Big Cells Cleave as Fast as Small Ones: the Physics of Cytokinesis

In an important article, Turlier et al. (1) present a predictive physical theory of the dynamics of cytokinesis, the cleaving in two of a eukaryotic cell.

Cells divide through a sequence of delicately coordinated processes. The last stage, once genetic material has been partitioned into two sister chromatids, consists of the radial shrinkage of an equatorial band, called the cleavage furrow, defined by enhanced contractile activity of myosin motors on actin filaments. Biologists now know in remarkable detail the vast number of proteins responsible for ensuring the fidelity of this crucial process, whose failure can lead to the wrong number of chromosomes in a daughter cell, with disastrous consequences. In addition to these key descriptive and biochemical facts, an understanding of the process based in the mechanics of the membrane, cytoskeleton, and attendant machinery in the cell has long been sought, and it seems clear that Turlier et al. (1) have found it.

The spectacularly large deformations suffered by a dividing cell have naturally attracted the attention of mechanicians. In pioneering but hard-to-repeat (2,3) experiments by Spek (4) nearly a hundred years ago, a steady flux of sodium ions was introduced into two antipodal points on a stabilized spherical oil-drop in water. The resulting localized lowering of surface tension due to soap formation at the poles produced dramatic Marangoni flows toward the equator. The flows, along with enhanced tension at the equator, gave rise to a contractile furrow and sometimes to total cleavage. Early qualitative studies like these work primarily with the membrane and not the cytoskeleton, and underline the central role of a maintained tension inhomogeneity, regardless of whether this arises through reduction at the poles or enhancement at the equator. Recent approaches, e.g., that of Zumdieck et al. (5), concentrate exclusively on the contractile ring; some, such as that of Mendes Pinto et al. (6), appear to manage apparently without invoking force balance for the system as a whole, which is a little puzzling. Even in a hydrodynamic framework, rather detailed models are available (7) with a full accounting of conservation laws and many possible interactions.

In view of this background of earlier work, what is special about the work of Turlier et al. (1) is its simplicity. It ignores many things: a detailed description of filaments in the contractile ring; the orientation of actin filaments; the elasticity of the cortex; viscous flow in the bulk of the cytoplasm; adhesion of cells to each other or to surfaces; friction between the membrane and the actin cortex; and spatial variation of actin concentration in the plane of the membrane.

Both driving force and relaxation mechanisms are argued to be situated primarily in the actomyosin cortex, described as a thin viscous fluid layer with isotropic active stresses, and a thickness determined by the kinetics of polymerization and depolymerization of actin filaments. The initiation of the contractile ring, triggered in the cell by a RhoA signal, is imposed as an equatorial band of heightened intensity of active stress. Dissipation in the cortex is shown to dominate; the cytoplasm enters essentially as a constraint of constant volume.

This minimal investment yields important testable predictions as well as explanations of existing observations, necessarily through a numerical solution of the coupled dynamics of membrane shape, cortex thickness, and force balance:

• 1.Tension is proportional to the cortex thickness, which in turn scales as the ratio of polymerization velocity to depolymerization rate (8); increasing the latter will thin the cortex and thus slow down constriction.
• 2.The time to complete cytokinesis is independent of initial cell size—as documented in recent experiments (9)—provided only that the contractile ring width is proportional to cell size.
• 3.Too small a contractility in the ring will mean that the effective membrane surface tension, contributed by the cortex, will arrest cytokinesis before it is complete.

Some points that come to mind:

• 1.It is intriguing that to obtain a duration independent of cell size, the width of the RhoA-GTP zone had to be proportional to the initial cell size. Given that this isochrony is vital to the synchronization of daughter-cell development during embryogenesis, such proportionality would appear to be a selected property, whose genetic or regulatory basis might be worth exploring.
• 2.The theoretical treatment was simplified by the assumption that most of the dissipation takes place in the cortex. Cytoplasmic/cortical dissipation, as remarked by the authors, is given by the cytoplasm/cortex viscosity ratio η_cytoplasm/η_actomyosin times the ratio R/e₀ of cell size/cortex thickness. Because the cortex is thin, it would naïvely seem natural to work in the limit R ≫ e₀, suggesting the dominance of cytoplasmic dissipation, which would be wrong because it just happens that η_cytoplasm/η_actomyosin is enormous. This pitfall for the theoretician is a nice reminder that biological systems are finite.
• 3.It is remarkable and reassuring that two of the central ingredients of the theory of cytokinesis—a stress proportional to the alignment tensor and a timescale independent of spatial extent—are, respectively, the defining property of an active fluid and one of the most robust outcomes of the theory of the hydrodynamics of self-propelled suspensions (10,11). The timescale given by viscosity/active stress determines the growth rate of the generic instability of oriented active fluids (10–12).

This article offers an elegant example of the insights into biology that can follow from simple questions rooted in fundamental physics. Like all good theory, its ultimate correctness will be decided by experiment.