along the inner boundary to check this quantitatively (see Fig. S1). (d) Shown here is a display of rotation that is trivially generated by choosing uniform nonzero vθ at the outer boundary ((c and d) in Fig. 3). Biophysical Journal 2017 113, 2787-2795DOI: (10.1016/j.bpj.2017.10.011) Copyright © 2017 Terms and Conditions

Figure 7 The closure rate is robust with respect to various initial and boundary conditions listed in Fig. 3. Data for radius versus time, from all our simulations (open symbols), are compared with our experimental data (solid circle). The various initial and boundary conditions for the simulation runs are listed next to the respective open symbols. The deviation band results from averaging the experimental data over six independent movies. The band ends where the hole size becomes too small for a reliable size estimation. Biophysical Journal 2017 113, 2787-2795DOI: (10.1016/j.bpj.2017.10.011) Copyright © 2017 Terms and Conditions