How Models and Biological Experimentation Have Come Together to Reveal Mechanisms of Cytokinesis Daniel B

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REVIEW Unite to divide – how models and biological experimentation have come together to reveal mechanisms of cytokinesis Daniel B. Cortes1, Adriana Dawes2, Jian Liu3, Masoud Nickaeen4, Wanda Strychalski5 and Amy Shaub Maddox1,*

ABSTRACT these systems. Thus, cytokinesis can serve as a paradigm to Cytokinesis is the fundamental and ancient cellular process by which understand diverse behaviors of cellular motility. one cell physically divides into two. Cytokinesis in animal and fungal Mathematical modeling (see Glossary), combined with biological ‘ ’ cells is achieved by contraction of an actomyosin cytoskeletal ring experimentation (i.e. wet lab approaches including microscopy, assembled in the cell cortex, typically at the cell equator. Cytokinesis genetics, biochemistry and biophysics), has significantly advanced ‘ ’ is essential for the development of fertilized eggs into multicellular our understanding of cytokinesis. Herein, we use the word modeling organisms and for homeostatic replenishment of cells. Correct to collectively refer to diverse theoretical approaches, in which execution of cytokinesis is also necessary for genome stability and biological, biochemical and biophysical processes are described with the evasion of diseases including cancer. Cytokinesis has fascinated mathematical equations. These approaches, often historically rooted scientists for well over a century, but its speed and dynamics make in, and motivated by, problems in physics and chemistry, include experiments challenging to perform and interpret. The presence continuum mechanics modeling and agent-based modeling (see of redundant mechanisms is also a challenge to understand Glossary). The following references can serve as a starting point for cytokinesis, leaving many fundamental questions unresolved. For foundational reading, i.e. Frenkel and Smit, 2002; Goldstein et al., example, how does a disordered cytoskeletal network transform into 2001; Lai et al., 2009; Landau and Binder, 2000; Landau and Lifshitz, a coherent ring? What are the long-distance effects of localized 1960. Depending on the length- and time-scales of the subject of the contractility? Here, we provide a general introduction to ‘modeling for model (Fig. 1), a solution may be written explicitly as a function of biologists’, and review how agent-based modeling and continuum time and space (analytic solution) or approximated by using numerical mechanics modeling have helped to address these questions. algorithms (computational solution). Mathematical models make predictions by recapitulating KEY WORDS: Agent-based modeling, Cell division, Continuum biological observations on the basis of proposed physical, mechanics modeling, Contractility chemical and structural properties of the proteins, membrane and other components that underlie cellular processes. Modeling has Introduction been applied to understand the positioning (Mogilner et al., 2016), Cytokinesis is regulated spatially and temporally to accomplish the assembly, contraction and disassembly of the contractile ring during partitioning of the cytoplasm and segregated genome into two cytokinesis (see below for non-exhaustive referencing). Modeling – daughter cells (Rappaport, 1996). In animal cells, formation of the constrained by available experimental data – has also been used to actomyosin cytokinetic ring at the division plane (or cell equator) establish the physical plausibility of a hypothesis. Even when ideas is elicited through activation of the small GTPase RhoA by derived from modeling are currently inaccessible by biological microtubule-borne spindle-based signals in anaphase (reviewed by experimentation, such as the dynamic reorientation of cytoskeletal Mogilner et al., 2016). Active (GTP-bound) RhoA, elicits F-actin filaments within diffraction-limited features (Ennomani et al., 2016; nucleation, activation and filament formation of non-muscle myosin Zumdieck et al., 2007), they guide biological experimentation. In II (NMM-II) and recruitment of scaffold proteins. The resulting the most powerful application of modeling, predictions from a ensemble of the cytokinetic ring encircles the cell equator in a band model are experimentally tested and the results are iteratively or cord and constricts, drawing the associated plasma membrane used to modify model parameters (see Glossary) or evaluate the into a furrow and partitioning the daughter cell contents (Green feasibility of the model. et al., 2012). The core actomyosin machinery of the cytokinetic ring In this Review, we first discuss what biologists look for in models is shared among animal and fungal cells, and is a specialization of and what modelers look for in data obtained from biological the contractile actomyosin cytoskeleton that governs cell shape in 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- 1Department of Biology, University of North Carolina at Chapel Hill, 407 Fordham Hall, Chapel Hill, NC 27599, USA. 2Departments of Mathematics and of Molecular based or continuum-based modeling methods. Finally, we use the Genetics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, questions of cytoskeletal organization and long-range effects of Columbus, OH 43210, USA. 3National Heart, Lung and Blood Institute, Biochemistry and Biophysics Center, 50 South Drive, NIH, Bethesda, MD 20892, local force generation as examples of how modeling and biological USA. 4Richard D. Berlin Center for Cell Analysis and Modeling, University of measurements were used to collaborate to advance our mechanistic Connecticut Health Center, Department of Cell Biology, 263 Farmington Avenue, understanding of cytokinesis. Farmington, CT 06030-6406, USA. 5Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA. What biologists look for in a model In setting out to understand a model, a biologist is likely to consider *Author for correspondence ([email protected]) the reductionist nature, robustness and relevance of the model. A

A.S.M., 0000-0002-4671-2949 reductionist approach to modeling focuses only on elements that are Journal of Cell Science

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Glossary Active gel theory: the continuum mechanics theory depicting a viscoelastic Nematic: a mesomorphic state, in which the linear orientation of molecules material, with polar filaments that dynamically change through energy results in anisotropic properties. Polar molecules, such as F-actin, are consumption. For example, the actin cortex is rearranged through myosin aligned in parallel or anti-parallel. ATP hydrolysis. Periodic boundary conditions: the property of a modeled domain in Agent-based modeling: modeling in which the dynamics of individual which an edge connects to another (opposite) edge, as do the west and players are explicitly simulated and tracked. Stochasticity is calculated for east margins of a two-dimensional map of the surface of the Earth. The and applied to each individual element modeled. periodic boundary condition represents continuity, wrapping or infinite Analytic solution: the solution to an equation given as a specific formula dimensions, such that there are no free edges. For example, the cortex has rather than by a numerical simulation. no edges, therefore models of small patches of it employ periodic boundary Continuum mechanics modeling: depicts a potentially complex conditions to simulate an effectively infinite domain. heterogeneous structure as a continuous material rather than a Reynolds number: dimensionless ratio of inertial forces to viscous forces collection of discrete particles, by simplifying their length- and time- in a fluid medium. Biological molecules and polymers experience the scales. These models are limited to phenomena that occur at cytoplasm as a low Reynolds number milieu due to the high viscosity of dimensions larger than the simplified length and time-scales, typically cytoplasm and the small length scales of the cell. depicting structures of one micron or larger; e.g. the entire cell Steady state: a dynamic process at its equilibrium; i.e. over time, the cortex can be described by continuum mechanics modeling as a thin dynamics (or average thermal fluctuations) are either unchanged or very elastic shell. small compared with other processes. For example, cytosolic forms of the Elastic: describes a material that recovers to its starting configuration after Rho family of small GTPases diffuse very quickly, so that the spatial any applied stress is released. Elastic materials are characterized by an distribution of these proteins can be assumed to be uniform; the diffusion elastic modulus, a quantity that represents a measurement of an object’s process is then said to be at steady state. resistance to an applied stress. Stochasticity: randomness generated by an underlying probability Hybrid (or multi-scale) model: incorporates aspects of both agent-based distribution. and continuum approximations of different cellular processes into one unified Strain: dimensionless quantity of the deformation of a material relative to a model. reference measurement (i.e. that of the same material at resting state). Langevin-type equation: force–balance equation that describes the Stress: force per unit area in 3D. change of one or more variables over time. Inherent to the equation is its Surface tension: describes the tendency of a fluid to minimize its surface- stochasticity, stemming from collisions among particles. area-to-volume ratio as force per unit length. Mathematical modeling: used here as a general term to refer Term: the part of an equation that describes a process or an effect, i.e. collectively to diverse theoretical approaches, in which biological, diffusion term, which describes the movement of a chemical species from biochemical and biophysical processes are described with mathematical an area of higher to lower concentration. equations. Viscoelastic: a material that exhibits both viscous and elastic behavior, Mean field: the ensemble average. depending on the duration over which a force is applied. The viscoelastic Mesoscale: on the threshold of currently resolvable length (∼100 nm), and modulus measures the ratio of stress to strain after applying oscillatory occurring at durations that are between those of typical single-molecule stresses. and cell biological studies (hundreds of milliseconds). Viscous: describes a material that remains deformed after an applied stress Model parameter: components of an equation that represent inherent is released. Viscosity is the resistance to flow. The viscosity coefficient properties of the subject of the model. For example, the binding range of a relates the velocity gradient of a fluid to its viscous stress. The viscous drag is crosslinker in an agent-based model or the elastic modulus of a material in the resistive force caused by viscosity. The irreversible process by which a continuum mechanics model. kinetic energy is transformed into heat is known as viscous dissipation. thought to be essential for the simulated phenomena. For example, the output of the model. Alterations should allow the output to, at models focused on dynamics of global cell shape during cytokinesis least qualitatively, resemble the effects of the corresponding might focus on terms (see Glossary), such as membrane tension and biological perturbations, and to remain within biologically cortical stiffness, while reducing contractility to minimal terms. meaningful ranges and physically reasonable limits. For example, Simplifying models in this way tends to mean that they are more a robust model of molecular motors should result in filament sliding phenomenological, contain as few terms as possible – such that within a milieu demonstrating a viscosity that is relevant to both the there is no 1:1 correspondence between biological component and cellular and the in vitro settings. model parameter – and are more amenable to analytical solutions. A biologist will also consider the relevance of a model regarding a For example, measurements of cortical mechanics made by variation of cell shapes and sizes. The applicability of a single model micropipette aspiration are interpreted by a simple model derived to several cell types is likely to be limited to cell-intrinsic from Laplace’s law, which relates the pressure change across an components. For example, the molecular make-up and material interface (the cell membrane) to surface tension (see Glossary) and properties of the actin cytoskeleton are likely to be shared among curvature of the interface. Since the pressure inside the micropipette animal cell types. Non-autonomous properties of cells, such as cell is known and the radius of the cell deformation is measured, cortical wall, and intercellular and substrate adhesion impact the mechanical tension can then be computed from the equation for Laplace’slaw and molecular requirements of cytokinesis (Bourdages et al., 2014; (Hochmuth, 2000; Tinevez et al., 2009). The contributions of Founounou et al., 2013; Guillot and Lecuit, 2013; Herszterg et al., contractility, cytoskeletal-membrane linkage and membrane 2014; Morais-de-Sa and Sunkel, 2013; Pinheiro et al., 2017). The composition are neither measured nor taken into account but influence of such cell type-specific effects can be tested by including rather summed up as ‘cortical tension’ (Evans and Robinson, 2018; specific considerations to the model, as was done when investigating Hochmuth, 2000). The approach has been extended to include the effect of adhesion and traction-mediated protrusive forces on cytoskeletal elasticity so that the cell’s elastic modulus of the cell furrow ingression in Dictyostelium cells (Poirier et al., 2012). (see ‘Elastic’ in Glossary) can be measured (Hochmuth, 2000). Depending on the availability of biological, biochemical and Models should be robust to perturbations, such that a variation in biophysical measurements, the relevance of a model may be the amount of a component or force should only quantitatively alter assessed on the basis of agreement between these measured values Journal of Cell Science

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Rho-GTP a F-actin b d Anillin Crosslinker Cytokinetic ring

c e Motor protein

Length (m) 10−10 10−9 10−8 10−7 10−6 10−5 Nanoscale Mesoscale Microscale

Duration (sec) 10−6 10−5 10−4 10−3 10−2 0.1 0 10 100

Common modeling Molecular Agent basedHybrids Continuum mechanics approach: dynamics

Number of 100,000, depicted as 1−10s10s−1000s 1000s−100,000s elements: one or more materials

Stochasticity is Individual elements calculated for: The ensemble

Key

Interaction between protein Spacing of Speed of Laser cutting and plasma membrane filament bundles motor protein followed by recoil Furrow ingression

Fig. 1. The machinery of cytokinesis has been studied 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. and the underlying parameters of the model. In some cases, actual precision are ideal, but it is important to keep in mind that biological values of model parameters vary greatly between cell types or with measurements themselves are likely to perturb the native state of the the method of measurement, even for high-accuracy measurements. system. For instance, protein abundance data on the basis of For example, experimental measurements of the cytoplasmic fluorescence intensity calculations are confounded by the incomplete viscosity coefficient vary from 3×10−3 Pa s (Mastro et al., 1984) quantum efficiency of digital cameras and imperfect behavior of to 2×10−1 Pa s (Daniels et al., 2006; Kreis et al., 1982). Such fluorescent probes (Heppert et al., 2018; Ulbrich and Isacoff, 2007). disparity might reflect the different timescales at which the viscosity The biophysical properties of molecular motors are often determined measurements were performed and/or the true difference in the for single motor domains, whereas many form ensembles in vivo (see cytosols of different species. Furthermore, the viscosity a particle Guo and Guilford, 2006; Stam et al., 2015 for examples). experiences also depends on its size and shape, i.e. a myosin In addition, measuring biophysical parameters, e.g. viscoelastic filament experiences a higher viscosity than a GFP protein moduli (see ‘viscoelastic’ in Glossary), requires calculations that are (Kalwarczyk et al., 2011). In addition, the volume of two popular based the assumption that biological materials, including the model cell types – fission yeast and the Xenopus zygote – differs cytoskeleton and the extracellular matrix, respond to a given 100,000-fold (Danilchik, 2011; Heald and Gibeaux, 2018; deformation in a known or predictable fashion. For example, to Zakhartsev and Reuss, 2018). Variation exists also at the calculate the traction force exerted by a cell from the displacement of molecular level, with the number of motor subunits in non-muscle beads embedded in a flexible substrate, one commonly employed myosin II (NMM-II) ensembles ranging from eight in fission yeast model assumes that the thickness of the substrate is semi-infinite to ≤30 in mammalian cells (Niederman and Pollard, 1975; Sinard (Dembo et al., 1996; Dembo and Wang, 1999), whereas another et al., 1989; Verkhovsky, 1995; Laplante et al., 2016; Wu and model takes into account the finite dimensions of the substrate Pollard, 2005). Thus, there is merit to ensure that biological models (del Alamo et al., 2007) in order to more accurately account for are robust, such that a variation of some biophysical parameters still the underlying geometry. Micro-pipet aspiration is frequently used yields similar observable behaviors and emergent properties. to measure the mechanical properties of various cellular components including the membrane-associated actomyosin cortical What modelers look for in biological measurements cytoskeleton (the ‘cortex’), but interpreting the results as a From the perspective of a modeler, several features are desirable in characterization of the membrane or the cortex, or both, requires biological measurements. It goes without saying that accuracy and distinct biophysical models for each object (Brugues et al., 2010; Dai Journal of Cell Science

3 REVIEW Journal of Cell Science (2018) 131, jcs203570. doi:10.1242/jcs.203570 et al., 1999; Hochmuth, 2000). Physical assumptions also affect the interpretation of biological measurements when modeling the Box 1. Bottom-up approaches mechanics of the entire cell, whether it is considered as a liquid A bottom-up approach relies on agent-based models that simulate the drop with a surface tension (see Glossary), a liquid drop with a microscopic and elementary dynamics of individual molecules, and their viscoelastic shell or as an elastic solid (Hochmuth, 2000; Yang et al., physical and chemical interactions (e.g. positions, conformational 2008). Modeling the cell as a liquid drop assumes not only that surface changes, mechanical and chemical states, and their degree of change) (Frenkel and Smit, 2002; Landau and Binder, 2000). Whereas tension comprises both membrane tension and cortical tension (due to molecular dynamics simulations are used for one or a few molecules, actomyosin contractility) but also that the cortical tension is uniform, agent-based modeling typically describes hundreds of interacting whereas – in reality – it is heterogenous (Hochmuth, 2000; Mayer players. Their behaviors are simulated by using a high number of time et al., 2010; Tinevez et al., 2009; Yang et al., 2008). This biophysical steps to bridge their minuscule and rapid molecular events to the cellular property is calculated from how the cortex recoils upon being cut with phenomena that occur over longer periods and greater distances. Such a laser; however, such calculations themselves are done using models simulations generate a macroscopic or mesoscopic picture of the complex phenomena that emerge from a molecular ensemble. Bottom- that make assumptions. For example, these models describe the cell as up approaches are conceptually straightforward, especially when input a periodic thin film of contractile viscous (i.e. viscoelastic) material molecular interactions and behaviors are known. However, these (see Glossary) but only take into account interactions and flows over a molecular details are frequently unknown even for the actin finite length (Mayer et al., 2010). Calculations used to determine cytoskeleton, one of the best-studied cellular machines. Depending on forces from traction force microscopy, micro-pipet aspiration and the dimensional and spatial constraints of the model and its subject, whole-cell modeling all underscore the principle that measurements volumetric exclusions and steric interactions should also be considered. If not accounted for correctly, such interactions can result in physical from biological experimentation are interpreted after having used a entanglement or crowding, which can result in simulation artifacts. In model based on assumptions. some cases, such as when representing a 3D collection of objects by Often, the measurements needed for a model are unknown in the modeling a projection into 2D space, these constraints can be ignored to specific system being modeled. When this is the case, one of several some extent or entirely (Belmonte et al., 2017; Nedelec and Foethke, strategies is taken: (i) values are estimated from cells with similar 2007). Since agent-based modeling takes into account the physical cell biological or physical properties, such as species and size, properties and inherent stochasticity of objects, it can be used to depict respectively; (ii) the sensitivity of the model output to a model small numbers of objects that may not behave according to the mean- field description. For larger collections of objects, each undergoing parameter range is tested over several orders of magnitude, which additional interaction that occurs at increased length- or timescales, a guides the prioritization of which parameters need to be precisely model can be simplified to mean-field approximations that are determined; or (iii) one parameter – which can be estimated with computationally less costly. In these cases, continuum modeling can high confidence – is used to normalize the others so that only the prove more powerful (see main text for specific examples and associated ratios of model components are considered. references). 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 to considerable noise and it cannot be assumed that they behave be taken into account. Below, we introduce several classes of model according to the ensemble average, i.e. mean field (see Glossary). and summarize how recent modeling approaches have contributed The dynamic behavior of a system can be expected to occasionally – to our knowledge of cytokinesis. but significantly – deviate from mean-field behavior when there are <100 hundred players (Kampen, 2007). For many cellular events Choosing among different modeling approaches relevant to cytokinesis, the copy number of each given factor is in Each modeling strategy has its own advantages and limitations. Two the range of 10–1000 (Wu and Pollard, 2005). Therefore, hybrid – main factors that influence the choice of modeling strategy are the or multiscale – models are currently being developed to describe the spatial scale and the duration of the process under study (Fig. 1). At mesoscale (see Glossary) dynamics of cytokinesis. For instance, one end of the biological spectrum is the dynamics of individual multiscale models that combine continuum mechanics modeling of molecular events, which typically occur in milliseconds or less and the cytokinetic ring with agent-based modeling (see Glossary) of across nanometers. To model these events, molecular dynamics actin filaments (Biron et al., 2005; Oelz et al., 2015; Zumdieck et al., modeling is conventionally applied. However, cytokinesis involves 2007) are being applied to address the mechanical and molecular the collective behavior of thousands to hundreds-of-thousands of origins of forces in cytokinesis. As outlined below, multiscale constituent molecules; over longer timescales (seconds to minutes) models are conceptually attractive and have offered unique insights and length scales (hundreds of nanometers to microns), their but, owing to the inherently different strategies to include and dynamics can be averaged. illustrate stochasticity (see Glossary) in agent-based versus Broadly speaking, there are two approaches to model a cellular continuum models, theoretical strategies to incorporate the output process, ‘bottom-up’ strategies that typically constitute agent-based of one as the input into the other are still being developed. For models of a group of specific molecular players (see below and example, it has not yet been established how to use the explicit Box 1), or ‘top-down’ approaches, such as continuum models that results of cytoskeletal alignment generated through agent-based depict the collective behavior of large molecular ensembles (see modeling, such as by using the cytoskeleton simulation engine below and Box 2). An important distinction between bottom-up and Cytosim (Nedelec and Foethke, 2007), as input for the degree of top-down modeling strategies is how they account for and describe local order within nematic (see Glossary) active gels of continuum the stochastic fluctuations that are ubiquitous in biology; bottom-up mechanics models (see below). approaches explicitly describe the effects of stochastic fluctuations, whereas most top-down approaches describe the average behavior Agent-based models of cytokinesis of each element. Agent-based models explicitly simulate and track the dynamics of Accounting for stochastic fluctuations is very important when individual players. For the cytokinetic ring, these players may include modeling a small number of molecules, as their behavior is subject actin monomers or filaments, motor proteins and crosslinkers. Agent- Journal of Cell Science

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Other agent-based models depict 2D domains with periodic Box 2. Top-down approaches boundary conditions (representing continuity or wrapping; see Top-down approaches, such as continuum modeling, assume that the Glossary), or explicitly 3D domains (Bidone et al., 2014). Below, detailed individual behavior of molecules can be smoothed into an we present findings that exemplify the use of agent-based modeling ensemble average (Goldstein et al., 2001; Lai et al., 2009; Landau and to study cytoskeletal dynamics in cytokinesis. 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. Continuum mechanics models of cytokinesis These concentration fields interact according to the principles of Models formulated using continuum mechanics treat a cellular mechanics (e.g. forces and velocities) and chemistry (e.g. on- and off- feature, such as the cortex or membrane, as a continuous material rates), both of which reflect bulk behaviors at long timescales. rather than as a collection of discrete particles. Equations that Consequently, the spatial-temporal resolution of continuum models is describe the dynamic material properties of cellular components in much lower (in the order of microns and minutes) than that of molecular continuum mechanics models entail both force balance and an dynamics and agent-based models (which depict processes at nanometers and milliseconds scales). As such, the number of model underlying assumption that the effects of stochastic fluctuations components is greatly reduced compared to agent-based models and is are relatively small, usually due to covering greater distances and more amenable to mathematical analysis. Because continuum models longer periods of time, i.e. modeling the entire cell cortex do not consider the interactions of individual molecules or particles, they (16,000 µm2 and ∼minutes) (Turlier et al., 2014) versus agent- typically do not encounter entanglement or dynamically arrested states. based modeling of a small patch of cortex (32 µm2 and ∼seconds) Furthermore, because a fundamental assumption of continuum (Bidone et al., 2017). Continuum mechanics models of the cortex modeling is that it represents the ensemble average, such top-down approaches are most suitable to study a mean-field system, wherein take into account whether the cellular component being modeled stochastic fluctuations introduce only minor modifications to the average is viscous (remains deformed after an applied stress is released) or behavior. By the same token, agent-based models – due to their discrete elastic (recovers to its starting configuration after stress is nature – are well-suited to describe the effects of stochastic fluctuations released), or both. It is routine in continuum modeling to but are often restricted to small space and short timescales. (Please see simplify equations in order to obtain a tractable and predictive main text for specific examples and associated references). 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 based models define how each component moves and interacts to be reduced (i.e. from 2D or 3D to 1D), and (ii) assuming the stochastically with its environment according to a set of rules, such as processes described by the model equations are in steady state (see the stiffness of a fiber or the force required to detach a protein from Glossary). a binding partner. These rules are based either on biological, Although cytokinesis involves numerous cellular and biochemical and biophysical measurements or on hypotheses. To extracellular structures (including the extracellular matrix, plasma model the dynamics of the cytokinetic ring, most models assume that membrane, cortical cytoskeleton, cytoplasm, spindle and forces are balanced at any given point in time, e.g. by implementing a chromatin), most models depict only one or a subset of them. By Langevin-type equation (see Glossary). Cellular models lack considering the inclusion of a force or dynamics component into a acceleration, inertia and momentum, since small components (e.g. continuum model, its magnitude, relative to other factors, is motor proteins or cytoskeletal filaments) experience the cytoplasm as evaluated. For example, an estimation of viscous dissipation or the having a relatively high viscosity, which translates into a low transformation of kinetic energy into heat revealed that dissipation Reynolds number (see Glossary) regime. Although a viscous drag in the cytoplasm is negligible compared to that of the cortex; thus, force is ubiquitous in the equations of several models, the inclusion of only the cortex was modeled (Turlier et al., 2014). additional sources of force differs widely but often includes the When choosing which cellular components to include in activity of motor proteins and the resistance of filaments to bending. continuum models, modelers often explicitly include the Agent-based models are often used to depict reaction kinetics and the actomyosin cortex since it is the primary source of force polymer dynamics of cytoskeletal components (see for example, generation. Continuum models of cytokinesis have depicted the Mendes Pinto et al., 2012; Vavylonis et al., 2008). Agent-based contractile ring either explicitly as a distinct model component (Biron models can depict cytoskeletal components as a series of repeating et al., 2005; Sain et al., 2015; Zumdieck et al., 2007) or implicitly, by nodes or segments, in order to minimize computation time and, thus, considering the effect of its localized contractile forces on the rest of facilitate investigation to a larger degree than such modeling would the cell (Gladilin et al., 2015; Koyama et al., 2012; Poirier et al., otherwise permit; for example, Cytosim depicts cytoskeletal 2012; Turlier et al., 2014). The cytoplasm is often considered by filaments as a series of segments (Nedelec and Foethke, 2007). A using one – or a combination – of the following: a constant volume major limitation of agent-based modeling of cytokinesis is the lack of constraint that represents the incompressibility of cytoplasm detailed measurements of the abundance of cytokinetic ring (Koyama et al., 2012; Poirier et al., 2012; Sain et al., 2015; Turlier components, their dynamics and interactions, which necessitates et al., 2014), a viscous element in a linear viscous/viscoelastic model assumptions and the estimation of the model parameter values related (Poirier et al., 2012; Zumdieck et al., 2007) or a viscous fluid (Zhao to these characteristics. In addition, computation timescales with the and Wang, 2016). These approaches all result in conservation of cell number of individual components simulated, exerting practical limits volume but differ in that when the cytoplasm is modeled as a viscous on the duration and area or volume that can be simulated. fluid, simulations output cytoplasmic velocity and, thus, the effects of Despite these limitations, agent-based models have been velocity on flow-based transport and local biochemistry can be successfully implemented to depict several aspects of cytokinesis. considered (Zhao and Wang, 2016). The cell membrane is not always For example, agent-based modeling has been implemented in 2D included in the modeling of cytokinesis but, when included, it is often domains that represent a patch of the cortex (Bidone et al., 2017), or considered as an elastic material with surface tension (Dorn et al., 2D sections that represent either a cross-section, or the entirety, of a 2016; Poirier et al., 2012; Sain et al., 2015; Zhao and Wang, 2016) or cytokinetic ring (Ennomani et al., 2016; Mendes Pinto et al., 2012). bending elasticity (Dorn et al., 2016; Koyama et al., 2012). Other Journal of Cell Science

5 REVIEW Journal of Cell Science (2018) 131, jcs203570. doi:10.1242/jcs.203570 models of cytokinesis have directly depicted the actomyosin cortex components have been used to simulate the rearrangements that (Poirier et al., 2012; Turlier et al., 2014) and the vitelline membrane transform an isotropic meshwork into a ring (Bidone et al., 2017; (Gladilin et al., 2015), the nucleus, and the extracellular matrix (Zhao Descovich et al., 2018; Vavylonis et al., 2008). We present, as an and Wang, 2016). example, the work by Vavylonis and colleagues, who asked whether Insights into the material properties of cellular structures have actin dynamics and NMM-II pulling forces are necessary and come from a number of experimental approaches. For example, a sufficient for ring coalescence (Vavylonis et al., 2008). Their relatively non-invasive measurement that reflects the material stochastic model of a 2D surface comprising periodic boundaries properties of a structure is of its turnover rate, as measured using (Fig. 2, dashed vertical lines) recapitulates ring formation in the fluorescence recovery after photobleaching (FRAP). FRAP cylindrical geometry of a fission yeast cell. Many of the measurements have demonstrated that actomyosin is turned over measurements that went into this model came from extensive cell within tens of seconds (Fritzsche et al., 2016, 2013; Guha et al., biological and biochemical work with fission yeast cytokinetic rings 2005; Murthy and Wadsworth, 2005), whereas cytokinesis usually and constituent proteins (Wu and Pollard, 2005). In the simulation, lasts much longer (tens of minutes). The demonstration that the surface-bound formin nodes are randomly distributed and two cytoskeleton is constantly renewed in the actomyosin cortex suggests filaments allowed to grow from each node. The random growth of that any elastic stress is released upon renewal of the layer and, filaments and their depolymerization dynamics is dictated by therefore, that the actomyosin cortex is viscous with negligible probability distributions based on experimental measurements. elasticity (Turlier et al., 2014). These assumptions are only valid Depending on the distance between neighboring nodes, overlapping when the changes of cortical shape during cytokinesis are much filaments ends emanating from the nodes have some probability to slower than the turnover of ring components; so, whenever possible, interact, thus pulling nodes together through NMM-II-based forces. these two measurements should be made and compared. The forces on each node, including drag and repulsion between In sum, continuum models are powerful in that they provide nodes, are summed up and the nodes move according to a force– estimates for biophysical parameters and highlight the relative balance equation (Fig. 2, top equation). Nodes coalesce and became contribution of different mechanical components. circumferentially align, leading to a local increase in the density of actin filaments. Vavylonis and colleagues concluded that the ring Probing cytokinesis with modeling assembles through stochastic search of elongating filaments and Below, we ask two questions in the field of cytokinesis to highlight motor proteins for each other. Elongating filaments and motor how modeling and biological experimentation have synergized to proteins are then captured upon reaching a certain proximity, address them. First, we discuss how the actomyosin cytoskeleton is followed by reeling in of filaments through motor proteins and the initially disorganized and then becomes circumferentially aligned stochastic release of filaments through motor proteins (dubbed the during cytokinesis in many cell types. An agent-based model ‘search–capture-pull–release’ model) (Vavylonis et al., 2008). depicting F-actin-like fibers and NMM-II-like pulling forces incorporated cell biological and biochemical measurements of A testable prediction fission yeast, and demonstrated which aspects of polymer The model described by Vavylonis et al. predicted that the force dynamics and interaction are sufficient for cytoskeletal remodeling exerted on actin filaments through NMM-II suppresses the growth (Vavylonis et al., 2008). Second, we focus on how cytokinetic ring rate of actin filaments that is regulated by formins within the nodes contractility is mechanically coupled to the cortex throughout the cell (Vavylonis et al., 2008). In fact, in vitro reconstitution assays later (He and Dembo, 1997; Reymann et al., 2016; Turlier et al., 2014; revealed that tugging on actin by myosin, indeed, decreases the rate of White and Borisy, 1983). We review how continuum mechanics actin polymerization that is induced by the fission yeast formin Cdc12 modeling examines the effects of local tension on the cortex, (Zimmermann et al., 2017); thus, lending support to the model. focusing on how autocatalysis emerges among contractility, cortical flow and the accumulation of force-generating material. Throughout, What are the long-range effects of localized contractility? we highlight the predictions that result from using these models, The subject which biologists may test by making key measurements. Since many Cytokinesis is the remodeling of a large part of the cell cortex; thus, cellular processes involve contractility and long-distance mechanical its high number of constituent components not only act locally but communication, we hope that these summaries prove useful to cell are also subject to long-range effects. Several studies, some of biologists also when studying other aspects of cellular behaviors. which are discussed below, have explored how the cortex behaves at a distance when one part is more contractile than others. How does the cytokinetic ring become organized? The subject The modeling Constituent filaments of the cytokinetic ring are initially randomly An early exploration of this subject through modeling on the basis of aligned at the cortical equator before cytokinesis and gradually continuum mechanics was a depiction of a cell comprising the cortex become circumferentially aligned during cytokinesis (Beach et al., and containing two signaling centers that represented the asters of the 2014; Descovich et al., 2018; Fenix et al., 2016; Reymann et al., anaphase spindle (White and Borisy, 1983). The cortex experiences 2016; Schroeder, 1973). In fission yeast, this reorganization occurs tension, whose magnitude is defined by the distance between the when randomly oriented actin filaments, anchored at formin nodes, cortex and the center of the asters, and by the concentration of cortical elongate and interact with actin filaments that are nucleated through contractile elements. Under the starting condition, which represents another node, via NMM-II, which is thought to exert pulling forces pre-anaphase, contractile elements are uniformly distributed (Pollard and Wu, 2010). throughout the cell surface. When contractile elements are allowed to flow along the gradient of cortical tension, aster separation causes The modeling them to flow towards and become aligned at the equator, where their To determine the mechanism and dynamics of node coalescence enrichment and alignment potentiates contractility (Bray and White, and circumferential alignment, agent-based models of cytoskeletal 1988; Fishkind and Wang, 1993; Franke et al., 1976; White and Journal of Cell Science

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AB a Total force Node velocity= Friction coefficient Thermal energy Friction coefficient= Diffusion constant

b Relevant model parameters:

Node velocity Total force* Friction coefficient* c Thermal energy* Diffusion constant Filament growth rate* Filament capture radius*

Key Formin nodes Filament growth rate Actin-like fibers Filament capture radius

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.

Borisy, 1983). This mechanical positive feedback has been dubbed colleagues assumed the system to be at steady state, and depicted ‘autocatalysis’ (He and Dembo, 1997). More recently, the cortex was cytoskeletal alignment in one dimension (Reymann et al., 2016). modeled as a 3D shell of viscous, active material, without The active gel is subject to elastic, viscous and active forces that representing cytoskeletal filament organization (Fig. 3A) (Turlier represent stresses generated by energy (ATP) consumption, such as et al., 2014). Localized contractility (Fig. 3) triggers ‘autocatalysis’ that produced by actomyosin contractility (Dorn et al., 2016; Mayer through positive feedback between cortical flow, local cortical et al., 2010; Reymann et al., 2016; Salbreux et al., 2009) (Fig. 3B). thickening and the generation of tension. This modeling results in Contractile active forces in the cortex could contribute in several biologically relevant axisymmetric cell shapes (Turlier et al., 2014). ways: by driving flows that, in turn, cause compression, by actively bundling and aligning F-actin, and by passively crosslinking F-actin Testable predictions bundles. Reymann and colleagues tested the effects of adding a The above models initially predicted that cortical flows augment term that represents the active bundling by myosin, and found that equatorial forces. Subsequently, cortical flows have been myosin is not important for flow-based compression and F-actin experimentally observed in various cell types, including Xenopus alignment during cytokinesis (Reymann et al., 2016). Instead, oocytes (Benink et al., 2000), rat kidney cells (Cao and Wang, 1990) Reymann’s work, together with previous implementations of active and C. elegans zygotes (Mayer et al., 2010). gel theory (Kruse et al., 2005; Mayer et al., 2010; Salbreux et al., 2009), concluded that flow-based cortical compression and the An updated subject resulting alignment are dominant contributors to furrow ingression In a recent investigation regarding long-range effects of equatorial (Salbreux et al., 2009). Since actomyosin alignment has been contractility, Reymann and colleagues measured cortical flows and equated with the generation of anisotropic cortical tension (White F-actin alignment, and explored how these and other factors relate to and Borisy, 1983), the results of Reymann and colleagues suggest furrowing (Reymann et al., 2016). that autocatalysis also exists among cortical flow, compression and alignment, and tension generation. The modeling Active gel theory was employed to test whether the experimentally Testable predictions measured F-actin alignment is sufficient for the anisotropic cortical Future directions inspired by the current continuum modeling tension and measured furrowing dynamics (Reymann et al., 2016). include investigations on how cortical actomyosin abundance and Active gel theory models the contractile cortex as a thin layer of a dynamics relate to biophysical parameters, such as tension. viscous, active gel that is nematic (see Glossary), comprising polar Specifically, contractility is not highest with maximal (or components, such as F-actin (Kruse et al., 2005; Prost et al., 2015; minimal) motor and crosslinker abundance but, rather, with

Salbreux and Jülicher, 2017; Salbreux et al., 2009). Reymann and intermediate amounts (the ‘Goldilocks’ effect) (Descovich et al., Journal of Cell Science

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A iiiiii

Cell membrane Relevant model parameters: Cortical thickness =+ Cortical tension active tension viscous tension Local actomyosin concentration ATP free energy Cortical viscosity Directional strain rates

B i ii iii

Relevant model parameters: Local myosin concentration Cortical tension = Isotropic active ++anisotropic active viscous tension Local cytoskeletal order parameter (Alignment) tension tension Viscosity of the active gel Velocity gradient

Alignment =+advection compression + filament−filament + myosin activity Relevant model parameters: interactions Nematic order parameter Rate of turnover to an isotropic state Effect of compression on nematic ordering* flow field; compression field Lengthscale of local filament orientation* Local myosin concentration

Key Localized Degree Cortical contractility contractility Cortical flow of circumferential alignment

Fig. 3. Continuum mechanics models underscore 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.

2018; Ding et al., 2017; Ennomani et al., 2016; Li et al., 2016). contractility at the point of intermediate crosslinker concentration, Therefore, it is unclear how the relationship between NMM-II supporting the idea that the optimal concentration of actin filaments concentration and contractile force can be accurately depicted by and crosslinkers for efficient contraction is, indeed, intermediate agent-based modeling. However, several recent agent-based (Belmonte et al., 2017; Descovich et al., 2018; Ding et al., 2017; models have recapitulated the emergent phenomenon of maximal Ennomani et al., 2016; Hiraiwa and Salbreux, 2016). Journal of Cell Science

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Conclusions and perspectives Danilchik, M. V. (2011). Manipulating and imaging the early Xenopus laevis Although simple models with limited components have brought embryo. Methods Mol. Biol. 770, 21-54. del Alamo, J. C., Meili, R., Alonso-Latorre, B., Rodrıguez-Rodŕ ıguez,́ J., us a long way, the ultimate goal of theoretically describing Aliseda, A., Firtel, R. A. and Lasheras, J. C. (2007). Spatio-temporal analysis of cytokinesis and other cellular processes is to comprehensively eukaryotic cell motility by improved force cytometry. Proc. Natl Acad. Sci. USA model the entire quantity of salient components. With more and 104, 13343-13348. more essential factors emerging (Kiyomitsu and Cheeseman, 2013; Dembo, M. and Wang, Y.-L. (1999). Stresses at the cell-to-substrate interface during locomotion of fibroblasts. Biophys. J. 76, 2307-2316. Rodrigues et al., 2015), the data obtained from continually Dembo, M., Oliver, T., Ishihara, A. and Jacobson, K. (1996). Imaging the traction improving biological experimentation will allow the more precise stresses exerted by locomoting cells with the elastic substratum method. Biophys. characterization of their physical and chemical properties, as well as J. 70, 2008-2022. their coordination and interactions. These experimental advances Descovich, C. P., Cortes, D. B., Ryan, S., Nash, J., Zhang, L., Maddox, P. S., Nedelec, F. and Maddox, A. S. (2018). Cross-linkers both drive and brake will invite the integration of such new information into more cytoskeletal remodeling and furrowing in cytokinesis. Mol. Biol. Cell 29, 622-631. realistic theoretical models. Hybrid models (see Glossary), which Ding, W. Y., Ong, H. T., Hara, Y., Wongsantichon, J., Toyama, Y., Robinson, incorporate both agent-based and continuum approximations of R. C., Nédélec, F. and Zaidel-Bar, R. (2017). Plastin increases cortical different cellular processes into a single model, are anticipated to connectivity to facilitate robust polarization and timely cytokinesis. J. Cell Biol. 216, 1371-1386. advance the field by providing the benefits of both approaches, Dorn, J. F., Zhang, L., Phi, T.-T., Lacroix, B., Maddox, P. S., Liu, J. and Maddox, while compensating for each other’s limitations. These next- A. S. (2016). A theoretical model of cytokinesis implicates feedback between generation models will have an enhanced predictive capability membrane curvature and cytoskeletal organization in asymmetric cytokinetic furrowing. Mol. Biol. Cell 27, 1286-1299. and, thus, stimulate further experimentation. Ultimately, it is the Ennomani, H., Letort, G., Guérin, C., Martiel, J.-L., Cao, W., Nédélec, F., De La synergy and the mutual challenge between experiment and theory Cruz, E. M., Théry, M. and Blanchoin, L. (2016). Architecture and connectivity that will push the field towards a complete mechanistic govern actin network contractility. Curr. Biol. 26, 616-626. understanding of cytokinesis. Evans, J. P. and Robinson, D. N. (2018). Micropipette aspiration of oocytes to assess cortical tension. Methods Mol. Biol. 1818, 163-171. Acknowledgements Fenix, A. M., Taneja, N., Buttler, C. A., Lewis, J., Van Engelenburg, S. B., Ohi, R. and Burnette, D. T. (2016). Expansion and concatenation of nonmuscle myosin The authors acknowledge the work on many additional interesting and important IIA filaments drive cellular contractile system formation during interphase and questions in cytokinesis that we did not have space to include here. The authors mitosis. Mol. Biol. Cell 27, 1465-1478. thank Ted Salmon, Jim Sellers, Liz Haswell and the three anonymous referees for Fishkind, D. J. and Wang, Y. L. (1993). Orientation and three-dimensional careful reading of this manuscript and insightful comments. This review is dedicated organization of actin filaments in dividing cultured cells. J. Cell Biol. 123, 837-848. to the memory of Karen Louise Dawes. Founounou, N., Loyer, N. and Le Borgne, R. (2013). Septins regulate the contractility of the actomyosin ring to enable adherens junction remodeling during Competing interests cytokinesis of epithelial cells. Dev. Cell 24, 242-255. The authors declare no competing or financial interests. Franke, W. W., Rathke, P. C., Seib, E., Trendelenburg, M. F., Osborn, M. and Weber, K. (1976). Distribution and mode of arrangement of microfilamentous Funding structures and actin in the cortex of the amphibian oocyte. Cytobiologie 14, The authors were supported by the National Science Foundation [MCB-1411898 111-130. to Wallace Marshall supported a QCBNet Hackathon that brought us together; Frenkel, D. and Smit, B. (2002). 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