- •Contents
- •Contributors
- •Preface
- •1 Introduction, with the biological basis for cell mechanics
- •Introduction
- •The role of cell mechanics in biological function
- •Maintenance of cell shape
- •Cell migration
- •Mechanosensing
- •Stress responses and the role of mechanical forces in disease
- •Active cell contraction
- •Structural anatomy of a cell
- •The extracellular matrix and its attachment to cells
- •Transmission of force to the cytoskeleton and the role of the lipid bilayer
- •Intracellular structures
- •Overview
- •References
- •2 Experimental measurements of intracellular mechanics
- •Introduction
- •Forces to which cells are exposed in a biological context
- •Methods to measure intracellular rheology by macrorheology, diffusion, and sedimentation
- •Whole cell aggregates
- •Sedimentation of particles
- •Diffusion
- •Mechanical indentation of the cell surface
- •Glass microneedles
- •Cell poker
- •Atomic force microscopy
- •Mechanical tension applied to the cell membrane
- •Shearing and compression between microplates
- •Optical traps
- •Magnetic methods
- •Twisting of magnetized particles on the cell surface and interior
- •Passive microrheology
- •Optically detected individual probes
- •One-particle method
- •Two-particle methods
- •Dynamic light scattering and diffusing wave spectroscopy
- •Fluorescence correlation spectroscopy
- •Optical stretcher
- •Acoustic microscopy
- •Outstanding issues and future directions
- •References
- •3 The cytoskeleton as a soft glassy material
- •Introduction
- •Magnetic Twisting Cytometry (MTC)
- •Measurements of cell mechanics
- •The structural damping equation
- •Reduction of variables
- •Universality
- •Scaling the data
- •Collapse onto master curves
- •Theory of soft glassy rheology
- •What are soft glassy materials
- •Sollich’s theory of SGMs
- •Soft glassy rheology and structural damping
- •Open questions
- •Biological insights from SGR theory
- •Malleability of airway smooth muscle
- •Conclusion
- •References
- •4 Continuum elastic or viscoelastic models for the cell
- •Introduction
- •Purpose of continuum models
- •Principles of continuum models
- •Boundary conditions
- •Mechanical and material characteristics
- •Example of studied cell types
- •Blood cells: leukocytes and erythrocytes
- •Limitations of continuum model
- •Conclusion
- •References
- •5 Multiphasic models of cell mechanics
- •Introduction
- •Biphasic poroviscoelastic models of cell mechanics
- •Analysis of cell mechanical tests
- •Micropipette aspiration
- •Cells
- •Biphasic properties of the pericellular matrix
- •Indentation studies of cell multiphasic properties
- •Analysis of cell–matrix interactions using multiphasic models
- •Summary
- •References
- •6 Models of cytoskeletal mechanics based on tensegrity
- •Introduction
- •The cellular tensegrity model
- •The cellular tensegrity model
- •Do living cells behave as predicted by the tensegrity model?
- •Circumstantial evidence
- •Prestress-induced stiffening
- •Action at a distance
- •Do microtubules carry compression?
- •Summary
- •Examples of mathematical models of the cytoskeleton based on tensegrity
- •The cortical membrane model
- •Tensed cable nets
- •Cable-and-strut model
- •Summary
- •Tensegrity and cellular dynamics
- •Conclusion
- •Acknowledgement
- •References
- •7 Cells, gels, and mechanics
- •Introduction
- •Problems with the aqueous-solution-based paradigm
- •Cells as gels
- •Cell dynamics
- •Gels and motion
- •Secretion
- •Muscle contraction
- •Conclusion
- •Acknowledgement
- •References
- •8 Polymer-based models of cytoskeletal networks
- •Introduction
- •The worm-like chain model
- •Force-extension of single chains
- •Dynamics of single chains
- •Network elasticity
- •Nonlinear response
- •Discussion
- •References
- •9 Cell dynamics and the actin cytoskeleton
- •Introduction: The role of actin in the cell
- •Interaction of the cell cytoskeleton with the outside environment
- •The role of cytoskeletal structure
- •Actin mechanics
- •Actin dynamics
- •The emergence of actin dynamics
- •The intrinsic dynamics of actin
- •Regulation of dynamics by actin-binding proteins
- •Capping protein: ‘decommissioning’ the old
- •Gelsolin: rapid remodeling in one or two steps
- •β4-thymosin: accounting (sometimes) for the other half
- •Dynamic actin in crawling cells
- •Actin in the leading edge
- •Monomer recycling: the other ‘actin dynamics’
- •The biophysics of actin-based pushing
- •Conclusion
- •Acknowledgements
- •References
- •10 Active cellular protrusion: continuum theories and models
- •Cellular protrusion: the standard cartoon
- •The RIF formalism
- •Mass conservation
- •Momentum conservation
- •Boundary conditions
- •Cytoskeletal theories of cellular protrusion
- •Network–membrane interactions
- •Network dynamics near the membrane
- •Special cases of network–membrane interaction: polymerization force, brownian and motor ratchets
- •Network–network interactions
- •Network dynamics with swelling
- •Other theories of protrusion
- •Numerical implementation of the RIF formalism
- •An example of cellular protrusion
- •Protrusion driven by membrane–cytoskeleton repulsion
- •Protrusion driven by cytoskeletal swelling
- •Discussion
- •Conclusions
- •References
- •11 Summary
- •References
- •Index
Active cellular protrusion: continuum theories and models |
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Constitutive equations
The mass and momentum conservation equations are cast in a very general framework that needs to be further constrained to provide closure of the system. These additional prescriptions (the constitutive equations) embody the biological specifications of the cell. For instance, it is likely that the viscosity ν will depend on the cytoskeletal density: a law such as
ν = ν0θn |
(10.10) |
prescribing a well-defined linear relation between viscosity and network concentration is such a constitutive relation. In principle, this could be verified empirically by investigating the rheology of the cytoplasm at various cytoskeletal concentrations. However, in general such experimental evidence is sparse and often difficult to interpret. One is thus usually reduced to educated guesses for the constitutive laws that govern J (the network formation or cytoskeletal polymerization rate), H (the resistance to solvent flow through the network), (the network stress due to elasticity and static interactions), ν (the network viscosity), and γ (the tension of the cortical membrane). Conversely, the main advantage of this formalism is that it is sufficiently general to accommodate most theories of protrusions: as we shall see below, it all depends on the proper adjustment of the constitutive equations.
Cytoskeletal theories of cellular protrusion
As has been touched on, it appears that polymerization of large amounts of actin in the vicinity of a membrane causes outward force and protrusion. It also appears that this phenomenon is probably not directly dependent on molecular motors such as myosins, especially as their contractile activity tends to ‘pull’ rather than ‘push’ the cytoskeleton. This has led to theories of protrusion such as the Brownian ratchet model, in which the free energy released by the addition of monomers to a filament is transduced to generate a pressure against a membrane that sterically interferes with the reaction. Without going into the specifics, however, it is clear that such cytoskeletal theories of protrusion can be categorized into two classes:
Network–membrane interaction theories in which the cytoskeleton and the membrane repel one another through a force field. The classic Brownian ratchet model belongs to this class, as it relies on the hard-core potential of actin monomers pushing on the membrane (Peskin et al. 1993).
Network–network interaction theories in which the cytoskeleton interacts with itself, resulting in a repulsive force. This could be due to electrostatic interactions (actin is negatively charged) or thermal agitation.
In what follows we shall formalize these classes of theory in a way that enables linkage to the RIF approach.
We wish to emphasize that we are only discussing free protrusions – that is, protrusions that emerge from the cell body without adhesion to an external substrate. When adhesion occurs, additional classes of theory become tenable, but these are not considered here.