Cytokinesis is the fundamental and ancient cellular process by which one cell physically divides into two. Cytokinesis in animal and fungal cells is achieved by contraction of an actomyosin cytoskeletal ring assembled in the cell cortex, typically at the cell equator. Cytokinesis is essential for the development of fertilized eggs into multicellular organisms and for homeostatic replenishment of cells. Correct execution of cytokinesis is also necessary for genome stability and the evasion of diseases including cancer. Cytokinesis has fascinated scientists for well over a century, but its speed and dynamics make experiments challenging to perform and interpret. The presence of redundant mechanisms is also a challenge to understand cytokinesis, leaving many fundamental questions unresolved. For example, how does a disordered cytoskeletal network transform into a coherent ring? What are the long-distance effects of localized contractility? Here, we provide a general introduction to ‘modeling for biologists’, and review how agent-based modeling and continuum mechanics modeling have helped to address these questions.

Cytokinesis is regulated spatially and temporally to accomplish the partitioning of the cytoplasm and segregated genome into two daughter cells (Rappaport, 1996). In animal cells, formation of the actomyosin cytokinetic ring at the division plane (or cell equator) is elicited through activation of the small GTPase RhoA by microtubule-borne spindle-based signals in anaphase (reviewed by Mogilner et al., 2016). Active (GTP-bound) RhoA, elicits F-actin nucleation, activation and filament formation of non-muscle myosin II (NMM-II) and recruitment of scaffold proteins. The resulting ensemble of the cytokinetic ring encircles the cell equator in a band or cord and constricts, drawing the associated plasma membrane into a furrow and partitioning the daughter cell contents (Green et al., 2012). The core actomyosin machinery of the cytokinetic ring is shared among animal and fungal cells, and is a specialization of the contractile actomyosin cytoskeleton that governs cell shape in these systems. Thus, cytokinesis can serve as a paradigm to understand diverse behaviors of cellular motility.

Glossary

Active gel theory: the continuum mechanics theory depicting a viscoelastic material, with polar filaments that dynamically change through energy consumption. For example, the actin cortex is rearranged through myosin ATP hydrolysis.

Agent-based modeling: modeling in which the dynamics of individual players are explicitly simulated and tracked. Stochasticity is calculated for and applied to each individual element modeled.

Analytic solution: the solution to an equation given as a specific formula rather than by a numerical simulation.

Continuum mechanics modeling: depicts a potentially complex heterogeneous structure as a continuous material rather than a collection of discrete particles, by simplifying their length- and time-scales. These models are limited to phenomena that occur at dimensions larger than the simplified length and time-scales, typically depicting structures of one micron or larger; e.g. the entire cell cortex can be described by continuum mechanics modeling as a thin elastic shell.

Elastic: describes a material that recovers to its starting configuration after any applied stress is released. Elastic materials are characterized by an elastic modulus, a quantity that represents a measurement of an object's resistance to an applied stress.

Hybrid (or multi-scale) model: incorporates aspects of both agent-based and continuum approximations of different cellular processes into one unified model.

Langevin-type equation: force–balance equation that describes the change of one or more variables over time. Inherent to the equation is its stochasticity, stemming from collisions among particles.

Mathematical modeling: used here as a general term to refer collectively to diverse theoretical approaches, in which biological, biochemical and biophysical processes are described with mathematical equations.

Mean field: the ensemble average.

Mesoscale: on the threshold of currently resolvable length (∼100 nm), and occurring at durations that are between those of typical single-molecule and cell biological studies (hundreds of milliseconds).

Model parameter: components of an equation that represent inherent properties of the subject of the model. For example, the binding range of a crosslinker in an agent-based model or the elastic modulus of a material in a continuum mechanics model.

Nematic: a mesomorphic state, in which the linear orientation of molecules results in anisotropic properties. Polar molecules, such as F-actin, are aligned in parallel or anti-parallel.

Periodic boundary conditions: the property of a modeled domain in which an edge connects to another (opposite) edge, as do the west and east margins of a two-dimensional map of the surface of the Earth. The periodic boundary condition represents continuity, wrapping or infinite dimensions, such that there are no free edges. For example, the cortex has no edges, therefore models of small patches of it employ periodic boundary conditions to simulate an effectively infinite domain.

Reynolds number: dimensionless ratio of inertial forces to viscous forces in a fluid medium. Biological molecules and polymers experience the cytoplasm as a low Reynolds number milieu due to the high viscosity of cytoplasm and the small length scales of the cell.

Steady state: a dynamic process at its equilibrium; i.e. over time, the dynamics (or average thermal fluctuations) are either unchanged or very small compared with other processes. For example, cytosolic forms of the Rho family of small GTPases diffuse very quickly, so that the spatial distribution of these proteins can be assumed to be uniform; the diffusion process is then said to be at steady state.

Stochasticity: randomness generated by an underlying probability distribution.

Strain: dimensionless quantity of the deformation of a material relative to a reference measurement (i.e. that of the same material at resting state).

Stress: force per unit area in 3D.

Surface tension: describes the tendency of a fluid to minimize its surface-area-to-volume ratio as force per unit length.

Term: the part of an equation that describes a process or an effect, i.e. diffusion term, which describes the movement of a chemical species from an area of higher to lower concentration.

Viscoelastic: a material that exhibits both viscous and elastic behavior, depending on the duration over which a force is applied. The viscoelastic modulus measures the ratio of stress to strain after applying oscillatory stresses.

Viscous: describes a material that remains deformed after an applied stress is released. Viscosity is the resistance to flow. The viscosity coefficient relates the velocity gradient of a fluid to its viscous stress. The viscous drag is the resistive force caused by viscosity. The irreversible process by which kinetic energy is transformed into heat is known as viscous dissipation.

Mathematical modeling (see Glossary), combined with biological experimentation (i.e. ‘wet lab’ approaches including microscopy, genetics, biochemistry and biophysics), has significantly advanced our understanding of cytokinesis. Herein, we use the word ‘modeling’ to collectively refer to diverse theoretical approaches, in which biological, biochemical and biophysical processes are described with mathematical equations. These approaches, often historically rooted in, and motivated by, problems in physics and chemistry, include continuum mechanics modeling and agent-based modeling (see Glossary). The following references can serve as a starting point for foundational reading, i.e. Frenkel and Smit, 2002; Goldstein et al., 2001; Lai et al., 2009; Landau and Binder, 2000; Landau and Lifshitz, 1960. Depending on the length- and time-scales of the subject of the model (Fig. 1), a solution may be written explicitly as a function of time and space (analytic solution) or approximated by using numerical algorithms (computational solution).

Fig. 1.

The machinery of cytokinesis has beenstudied and modeled at both ends of the spatial and temporal scales. Several key aspects of cytokinesis have been studied at the nanoscale, including (a) the interactions between plasma membrane and proteins (black arrow), (b) the cytoskeletal architecture, such as filament bundle spacing and; (c) the speed of motor proteins. Furthermore, the cytokinetic ring (red) is also studied at the microscale, for example, (d) testing the response of the cell to mechanical perturbations, such as laser cutting, and (e) measuring furrow ingression speed. Indicated below are length (in meters), duration (in seconds), modeling approach, typical number of elements (components) per modeling approach, and stochasticity.

Fig. 1.

The machinery of cytokinesis has beenstudied and modeled at both ends of the spatial and temporal scales. Several key aspects of cytokinesis have been studied at the nanoscale, including (a) the interactions between plasma membrane and proteins (black arrow), (b) the cytoskeletal architecture, such as filament bundle spacing and; (c) the speed of motor proteins. Furthermore, the cytokinetic ring (red) is also studied at the microscale, for example, (d) testing the response of the cell to mechanical perturbations, such as laser cutting, and (e) measuring furrow ingression speed. Indicated below are length (in meters), duration (in seconds), modeling approach, typical number of elements (components) per modeling approach, and stochasticity.

Mathematical models make predictions by recapitulating biological observations on the basis of proposed physical, chemical and structural properties of the proteins, membrane and other components that underlie cellular processes. Modeling has been applied to understand the positioning (Mogilner et al., 2016), assembly, contraction and disassembly of the contractile ring during cytokinesis (see below for non-exhaustive referencing). Modeling – constrained by available experimental data – has also been used to establish the physical plausibility of a hypothesis. Even when ideas derived from modeling are currently inaccessible by biological experimentation, such as the dynamic reorientation of cytoskeletal filaments within diffraction-limited features (Ennomani et al., 2016; Zumdieck et al., 2007), they guide biological experimentation. In the most powerful application of modeling, predictions from a model are experimentally tested and the results are iteratively used to modify model parameters (see Glossary) or evaluate the feasibility of the model.

In this Review, we first discuss what biologists look for in models and what modelers look for in data obtained from biological experiments. Next, we consider how to choose the best modeling approach dependent on scales of length and time (Fig. 1). Then we consider in more detail the key criteria for choosing either agent-based or continuum-based modeling methods. Finally, we use the questions of cytoskeletal organization and long-range effects of local force generation as examples of how modeling and biological measurements were used to collaborate to advance our mechanistic understanding of cytokinesis.

In setting out to understand a model, a biologist is likely to consider the reductionist nature, robustness and relevance of the model. A reductionist approach to modeling focuses only on elements that are thought to be essential for the simulated phenomena. For example, models focused on dynamics of global cell shape during cytokinesis might focus on terms (see Glossary), such as membrane tension and cortical stiffness, while reducing contractility to minimal terms. Simplifying models in this way tends to mean that they are more phenomenological, contain as few terms as possible – such that there is no 1:1 correspondence between biological component and model parameter – and are more amenable to analytical solutions. For example, measurements of cortical mechanics made by micropipette aspiration are interpreted by a simple model derived from Laplace's law, which relates the pressure change across an interface (the cell membrane) to surface tension (see Glossary) and curvature of the interface. Since the pressure inside the micropipette is known and the radius of the cell deformation is measured, cortical tension can then be computed from the equation for Laplace's law (Hochmuth, 2000; Tinevez et al., 2009). The contributions of contractility, cytoskeletal-membrane linkage and membrane composition are neither measured nor taken into account but rather summed up as ‘cortical tension’ (Evans and Robinson, 2018; Hochmuth, 2000). The approach has been extended to include cytoskeletal elasticity so that the cell's elastic modulus of the cell (see ‘Elastic’ in Glossary) can be measured (Hochmuth, 2000).

Models should be robust to perturbations, such that a variation in the amount of a component or force should only quantitatively alter the output of the model. Alterations should allow the output to, at least qualitatively, resemble the effects of the corresponding biological perturbations, and to remain within biologically meaningful ranges and physically reasonable limits. For example, a robust model of molecular motors should result in filament sliding within a milieu demonstrating a viscosity that is relevant to both the cellular and the in vitro settings.

A biologist will also consider the relevance of a model regarding a variation of cell shapes and sizes. The applicability of a single model to several cell types is likely to be limited to cell-intrinsic components. For example, the molecular make-up and material properties of the actin cytoskeleton are likely to be shared among animal cell types. Non-autonomous properties of cells, such as cell wall, and intercellular and substrate adhesion impact the mechanical and molecular requirements of cytokinesis (Bourdages et al., 2014; Founounou et al., 2013; Guillot and Lecuit, 2013; Herszterg et al., 2014; Morais-de-Sa and Sunkel, 2013; Pinheiro et al., 2017). The influence of such cell type-specific effects can be tested by including specific considerations to the model, as was done when investigating the effect of adhesion and traction-mediated protrusive forces on furrow ingression in Dictyostelium cells (Poirier et al., 2012).

Depending on the availability of biological, biochemical and biophysical measurements, the relevance of a model may be assessed on the basis of agreement between these measured values and the underlying parameters of the model. In some cases, actual values of model parameters vary greatly between cell types or with the method of measurement, even for high-accuracy measurements. For example, experimental measurements of the cytoplasmic viscosity coefficient vary from 3×10−3 Pa s (Mastro et al., 1984) to 2×10−1 Pa s (Daniels et al., 2006; Kreis et al., 1982). Such disparity might reflect the different timescales at which the viscosity measurements were performed and/or the true difference in the cytosols of different species. Furthermore, the viscosity a particle experiences also depends on its size and shape, i.e. a myosin filament experiences a higher viscosity than a GFP protein (Kalwarczyk et al., 2011). In addition, the volume of two popular model cell types – fission yeast and the Xenopus zygote – differs 100,000-fold (Danilchik, 2011; Heald and Gibeaux, 2018; Zakhartsev and Reuss, 2018). Variation exists also at the molecular level, with the number of motor subunits in non-muscle myosin II (NMM-II) ensembles ranging from eight in fission yeast to ≤30 in mammalian cells (Niederman and Pollard, 1975; Sinard et al., 1989; Verkhovsky, 1995; Laplante et al., 2016; Wu and Pollard, 2005). Thus, there is merit to ensure that biological models are robust, such that a variation of some biophysical parameters still yields similar observable behaviors and emergent properties.

From the perspective of a modeler, several features are desirable in biological measurements. It goes without saying that accuracy and precision are ideal, but it is important to keep in mind that biological measurements themselves are likely to perturb the native state of the system. For instance, protein abundance data on the basis of fluorescence intensity calculations are confounded by the incomplete quantum efficiency of digital cameras and imperfect behavior of fluorescent probes (Heppert et al., 2018; Ulbrich and Isacoff, 2007). The biophysical properties of molecular motors are often determined for single motor domains, whereas many form ensembles in vivo (see Guo and Guilford, 2006; Stam et al., 2015 for examples).

In addition, measuring biophysical parameters, e.g. viscoelastic moduli (see ‘viscoelastic’ in Glossary), requires calculations that are based the assumption that biological materials, including the cytoskeleton and the extracellular matrix, respond to a given deformation in a known or predictable fashion. For example, to calculate the traction force exerted by a cell from the displacement of beads embedded in a flexible substrate, one commonly employed model assumes that the thickness of the substrate is semi-infinite (Dembo et al., 1996; Dembo and Wang, 1999), whereas another model takes into account the finite dimensions of the substrate (del Alamo et al., 2007) in order to more accurately account for the underlying geometry. Micro-pipet aspiration is frequently used to measure the mechanical properties of various cellular components including the membrane-associated actomyosin cortical cytoskeleton (the ‘cortex’), but interpreting the results as a characterization of the membrane or the cortex, or both, requires distinct biophysical models for each object (Brugues et al., 2010; Dai et al., 1999; Hochmuth, 2000). Physical assumptions also affect the interpretation of biological measurements when modeling the mechanics of the entire cell, whether it is considered as a liquid drop with a surface tension (see Glossary), a liquid drop with a viscoelastic shell or as an elastic solid (Hochmuth, 2000; Yang et al., 2008). Modeling the cell as a liquid drop assumes not only that surface tension comprises both membrane tension and cortical tension (due to actomyosin contractility) but also that the cortical tension is uniform, whereas – in reality – it is heterogenous (Hochmuth, 2000; Mayer et al., 2010; Tinevez et al., 2009; Yang et al., 2008). This biophysical property is calculated from how the cortex recoils upon being cut with a laser; however, such calculations themselves are done using models that make assumptions. For example, these models describe the cell as a periodic thin film of contractile viscous (i.e. viscoelastic) material (see Glossary) but only take into account interactions and flows over a finite length (Mayer et al., 2010). Calculations used to determine forces from traction force microscopy, micro-pipet aspiration and whole-cell modeling all underscore the principle that measurements from biological experimentation are interpreted after having used a model based on assumptions.

Often, the measurements needed for a model are unknown in the specific system being modeled. When this is the case, one of several strategies is taken: (i) values are estimated from cells with similar cell biological or physical properties, such as species and size, respectively; (ii) the sensitivity of the model output to a model parameter range is tested over several orders of magnitude, which guides the prioritization of which parameters need to be precisely determined; or (iii) one parameter – which can be estimated with high confidence – is used to normalize the others so that only the ratios of model components are considered.

In summary, although precise measurements and well-informed estimates contribute to the construction of any mathematical model, the source and limitations of these measurements or estimates must be taken into account. Below, we introduce several classes of model and summarize how recent modeling approaches have contributed to our knowledge of cytokinesis.

Each modeling strategy has its own advantages and limitations. Two main factors that influence the choice of modeling strategy are the spatial scale and the duration of the process under study (Fig. 1). At one end of the biological spectrum is the dynamics of individual molecular events, which typically occur in milliseconds or less and across nanometers. To model these events, molecular dynamics modeling is conventionally applied. However, cytokinesis involves the collective behavior of thousands to hundreds-of-thousands of constituent molecules; over longer timescales (seconds to minutes) and length scales (hundreds of nanometers to microns), their dynamics can be averaged.

Broadly speaking, there are two approaches to model a cellular process, ‘bottom-up’ strategies that typically constitute agent-based models of a group of specific molecular players (see below and Box 1), or ‘top-down’ approaches, such as continuum models that depict the collective behavior of large molecular ensembles (see below and Box 2). An important distinction between bottom-up and top-down modeling strategies is how they account for and describe the stochastic fluctuations that are ubiquitous in biology; bottom-up approaches explicitly describe the effects of stochastic fluctuations, whereas most top-down approaches describe the average behavior of each element.

Box 1. Bottom-up approaches

A bottom-up approach relies on agent-based models that simulate the microscopic and elementary dynamics of individual molecules, and their physical and chemical interactions (e.g. positions, conformational changes, mechanical and chemical states, and their degree of change) (Frenkel and Smit, 2002; Landau and Binder, 2000). Whereas molecular dynamics simulations are used for one or a few molecules, agent-based modeling typically describes hundreds of interacting players. Their behaviors are simulated by using a high number of time steps to bridge their minuscule and rapid molecular events to the cellular phenomena that occur over longer periods and greater distances. Such simulations generate a macroscopic or mesoscopic picture of the complex phenomena that emerge from a molecular ensemble. Bottom-up approaches are conceptually straightforward, especially when input molecular interactions and behaviors are known. However, these molecular details are frequently unknown even for the actin cytoskeleton, one of the best-studied cellular machines. Depending on the dimensional and spatial constraints of the model and its subject, volumetric exclusions and steric interactions should also be considered. If not accounted for correctly, such interactions can result in physical entanglement or crowding, which can result in simulation artifacts. In some cases, such as when representing a 3D collection of objects by modeling a projection into 2D space, these constraints can be ignored to some extent or entirely (Belmonte et al., 2017; Nedelec and Foethke, 2007). Since agent-based modeling takes into account the physical properties and inherent stochasticity of objects, it can be used to depict small numbers of objects that may not behave according to the mean-field description. For larger collections of objects, each undergoing additional interaction that occurs at increased length- or timescales, a model can be simplified to mean-field approximations that are computationally less costly. In these cases, continuum modeling can prove more powerful (see main text for specific examples and associated references).

Box 2. Top-down approaches

Top-down approaches, such as continuum modeling, assume that the detailed individual behavior of molecules can be smoothed into an ensemble average (Goldstein et al., 2001; Lai et al., 2009; Landau and Lifshitz, 1960). Rather than describing each molecular structure and its dynamics as in agent-based modeling, the variables in a top-down approach are typically concentration fields or the densities of players. These concentration fields interact according to the principles of mechanics (e.g. forces and velocities) and chemistry (e.g. on- and off-rates), both of which reflect bulk behaviors at long timescales. Consequently, the spatial-temporal resolution of continuum models is much lower (in the order of microns and minutes) than that of molecular dynamics and agent-based models (which depict processes at nanometers and milliseconds scales). As such, the number of model components is greatly reduced compared to agent-based models and is more amenable to mathematical analysis. Because continuum models do not consider the interactions of individual molecules or particles, they typically do not encounter entanglement or dynamically arrested states. Furthermore, because a fundamental assumption of continuum modeling is that it represents the ensemble average, such top-down approaches are most suitable to study a mean-field system, wherein stochastic fluctuations introduce only minor modifications to the average behavior. By the same token, agent-based models – due to their discrete nature – are well-suited to describe the effects of stochastic fluctuations but are often restricted to small space and short timescales. (Please see main text for specific examples and associated references).

Accounting for stochastic fluctuations is very important when modeling a small number of molecules, as their behavior is subject to considerable noise and it cannot be assumed that they behave according to the ensemble average, i.e. mean field (see Glossary). The dynamic behavior of a system can be expected to occasionally – but significantly – deviate from mean-field behavior when there are <100 hundred players (Kampen, 2007). For many cellular events relevant to cytokinesis, the copy number of each given factor is in the range of 10–1000 (Wu and Pollard, 2005). Therefore, hybrid – or multiscale – models are currently being developed to describe the mesoscale (see Glossary) dynamics of cytokinesis. For instance, multiscale models that combine continuum mechanics modeling of the cytokinetic ring with agent-based modeling (see Glossary) of actin filaments (Biron et al., 2005; Oelz et al., 2015; Zumdieck et al., 2007) are being applied to address the mechanical and molecular origins of forces in cytokinesis. As outlined below, multiscale models are conceptually attractive and have offered unique insights but, owing to the inherently different strategies to include and illustrate stochasticity (see Glossary) in agent-based versus continuum models, theoretical strategies to incorporate the output of one as the input into the other are still being developed. For example, it has not yet been established how to use the explicit results of cytoskeletal alignment generated through agent-based modeling, such as by using the cytoskeleton simulation engine Cytosim (Nedelec and Foethke, 2007), as input for the degree of local order within nematic (see Glossary) active gels of continuum mechanics models (see below).

Agent-based models explicitly simulate and track the dynamics of individual players. For the cytokinetic ring, these players may include actin monomers or filaments, motor proteins and crosslinkers. Agent-based models define how each component moves and interacts stochastically with its environment according to a set of rules, such as the stiffness of a fiber or the force required to detach a protein from a binding partner. These rules are based either on biological, biochemical and biophysical measurements or on hypotheses. To model the dynamics of the cytokinetic ring, most models assume that forces are balanced at any given point in time, e.g. by implementing a Langevin-type equation (see Glossary). Cellular models lack acceleration, inertia and momentum, since small components (e.g. motor proteins or cytoskeletal filaments) experience the cytoplasm as having a relatively high viscosity, which translates into a low Reynolds number (see Glossary) regime. Although a viscous drag force is ubiquitous in the equations of several models, the inclusion of additional sources of force differs widely but often includes the activity of motor proteins and the resistance of filaments to bending. Agent-based models are often used to depict reaction kinetics and the polymer dynamics of cytoskeletal components (see for example, Mendes Pinto et al., 2012; Vavylonis et al., 2008). Agent-based models can depict cytoskeletal components as a series of repeating nodes or segments, in order to minimize computation time and, thus, facilitate investigation to a larger degree than such modeling would otherwise permit; for example, Cytosim depicts cytoskeletal filaments as a series of segments (Nedelec and Foethke, 2007). A major limitation of agent-based modeling of cytokinesis is the lack of detailed measurements of the abundance of cytokinetic ring components, their dynamics and interactions, which necessitates assumptions and the estimation of the model parameter values related to these characteristics. In addition, computation timescales with the number of individual components simulated, exerting practical limits on the duration and area or volume that can be simulated.

Despite these limitations, agent-based models have been successfully implemented to depict several aspects of cytokinesis. For example, agent-based modeling has been implemented in 2D domains that represent a patch of the cortex (Bidone et al., 2017), or 2D sections that represent either a cross-section, or the entirety, of a cytokinetic ring (Ennomani et al., 2016; Mendes Pinto et al., 2012). Other agent-based models depict 2D domains with periodic boundary conditions (representing continuity or wrapping; see Glossary), or explicitly 3D domains (Bidone et al., 2014). Below, we present findings that exemplify the use of agent-based modeling to study cytoskeletal dynamics in cytokinesis.

Models formulated using continuum mechanics treat a cellular feature, such as the cortex or membrane, as a continuous material rather than as a collection of discrete particles. Equations that describe the dynamic material properties of cellular components in continuum mechanics models entail both force balance and an underlying assumption that the effects of stochastic fluctuations are relatively small, usually due to covering greater distances and longer periods of time, i.e. modeling the entire cell cortex (16,000 µm2 and ∼minutes) (Turlier et al., 2014) versus agent-based modeling of a small patch of cortex (32 µm2 and ∼seconds) (Bidone et al., 2017). Continuum mechanics models of the cortex take into account whether the cellular component being modeled is viscous (remains deformed after an applied stress is released) or elastic (recovers to its starting configuration after stress is released), or both. It is routine in continuum modeling to simplify equations in order to obtain a tractable and predictive model that can be compared to experimental data. Examples of simplifications include (i) considering the cell to be cylindrical and axisymmetric, which allows the dimensionality of the system to be reduced (i.e. from 2D or 3D to 1D), and (ii) assuming the processes described by the model equations are in steady state (see Glossary).

Although cytokinesis involves numerous cellular and extracellular structures (including the extracellular matrix, plasma membrane, cortical cytoskeleton, cytoplasm, spindle and chromatin), most models depict only one or a subset of them. By considering the inclusion of a force or dynamics component into a continuum model, its magnitude, relative to other factors, is evaluated. For example, an estimation of viscous dissipation or the transformation of kinetic energy into heat revealed that dissipation in the cytoplasm is negligible compared to that of the cortex; thus, only the cortex was modeled (Turlier et al., 2014).

When choosing which cellular components to include in continuum models, modelers often explicitly include the actomyosin cortex since it is the primary source of force generation. Continuum models of cytokinesis have depicted the contractile ring either explicitly as a distinct model component (Biron et al., 2005; Sain et al., 2015; Zumdieck et al., 2007) or implicitly, by considering the effect of its localized contractile forces on the rest of the cell (Gladilin et al., 2015; Koyama et al., 2012; Poirier et al., 2012; Turlier et al., 2014). The cytoplasm is often considered by using one – or a combination – of the following: a constant volume constraint that represents the incompressibility of cytoplasm (Koyama et al., 2012; Poirier et al., 2012; Sain et al., 2015; Turlier et al., 2014), a viscous element in a linear viscous/viscoelastic model (Poirier et al., 2012; Zumdieck et al., 2007) or a viscous fluid (Zhao and Wang, 2016). These approaches all result in conservation of cell volume but differ in that when the cytoplasm is modeled as a viscous fluid, simulations output cytoplasmic velocity and, thus, the effects of velocity on flow-based transport and local biochemistry can be considered (Zhao and Wang, 2016). The cell membrane is not always included in the modeling of cytokinesis but, when included, it is often considered as an elastic material with surface tension (Dorn et al., 2016; Poirier et al., 2012; Sain et al., 2015; Zhao and Wang, 2016) or bending elasticity (Dorn et al., 2016; Koyama et al., 2012). Other models of cytokinesis have directly depicted the actomyosin cortex (Poirier et al., 2012; Turlier et al., 2014) and the vitelline membrane (Gladilin et al., 2015), the nucleus, and the extracellular matrix (Zhao and Wang, 2016).

Insights into the material properties of cellular structures have come from a number of experimental approaches. For example, a relatively non-invasive measurement that reflects the material properties of a structure is of its turnover rate, as measured using fluorescence recovery after photobleaching (FRAP). FRAP measurements have demonstrated that actomyosin is turned over within tens of seconds (Fritzsche et al., 2016, 2013; Guha et al., 2005; Murthy and Wadsworth, 2005), whereas cytokinesis usually lasts much longer (tens of minutes). The demonstration that the cytoskeleton is constantly renewed in the actomyosin cortex suggests that any elastic stress is released upon renewal of the layer and, therefore, that the actomyosin cortex is viscous with negligible elasticity (Turlier et al., 2014). These assumptions are only valid when the changes of cortical shape during cytokinesis are much slower than the turnover of ring components; so, whenever possible, these two measurements should be made and compared.

In sum, continuum models are powerful in that they provide estimates for biophysical parameters and highlight the relative contribution of different mechanical components.

Below, we ask two questions in the field of cytokinesis to highlight how modeling and biological experimentation have synergized to address them. First, we discuss how the actomyosin cytoskeleton is initially disorganized and then becomes circumferentially aligned during cytokinesis in many cell types. An agent-based model depicting F-actin-like fibers and NMM-II-like pulling forces incorporated cell biological and biochemical measurements of fission yeast, and demonstrated which aspects of polymer dynamics and interaction are sufficient for cytoskeletal remodeling (Vavylonis et al., 2008). Second, we focus on how cytokinetic ring contractility is mechanically coupled to the cortex throughout the cell (He and Dembo, 1997; Reymann et al., 2016; Turlier et al., 2014; White and Borisy, 1983). We review how continuum mechanics modeling examines the effects of local tension on the cortex, focusing on how autocatalysis emerges among contractility, cortical flow and the accumulation of force-generating material. Throughout, we highlight the predictions that result from using these models, which biologists may test by making key measurements. Since many cellular processes involve contractility and long-distance mechanical communication, we hope that these summaries prove useful to cell biologists also when studying other aspects of cellular behaviors.

The subject

Constituent filaments of the cytokinetic ring are initially randomly aligned at the cortical equator before cytokinesis and gradually become circumferentially aligned during cytokinesis (Beach et al., 2014; Descovich et al., 2018; Fenix et al., 2016; Reymann et al., 2016; Schroeder, 1973). In fission yeast, this reorganization occurs when randomly oriented actin filaments, anchored at formin nodes, elongate and interact with actin filaments that are nucleated through another node, via NMM-II, which is thought to exert pulling forces (Pollard and Wu, 2010).

The modeling

To determine the mechanism and dynamics of node coalescence and circumferential alignment, agent-based models of cytoskeletal components have been used to simulate the rearrangements that transform an isotropic meshwork into a ring (Bidone et al., 2017; Descovich et al., 2018; Vavylonis et al., 2008). We present, as an example, the work by Vavylonis and colleagues, who asked whether actin dynamics and NMM-II pulling forces are necessary and sufficient for ring coalescence (Vavylonis et al., 2008). Their stochastic model of a 2D surface comprising periodic boundaries (Fig. 2, dashed vertical lines) recapitulates ring formation in the cylindrical geometry of a fission yeast cell. Many of the measurements that went into this model came from extensive cell biological and biochemical work with fission yeast cytokinetic rings and constituent proteins (Wu and Pollard, 2005). In the simulation, surface-bound formin nodes are randomly distributed and two filaments allowed to grow from each node. The random growth of filaments and their depolymerization dynamics is dictated by probability distributions based on experimental measurements. Depending on the distance between neighboring nodes, overlapping filaments ends emanating from the nodes have some probability to interact, thus pulling nodes together through NMM-II-based forces. The forces on each node, including drag and repulsion between nodes, are summed up and the nodes move according to a force–balance equation (Fig. 2, top equation). Nodes coalesce and became circumferentially align, leading to a local increase in the density of actin filaments. Vavylonis and colleagues concluded that the ring assembles through stochastic search of elongating filaments and motor proteins for each other. Elongating filaments and motor proteins are then captured upon reaching a certain proximity, followed by reeling in of filaments through motor proteins and the stochastic release of filaments through motor proteins (dubbed the ‘search–capture-pull–release’ model) (Vavylonis et al., 2008).

Fig. 2.

Agent-based modeling has shed light on the mechanisms that organize the cytokinetic ring and generate the contractile force. (A) In the search-capture–pull-release model (Vavylonis et al., 2008), Actin-like fibers emanate from nodes within a region comprising periodic boundaries (a–d, dashed lines indicating boundaries). When growing actin fibers (a) come within range to interact with formin nodes (b), these nodes are crosslinked (c) and aligned (d). This, in turn, leads to alignment of the initially isotropic meshwork (top left) into a ring (bottom left). (B) Node velocity is calculated as a function of force and friction. Color of model parameters relate to the component of the equation to which they contribute. Parameters measured by biological experimentation are indicated by an asterisk; parameters fitted or calculated by using the model are unmarked. Model parameters relevant for a specific equation term (boxed) are color-coded to match the term shown in the equation.

Fig. 2.

Agent-based modeling has shed light on the mechanisms that organize the cytokinetic ring and generate the contractile force. (A) In the search-capture–pull-release model (Vavylonis et al., 2008), Actin-like fibers emanate from nodes within a region comprising periodic boundaries (a–d, dashed lines indicating boundaries). When growing actin fibers (a) come within range to interact with formin nodes (b), these nodes are crosslinked (c) and aligned (d). This, in turn, leads to alignment of the initially isotropic meshwork (top left) into a ring (bottom left). (B) Node velocity is calculated as a function of force and friction. Color of model parameters relate to the component of the equation to which they contribute. Parameters measured by biological experimentation are indicated by an asterisk; parameters fitted or calculated by using the model are unmarked. Model parameters relevant for a specific equation term (boxed) are color-coded to match the term shown in the equation.

A testable prediction

The model described by Vavylonis et al. predicted that the force exerted on actin filaments through NMM-II suppresses the growth rate of actin filaments that is regulated by formins within the nodes (Vavylonis et al., 2008). In fact, in vitro reconstitution assays later revealed that tugging on actin by myosin, indeed, decreases the rate of actin polymerization that is induced by the fission yeast formin Cdc12 (Zimmermann et al., 2017); thus, lending support to the model.

The subject

Cytokinesis is the remodeling of a large part of the cell cortex; thus, its high number of constituent components not only act locally but are also subject to long-range effects. Several studies, some of which are discussed below, have explored how the cortex behaves at a distance when one part is more contractile than others.

The modeling

An early exploration of this subject through modeling on the basis of continuum mechanics was a depiction of a cell comprising the cortex and containing two signaling centers that represented the asters of the anaphase spindle (White and Borisy, 1983). The cortex experiences tension, whose magnitude is defined by the distance between the cortex and the center of the asters, and by the concentration of cortical contractile elements. Under the starting condition, which represents pre-anaphase, contractile elements are uniformly distributed throughout the cell surface. When contractile elements are allowed to flow along the gradient of cortical tension, aster separation causes them to flow towards and become aligned at the equator, where their enrichment and alignment potentiates contractility (Bray and White, 1988; Fishkind and Wang, 1993; Franke et al., 1976; White and Borisy, 1983). This mechanical positive feedback has been dubbed ‘autocatalysis’ (He and Dembo, 1997). More recently, the cortex was modeled as a 3D shell of viscous, active material, without representing cytoskeletal filament organization (Fig. 3A) (Turlier et al., 2014). Localized contractility (Fig. 3) triggers ‘autocatalysis’ through positive feedback between cortical flow, local cortical thickening and the generation of tension. This modeling results in biologically relevant axisymmetric cell shapes (Turlier et al., 2014).

Fig. 3.

Continuum mechanics modelsunderscore the importance of cortical flows versus localized contractility and cytoskeletal organization. (A) Modeling the cell as a 3D shell of viscous, active material results in biologically relevant axisymmetric cell shapes (Turlier et al., 2014). Localized contractility (i) centered at the cell equator and with a Gaussian distribution (orange arrows) generates a shallow furrow (ii) and cortical flow (black arrows), which drives accumulation of tension-generating cortex (blue) at the equator and deepening of the furrow (iii) (He and Dembo, 1997; Turlier et al., 2014). In this case, cortical tension is calculated as a function of localized, cytokinetic ring-like tension and global cortical tension. Directional strain (see Glossary) rate relates to elastic force of the cortex. (B) By modeling the cortex of active, nematic material, Reymann and colleagues studied the interplay of cortical flows, cytoskeletal alignment and force generation (Reymann et al., 2016; Salbreux et al., 2009). Shown is an active gel representing the cortical cytoskeleton, which begins isotropic (i), until equatorial tension (orange arrows) drives gradual (ii, iii) circumferential alignment (blue shading) of constituent linear contractile elements. For this model, cortical tension is calculated as a function of several discrete sources of tension. The font color of the boxed model parameters relates to the components of the equation (terms) to which they contribute. Parameters measured by biological experimentation are unmarked; parameters fitted or calculated by using the model are indicated by an asterisk.

Fig. 3.

Continuum mechanics modelsunderscore the importance of cortical flows versus localized contractility and cytoskeletal organization. (A) Modeling the cell as a 3D shell of viscous, active material results in biologically relevant axisymmetric cell shapes (Turlier et al., 2014). Localized contractility (i) centered at the cell equator and with a Gaussian distribution (orange arrows) generates a shallow furrow (ii) and cortical flow (black arrows), which drives accumulation of tension-generating cortex (blue) at the equator and deepening of the furrow (iii) (He and Dembo, 1997; Turlier et al., 2014). In this case, cortical tension is calculated as a function of localized, cytokinetic ring-like tension and global cortical tension. Directional strain (see Glossary) rate relates to elastic force of the cortex. (B) By modeling the cortex of active, nematic material, Reymann and colleagues studied the interplay of cortical flows, cytoskeletal alignment and force generation (Reymann et al., 2016; Salbreux et al., 2009). Shown is an active gel representing the cortical cytoskeleton, which begins isotropic (i), until equatorial tension (orange arrows) drives gradual (ii, iii) circumferential alignment (blue shading) of constituent linear contractile elements. For this model, cortical tension is calculated as a function of several discrete sources of tension. The font color of the boxed model parameters relates to the components of the equation (terms) to which they contribute. Parameters measured by biological experimentation are unmarked; parameters fitted or calculated by using the model are indicated by an asterisk.

Testable predictions

The above models initially predicted that cortical flows augment equatorial forces. Subsequently, cortical flows have been experimentally observed in various cell types, including Xenopus oocytes (Benink et al., 2000), rat kidney cells (Cao and Wang, 1990) and C. elegans zygotes (Mayer et al., 2010).

An updated subject

In a recent investigation regarding long-range effects of equatorial contractility, Reymann and colleagues measured cortical flows and F-actin alignment, and explored how these and other factors relate to furrowing (Reymann et al., 2016).

The modeling

Active gel theory was employed to test whether the experimentally measured F-actin alignment is sufficient for the anisotropic cortical tension and measured furrowing dynamics (Reymann et al., 2016). Active gel theory models the contractile cortex as a thin layer of a viscous, active gel that is nematic (see Glossary), comprising polar components, such as F-actin (Kruse et al., 2005; Prost et al., 2015; Salbreux and Jülicher, 2017; Salbreux et al., 2009). Reymann and colleagues assumed the system to be at steady state, and depicted cytoskeletal alignment in one dimension (Reymann et al., 2016). The active gel is subject to elastic, viscous and active forces that represent stresses generated by energy (ATP) consumption, such as that produced by actomyosin contractility (Dorn et al., 2016; Mayer et al., 2010; Reymann et al., 2016; Salbreux et al., 2009) (Fig. 3B). Contractile active forces in the cortex could contribute in several ways: by driving flows that, in turn, cause compression, by actively bundling and aligning F-actin, and by passively crosslinking F-actin bundles. Reymann and colleagues tested the effects of adding a term that represents the active bundling by myosin, and found that myosin is not important for flow-based compression and F-actin alignment during cytokinesis (Reymann et al., 2016). Instead, Reymann's work, together with previous implementations of active gel theory (Kruse et al., 2005; Mayer et al., 2010; Salbreux et al., 2009), concluded that flow-based cortical compression and the resulting alignment are dominant contributors to furrow ingression (Salbreux et al., 2009). Since actomyosin alignment has been equated with the generation of anisotropic cortical tension (White and Borisy, 1983), the results of Reymann and colleagues suggest that autocatalysis also exists among cortical flow, compression and alignment, and tension generation.

Testable predictions

Future directions inspired by the current continuum modeling include investigations on how cortical actomyosin abundance and dynamics relate to biophysical parameters, such as tension. Specifically, contractility is not highest with maximal (or minimal) motor and crosslinker abundance but, rather, with intermediate amounts (the ‘Goldilocks’ effect) (Descovich et al., 2018; Ding et al., 2017; Ennomani et al., 2016; Li et al., 2016). Therefore, it is unclear how the relationship between NMM-II concentration and contractile force can be accurately depicted by agent-based modeling. However, several recent agent-based models have recapitulated the emergent phenomenon of maximal contractility at the point of intermediate crosslinker concentration, supporting the idea that the optimal concentration of actin filaments and crosslinkers for efficient contraction is, indeed, intermediate (Belmonte et al., 2017; Descovich et al., 2018; Ding et al., 2017; Ennomani et al., 2016; Hiraiwa and Salbreux, 2016).

Although simple models with limited components have brought us a long way, the ultimate goal of theoretically describing cytokinesis and other cellular processes is to comprehensively model the entire quantity of salient components. With more and more essential factors emerging (Kiyomitsu and Cheeseman, 2013; Rodrigues et al., 2015), the data obtained from continually improving biological experimentation will allow the more precise characterization of their physical and chemical properties, as well as their coordination and interactions. These experimental advances will invite the integration of such new information into more realistic theoretical models. Hybrid models (see Glossary), which incorporate both agent-based and continuum approximations of different cellular processes into a single model, are anticipated to advance the field by providing the benefits of both approaches, while compensating for each other's limitations. These next-generation models will have an enhanced predictive capability and, thus, stimulate further experimentation. Ultimately, it is the synergy and the mutual challenge between experiment and theory that will push the field towards a complete mechanistic understanding of cytokinesis.

The authors acknowledge the work on many additional interesting and important questions in cytokinesis that we did not have space to include here. The authors thank Ted Salmon, Jim Sellers, Liz Haswell and the three anonymous referees for careful reading of this manuscript and insightful comments. This review is dedicated to the memory of Karen Louise Dawes.

Funding

The authors were supported by the National Science Foundation [MCB-1411898 to Wallace Marshall supported a QCBNet Hackathon that brought us together; MCB-1616661 to A.S.M.; DMS-1554896 to A.D.], the National Institutes of Health [GM102390 to A.S.M.; GM103313 supports M.N.], the Simons Foundation [#429808 to W.S.], and the NIH intramural research program (J.L.). Deposited in PMC for release after 12 months.

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Competing interests

The authors declare no competing or financial interests.