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WIREs Syst Biol Med
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Computational modeling of single‐cell mechanics and cytoskeletal mechanobiology

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Cellular cytoskeletal mechanics plays a major role in many aspects of human health from organ development to wound healing, tissue homeostasis and cancer metastasis. We summarize the state‐of‐the‐art techniques for mathematically modeling cellular stiffness and mechanics and the cytoskeletal components and factors that regulate them. We highlight key experiments that have assisted model parameterization and compare the advantages of different models that have been used to recapitulate these experiments. An overview of feed‐forward mechanisms from signaling to cytoskeleton remodeling is provided, followed by a discussion of the rapidly growing niche of encapsulating feedback mechanisms from cytoskeletal and cell mechanics to signaling. We discuss broad areas of advancement that could accelerate research and understanding of cellular mechanobiology. A precise understanding of the molecular mechanisms that affect cell and tissue mechanics and function will underpin innovations in medical device technologies of the future. WIREs Syst Biol Med 2018, 10:e1407. doi: 10.1002/wsbm.1407 This article is categorized under: Models of Systems Properties and Processes > Mechanistic Models Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Cellular Models
Mechanical modelling techniques across spatial scales. Each panel shows the unloaded, initial geometry, and simulated deformation of the cell or cytoskeleton. (a) a continuum mechanics model of an actin cytoskeleton cortex (red), intermediate vimentin filaments (green) and the nucleus (Reprinted with permission from Ref 10. Copyright 2011 Elsevier Ltd); (b) a dissipative particle dynamics simulation of a red blood cell passing through a narrow slit (Reprinted with permission from Ref 11. Copyright 2016 National Academy of Sciences); (c) a coarse‐grained brownian dynamics simulation of a small portion of the cell actin cytoskeleton to study the role of individual cytoskeletal proteins to the emergent mechanical behaviour of the cytoskeleton. The image shows snapshots of the cytoskeleton at different time points (Reprinted with permission from Ref 12. Copyright 2016 Nature Publishing Group).
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(a) Synopsis of some of the types of manipulations that have been performed to manipulate cell tension and investigate its effects. (b) Illustration of micropipette aspiration used to increase plasma membrane tension (Reprinted with permission from Ref . Copyright 2012 Elsevier Inc.). (c) Images of an actin wave (circular dorsal ruffle) expanding and subsequently contracting (scale bar corresponds to 25 μm, reprinted with permission from Ref . Copyright 2015 Public Library of Science). (d) Image depicting a phase relationship where actin waves are preceded by and appear to suppress FBP17 (a downstream effector of Cdc42) activity, suggesting a negative influence of actin on its own signaling (Reprinted with permission from Ref . Copyright 2013 National Academy of Sciences).
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Signalling to cytoskeletal mechanics and back. (a) Well‐established biochemical pathways that link signalling molecules to the cytoskeletal machinery (Reprinted with permission from Ref . Copyright 2012 Frontiers) (b)‐(e) are new ideas of how cytoskeletal mechanics may feedback to signalling as well. The acronyms within each box refer to key signalling molecules involved in modulating cytoskeletal components such as actin (polymerisation, turnover) and myosin (contraction). ((b) reprinted with permission from Ref . Copyright 2013 National Academy of Sciences; (c) reprinted with permission from Ref . Copyright 2017 Public Library of Science; d reprinted with permission from Ref . Copyright 2016 Public Library of Congress).
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Measuring and modelling cell‐ECM tractions. (a) Measurement of bead displacements (left) and calculated peak strains (right) of a cell migrating inside a 3D ECM environment (Reprinted with permission from Ref . Copyright 2010 Nature Publishing Group Inc.). Scale bar represents 50 μm. (b) A simulation from a model of cell‐ECM interactions in which the ECM fibres are explicitly modelled as cylindrical segments (Reprinted with permission from Ref 160. Copyright 2012 Elsevier Inc.). The two spheres represent cells that are migrating through the ECM, remodelling the local ECM matrix in the process. (c) A conceptual diagram of a continuum model of a cell and its mechanical interaction with the ECM. The model incorporates the role of ECM stiffness in the traction forces felt at the cell adhesions (Reprinted with permission from Ref . Copyright 2013 Springer). (d) Cell doublet assays involve pulling two adhered cells apart using micropipette aspiration The fluorescence image shows cortical actin accumulation at the cell‐cell junction. The plot shows the temporal change in separation force as the cells are pulled apart (Reprinted with permission from Ref . Copyright 2004 Rockefeller University Press).
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Measuring and modelling actin polymerisation mechanics. (a) A method of measuring the force produced by actin polymerisation where polymerisation is promoted near an AFM tip. The deflection of the tip and imaging the growing actin gel provide sufficient data to parameterise a model of actin gel growth from actin polymerisation. (b) Plots that relate growth rates and protrusion force that can be used to parameterise a model of actin protrusion. Circles represent the experimental data and the curves are predictions from a macroscopic growth‐tensor model of the leading edge. (c) Schematic of mesoscopic and macroscopic concepts of the interpretation of the force generation due to actin polymerisation. In the mesoscopic model, actin filaments push and pull the membrane depending on their kinetics. One approach to macroscopic modelling of the forces due to actin polymerisation is to treat the polymerisation as a cell growth strain, G, that is applied on top of any mechanical strains due to external loads. ((a), (b) and macroscopic model of (c) reprinted with permission from Ref . Copyright 2009 Elsevier Inc. Mescoscale schematic in (c) reprinted with permission from Ref . Copyright 2012 Elsevier Inc.)
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Stiffness of microtubule and intermediate filament networks as functions of cross‐linker density. (a) Plot of microtubule network stiffness, calculated by dividing force F by the initial elastic jump distance d1, with three different densities of cross‐linkers (12.5%, 25% and 50%). The plot shows network stiffening at low forces and softening at high forces. Inset shows the crossover force Fc from stiffening to softening reginmes increases with crosslinking density (Reprinted with permissions from Ref . Copyright 2012 RSC Publishing); (b) Plots of stiffness of vimentin intermediate filament networks cross‐linked by different divalent ions. Plot on the left shows linear viscoelastic moduli in the absence of divalent ions (with vimentin concentration cv = 1 mg/ml, black open circles; cv = 2.5 mg/ml, gray open circles) and in the presence of Mg2+ (with molar ratio, RMg = 215; cv = 1 mg/ml black squares; cv = 2.5 mg/ml, gray squares) and Ca2+ (with molar ratio, RCa = 215; cv = 1 mg/ml, black triangles). G′ dominates over G″ and exhibits weak power‐law scaling with frequency, ω, having an exponent of 0.09. The networks in the presence of divalent ions are two to four times stiffer. Plot on the right summarizes the dependence of the elastic response, Go, of vimentin networks on RMg and cv. (Reprinted with permissions from Ref . Copyright 2010 Elsevier Inc.).
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The role of cytoskeletal proteins and their cross‐linkers: (a, b) Cytoskeletal networks made from combinations of different cross‐linkers exhibit different emergent mechanical behaviour (Reprinted with permission from Ref 101 and 102. Copyright 2005 Elsevier Inc. and Copyright 2009 Public Library of Science, respectively). (c) Myosin‐contractility of the cytoskeletal network is ineffective at very low or very high cross‐linker concentrations (Reprinted with permission from Ref 103. Copyright 2008 Elsevier Inc.). The shaded region indicates the myosin contractility feasibility. (d) Myosin exerts contractile stresses on the cytoskeletal network that reaches equilibrium after approximately 800 ms (Reprinted with permission from Ref 103. Copyright 2008 Elsevier Inc.).
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(a) Illustrates the fundamental structure of the three groups of filaments that constitute the cytoskeletal network (Reprinted with permission from Ref 83 and Ref 84. Copyright 2016 Cold Spring Harbor Laboratory Press and Copyright 2010 Elsevier Inc., respectively); (b) Stress‐strain relationship for reconstituted cytoskeletal networks purely made from one filament type (Reprinted with permission from Ref . Copyright 1991 Rockefeller University Press).
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Examples of experimental measurements and mechanical models of bulk mechanical properties of cells. (a) Mills et al. used optical tweezers to perform uniaxial extension and relaxation experiments on red blood cell to parameterize hyperelastic and viscoelastic constitutive equations of its mechanical behavior (Reprinted with permission from Ref 51. Copyright 2004 Tech Science Press); (b) Zhou et al. implemented a finite element model (top image in (b)) with a power‐law rheology based constitutive equation to capture the long‐time‐range soft‐glassy like response of cells as measured by creep tests using micropipette aspiration (middle of panel) (Reprinted with permission from Ref 45. Copyright 2012 Springer); (c) Herant and Dembo18 used the poroelastic continuum mechanics equations to account for the porous nature of the cytoskeletal network (image shown in the top panel of (c), reprinted with permission from Ref 18. Copyright 2010, Elsevier Inc.) and the movement of fluid through these pores. The influence of fluid reorganization during mechanical perturbations and the poroelastic nature of the cell have been recently experimentally measured (and fitted with a poroelasitc model) by Moeenderbary et al. 22 (middle and bottom panels, reprinted with permission from Ref 22. Copyright 2013 Nature Publishing Group).
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