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WIREs Syst Biol Med
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Tools for computational analysis of moving boundary problems in cellular mechanobiology

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Abstract A cell's ability to change shape is one of the most fundamental biological processes and is essential for maintaining healthy organisms. When the ability to control shape goes awry, it often results in a diseased system. As such, it is important to understand the mechanisms that allow a cell to sense and respond to its environment so as to maintain cellular shape homeostasis. Because of the inherent complexity of the system, computational models that are based on sound theoretical understanding of the biochemistry and biomechanics and that use experimentally measured parameters are an essential tool. These models involve an inherent feedback, whereby shape is determined by the action of regulatory signals whose spatial distribution depends on the shape. To carry out computational simulations of these moving boundary problems requires special computational techniques. A variety of alternative approaches, depending on the type and scale of question being asked, have been used to simulate various biological processes, including cell motility, division, mechanosensation, and cell engulfment. In general, these models consider the forces that act on the system (both internally generated, or externally imposed) and the mechanical properties of the cell that resist these forces. Moving forward, making these techniques more accessible to the non‐expert will help improve interdisciplinary research thereby providing new insight into important biological processes that affect human health. This article is categorized under: Cancer > Cancer>Computational Models Cancer > Cancer>Molecular and Cellular Physiology
Discrete mechanical models of cells. (a) The spring joining two masses at xi and xj induces a force if the distance between the two points is different than the resting length L. Whereas the magnitude of the force is proportional to the displacement away from the spring's resting point, the direction of the force is along the vector between the two points acting away from the deviation. Thus, when stretched, the force acts to move the two masses towards each other. The force shown, Fij is the force on mass at xi; there is an equivalent force on this mass pointing towards xi. On the right is shown the net force (red) of the three springs that act on the mass at xi. (b) A Maxwell mechanical element consists of a series connection of a spring and viscous damper. The graph on the right shows the strain when a constant force is first applied and then removed. Whereas the elastic component is responsible for the two instant jumps, the continuous lengthening is due to the viscous damper. (c) In a Kelvin–Voigt element, the damper and spring are in parallel. On the right is the corresponding strain. Note that in this case there are no instant jumps and there is a maximum strain. (d). Mechanical model combining a Kelvin–Voigt element in series with a second viscous damper, and its corresponding strain
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Simulations of cell division. (a) Simulation of cytokinesis using the LSM. The four rows depict the relative contribution of each force to during time (Reprinted from Poirier et al. (2012) with permission under the Creative Commons Attribution License). (b) Three‐dimensional IBM simulation of asymmetric cell division. The asymmetry is obtained by specifying varying positions for the asters, marked by the letter “Y” (Reprinted with permission from Li and Kim (2016). Copyright 2015 Elsevier Inc.). (c) Simulation of cytokinesis using PFM. These simulations show the nucleus (yellow spheres) surrounded by the cytoplasm. After splitting, the daughter cells round up under the influence of surface tension (Reprinted with permission from Zhao and Wang (2016b). Copyright © John Wiley & Sons, Ltd.)
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Cell shape changes. (a) Simulation of collective cell migration using PFM. The cells are restricted to moving inside the circle. As a result, the cells organize to move radially (Reprinted from Löber et al. (2015) with permission under a Creative Commons Attribution 4.0 International License). (b) Phagocytosis simulation. Snapshots of simulations of the engulfment of antibody‐coated beads (Reprinted from Herant, Lee, Dembo, and Heinrich (2011) with permission under a Creative Commons Attribution License). Note that the individual panels have been rearranged to fit the page better
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Cell migration. (a) Simulations of cell chemotactic movement using the LSM. This simulation compares the movement of a cell without (top) or with (bottom) an adaptation mechanism in response to a chemoattractant gradient. The color along the perimeter represents the activity of the species that regulates protrusive forces (Reprinted from Shi et al. (2013) with permission under the Creative Commons Attribution License). (b) Simulation of 3D cell migration using the IBM. Shown are instantaneous cell shapes at varying membrane stiffness viscosity ratios. The cells swim from left to right. Color bar represents activator concentration (Reprinted with permission from Campbell and Bagchi (2017). Copyright 2017 AIP Publishing). (c). Simulations of wave dynamics in giant Dictyostelium cells. As the wave of activity reaches the cell perimeter, it pushes the boundary eventually driving the cell to cytofission. The fragments then move in a highly persistent fashion (Reprinted from Flemming, Font, Alonso, and Beta (2020) with permission of the authors)
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Immersed boundary, level set and phase field methods. (a) This shows the two sets of nodes needed in the IBM. The Eulerian points are defined on a fixed grid and are used to track the fluid. The Lagrangian mesh is used to specify the cell's evolution. It can consist of membrane points alone, or membrane/cytosol points. The forces on the Lagrangian network are spread to the fluid based on their location. The fluid velocity is used to upgrade the location of the Lagrangian nodes. (b) Signed distance function used to define shape in the LSM. Along the Cartesian grid, the distance of every point to the initial shape (in black) is calculated with a positive or negative sign for points outside or inside, respectively, of the region. This forms the level set function ϕ(x, t) which is updated over time. (c) Example of a PFM φ (x, t) (left) and magnitude of the gradient ∥∇φ(x, t)∥. The cells in panels b and c have the same shape
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Finite element method. (a) Example of a meshing for two objects. The inner sphere represents a nucleus; the outer, which is only partially shown, is the cytoplasm. (b) Remeshing in a moving boundary. As the object deforms, the elements change shape. The three insets show how the elements change shape and size. How to account for these changes is a particularly challenging task in using FEM in moving boundary problems. (c) FE simulation result showing the stress on a cell that is being indented with a spherical indenter. The top plot shows the relative dimensions of the three objects included in the simulations: the cell body surrounding the nucleus and the spherical indenter. The bottom figure shows stress contour plots at an indentation depth of 1.5 μm (Reprinted with permission from Tang, Galluzzi, Zhang, Shen, and Stadler (2019). Copyright 2019 American Chemical Society)
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Continuum description of mechanics. (a) Coordinates for a three‐dimensional continuum representation. Stresses perpendicular to a face are defined as having two common indices; for example, a stress that is perpendicular to the face where the normal lies in the x‐direction, is given by σxx. If the force is tangential, mixed subscripts are used. For example, σxy denotes a force acting in the y‐direction along the face with normal in the x‐direction. (b) In the continuum description, a stress σxx causes the object to extend in the x‐direction, but also causes it to contract in the y‐ and z‐directions. The ratio of the respective changes in length is the Poisson ratio
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