FL50CH14-Mogilner ARI 11 November 2017 8:51 Annual Review of Fluid Mechanics Intracellular Fluid Mechanics: Coupling Cytoplasmic Flow with Active Cytoskeletal Gel Alex Mogilner1,2 and Angelika Manhart1 1Courant Institute of Mathematical Sciences, New York University, New York, NY 10012; email: [email protected], [email protected] 2Department of Biology, New York University, New York, NY 10012 ANNUAL REVIEWS Further Click here to view this article's online features: • Download figures as PPT slides • Navigate linked references • Download citations • Explore related articles • Search keywords Annu. Rev. Fluid Mech. 2018. 50:347–70 Keywords First published as a Review in Advance on October actin, active polar gel, blebs, cytoplasmic fluid flow, free moving boundary, Annu. Rev. Fluid Mech. 2018.50:347-370. Downloaded from www.annualreviews.org 13, 2017 myosin Access provided by New York University - Bobst Library on 01/10/18. For personal use only. The Annual Review of Fluid Mechanics is online at fluid.annualreviews.org Abstract https://doi.org/10.1146/annurev-fluid-010816- The cell is a mechanical machine, and continuum mechanics of the fluid 060238 cytoplasm and the viscoelastic deforming cytoskeleton play key roles in cell Copyright c 2018 by Annual Reviews. physiology. We review mathematical models of intracellular fluid mechanics, All rights reserved from cytoplasmic fluid flows, to the flow of a viscous active cytoskeletal gel, to models of two-phase poroviscous flows, to poroelastic models. We discuss application of these models to cell biological phenomena, such as organelle positioning, blebbing, and cell motility. We also discuss challenges of understanding fluid mechanics on the cellular scale. 347 FL50CH14-Mogilner ARI 11 November 2017 8:51 CELL MECHANICS The cell, the fundamental unit of life, is first and foremost a biochemical reactor and a self- replicating carrier of hereditary information (Alberts et al. 2014). The idea that the cell is also a mechanical machine changing shape and generating movements and forces is also well accepted and growing in recognition of its importance (Iskratsch et al. 2014). The mechanical aspects of cell life are central for cell motility, cell division, intracellular transport, and positioning of organelles, to name but a few relevant phenomena. The cytoskeleton, a dynamic network of polymers cross-linking proteins and molecular motors, is the main part of the cellular mechanical machine (Alberts et al. 2014). The discrete nature of the molecular elements of the cytoskeleton has drawn much attention to the view of the cell as a dynamic machine built from interconnected, solid, deformable and force-generating molecular units. However, on the scale of the whole cell, much insight comes from treating the cytoskeleton as a continuous gel (Pollack 2001). The cytoskeletal gel is highly nontrivial for two reasons: First, because it is active in the sense discussed below, and second, because its mechanical properties are very complex. Another important aspect of the cell is that it is filled with cytoplasm, a fluid that permeates the cytoskeletal scaffold. Organelles and large protein complexes either float in the cytoplasm or are embedded in the cytoskeletal mesh. There is a certain confusion related to the notion of the cytoplasm: The term is used both for the fluid fraction of the cell interior and the whole mixture of the fluid and cytoskeletal mesh in the cell. Also, there exists a mild controversy about the fluid versus gel-sol or colloidal nature of the cytoplasm (reviewed in Luby-Phelps 2013). To avoid confusion, in this review we try to avoid the term cytoplasm and use the term cytosol to describe the fluid fraction of the cytoplasm. The mechanical role of the cytosol is less well studied than that of the cytoskeleton. In recent years, though, it has become clear that the intimate connection between the interpenetrating cytoskeletal mesh and cytosol plays an important role in many cell mechanical phenomena. Quantitative analysis of the cell mechanics created exciting new computational problems, and, in turn, mathematical models of the cell fluid dynamics made a significant impact on understanding of experimental data. Here, we review mathematical and computational models of cell mechanics, starting with hydrostatics and Stokes flows of the cytosol in a cell volume devoid of a cytoskeleton and, in contrast, moving to models of the cell cytoskeleton as an active gel, ignoring the cytosol. Then we proceed to the full complexity of the two-phase flow models that consider interdependent dynamics of the cytoplasm and of an either viscous or elastic cytoskeletal gel. In this review, we discuss only the cell mechanics above the molecular scale. There is a great wealth of interesting fluid mechanics phenomena on the molecular scale (tens of nanometers), for Annu. Rev. Fluid Mech. 2018.50:347-370. Downloaded from www.annualreviews.org example, growth of a single actin filament in fast fluid flow (Carlier et al. 2014), as well as relevant Access provided by New York University - Bobst Library on 01/10/18. For personal use only. computational modeling (Goldstein 2016); here, we consider mechanics only on the scale of the whole cell (tens of microns). Similarly, we do not consider the fluid dynamics of multicellular groups on the tissue scale (Lee & Wolgemuth 2011, Peskin & McQueen 1995) or the fluid mechanics of blood flow (Kamm 2002). Many aspects of fluid mechanics on the cellular scale have been reviewed earlier, including multiphase flow theories and biofilms (Cogan & Guy 2010), mechanics of red blood cells and platelets, the response of endothelial cells to blood flow (Kamm 2002), cilia- or flagella-related fluid dynamics (Eloy & Lauga 2012, Goldstein 2016), leukocyte rolling under the action of flow and dynamic adhesions (Khismatullin & Truskey 2012), and cell swimming (Schwarz 2015). We touch only lightly on the electrolyte aspects of cell mechanics; readers can find deeper coverage of this topic by Tao et al. (2017). We discuss a few recent studies of cytoplasmic streaming; further and more extensive discussion can be gleaned from Goldstein & 348 Mogilner · Manhart FL50CH14-Mogilner ARI 11 November 2017 8:51 van de Meent (2015). Goldstein (2016) provided an elegant and exciting review of the history of fluid dynamics in biology. Even with all these restrictions, it is impossible to cite all computational studies of cell-related fluid dynamics, and for that, we apologize in advance. CHARACTERISTIC MECHANICAL SCALES IN THE CELL A majority of animal cells, which are the focus of our discussion, have a characteristic linear size in the range of tens of microns (Alberts et al. 2014). Characteristic rapid cell movement rates are on the order of ∼0.1–1 µm/s (Keren et al. 2009); hence, the characteristic relevant time is minutes. Fluid cytosolic flow rates, in the few cases when they were directly or indirectly measured, are ∼0.1–1 µm/s (Keren et al. 2009), and sometimes are greater in larger cells at ∼1–10 µm/s (Lewis et al. 2015). Slower flows on the order of ∼0.01 µm/s were also detected (Yi et al. 2011). Rates of molecular motors’ movement and of cytoskeletal fibers’ growth, which in most cases ultimately generate the flows, are also in the ∼0.1–1 µm/s range (Gross et al. 2000). Motor-generated rates of deformation and displacement of the cytoskeleton mesh are, again, on the same order or slower (Keren et al. 2009). Organelle movement in the cell is normally slow at ∼1 µm/min (Yi et al. 2011). On the microscopic (nanometer) scale, the viscosity of the cytosol is similar to that of water, ∼10−3 Pa·s. However, at greater scales, due to crowding of the cytoplasm and many complex physico-chemical factors, the effective viscosity of the cytoplasm is likely an order (or even two orders) of magnitude higher, ∼10−2–10−1 Pa·s (Luby-Phelps 1999). The mesh size of the actin network is in the range of tens of nanometers (Charras et al. 2005, Keren et al. 2009), and therefore the diffusion of a majority of globular proteins, ∼10 nm in size, is sensitive to the local conditions in the cell. For example, the effective diffusion coefficient of actin monomers could be as low as ∼1 µm2/s (Keren et al. 2009), more than an order of magnitude less than the diffusion coefficient of this protein in water. These characteristic scales lead to the extremely low Reynolds numbers, ∼10−5. Even in giant cells with faster flows, the Reynolds numbers are still much smaller than 1 (Goldstein 2016). Therefore, inertial terms can safely be neglected in all cell biology–relevant problems, and Stokes flow is an accurate approximation for the cytosol. In contrast, Peclet´ numbers characterizing the effectiveness of the convective mode of transport, as compared to diffusion, could be on the order of 1 (or greater, or smaller). Thus, effectiveness of the cytosolic flow for intracellular transport has to be investigated on a case-by-case basis. Fluid pressure inside the cell is normally ∼100 Pa higher with respect to that of the external fluid (Charras et al. 2008). This pressure is the result both of myosin-generated squeezing of Annu. Rev. Fluid Mech. 2018.50:347-370. Downloaded from www.annualreviews.org the cytosol by the contracting actomyosin mesh (Keren et al. 2009) and of the osmotic pressure generated by much higher ion concentration in the cell compared to the outside aqueous medium Access provided by New York University - Bobst Library on 01/10/18. For personal use only. (Tao et al. 2017); both factors could be on the same order of magnitude. Hydraulic permeability of the cytoskeletal mesh can be estimated by the mesh size, viscosity of the cytosol, and volume fraction of the cytosol. In cells, the latter varies significantly but is generally on the order of tens of percentage points (Luby-Phelps 1999), and hence, the hydraulic permeability of the cytoskeleton is on the order of 0.1 µm4/(pN·s) (Charras et al.