How Models and Biological Experimentation Have Come Together to Reveal Mechanisms of Cytokinesis Daniel B
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© 2018. Published by The Company of Biologists Ltd | Journal of Cell Science (2018) 131, jcs203570. doi:10.1242/jcs.203570 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 1 REVIEW Journal of Cell Science (2018) 131, jcs203570. doi:10.1242/jcs.203570 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