222M. Herant and M. Dembo

However, the cytoskeleton velocity field is different in a way that is intrinsic to the mechanism of protrusion invoked in each case. In the swelling model, there exists a stagnation point (or center of expansion) within the cell across which the cytoskeletal flow goes from centripetal to centrifugal. The distance of the stagnation point from the leading edge depends on the force resisting protrusion (for instance in the limit of a hard wall, the stagnation point is at the membrane), but in certain conditions (see Herant et al., 2003 for an example), it can lie far back from the leading edge. This is not the case in the membrane–cytoskeleton repulsion model, where the stagnation point lies in the “gap” region (Fig. 10-2) very near the membrane. Such distinction may be a way to experimentally establish the distinction between swelling and repulsion models of protrusion.

Finally, it would have been possible to compute models of alternative protrusion theories such as those in Fig. 10.4 using the RIF formalism. We do not do so here because these other theories are not currently favored.

Conclusions

For reasons of brevity, we have not included computational examples of protrusion models that fail to produce cell-like behavior. The fact that these tend to make only rare appearances in the literature can be misleading: in reality good models with good parameters are needles in haystacks (for example, see Drury and Dembo, 2001). This is mostly due to the fact that the spaces of inputs (model specifications, such as viscosity) and outputs (model behavior, such as cellular shape) that need to be explored are both extremely large, as expected in complex biological systems.

In general, our experience has been that in hindsight, it is not difficult to discover why it is that a particular model with a particular choice of parameters does not lead to the behavior one was hoping for. On the other hand, we have also found that a priori expectations about the kind of results a given model will produce are rarely precisely matched by a numerical simulation. As a result, the process of constructing mechanical models of cell behavior is one of iteration during which one refines one’s experience and intuition by running many numerical experiments. It is also our experience that this process often leads to deeper insights about the fundamental mechanisms at play in certain cellular events such as protrusions.

With this in mind, caution is in order when dealing with cartoon descriptions of the mechanics of living cells. As compelling as a mechanical diagram in the form of rods, ropes, pulleys, and motors working together may be, the actual implementation of such models within a quantitative model may not lead to what one was expecting! Thus, in a Popperian way, one of the principal benefits of a rigorous framework for cell mechanics is the ability to falsify invalid theories (Popper, 1968). This is why, although such frameworks can be difficult to develop and use, they are necessary to reach a real understanding of the mechanics of living cells.

We thank Juliet Lee as well as the editors for comments on an early version of this chapter. This work was supported by Whitaker biomedical engineering research grant RG-02-0714 to MH and NIH grant RO1-GM 61806 to MD.

References

Condeelis, J. (1993). Life at the Leading Edge: The Formation of Cell Protrusions. Annu. Rev. Cell Biol., 9, 411–44.

Dembo, M., & Harlow, F. (1986). Cell Motion, Contractile Networks, and the Physics of Interpenetrating Reactive Flow. Biophys. J., 50, 109–21.

Dickinson R., & Purich, D. L. (2002). Clamped-filament Elongation Model for Actin-Based Motors. Biophys. J., 82, 605–17.

Drury, J. L., & Dembo M. (2001). Aspiration of Human Neutrophils: Effects of Shear Thinning and Cortical Dissipation. Biophys. J., 81, 3166–77.

Gerbal, F., Laurent V., Ott A., Carlier M., Chaikin P., & Prost J. (2000). Measurement of the Elasticity of the Actin Tail of Listeria monocytogenes. Eur. Biophys. J., 29, 134–40.

Herant, M., Marganski, W. A., & Dembo, M. (2003). The Mechanics of Neutrophils: Synthetic Modeling of Three Experiments. Biophys. J., 84, 3389–413.

Herant, M., Heinrich, V., & Dembo, M. (2005). Mechanics of Neutrophil Phagocytosis: Behavior of the Cortical Tension. J. Cell Sci., 118, 1789–97.

Herant, M., Heinrich, V., & Dembo, M. (2006). Mechanics of Neutrophil Phagocytosis: Experiments and Quantitative Models. J. Cell Sci., 119, in press.

Hartwig, J. H., & Shevlin, P. (1986). The Architecture of Actin Filament and the Ultrastructural Location of Axtin-binding Protein in the Periphery of Lung Macrophages. J. Cell Biol., 103, 1007–20.

Hill, T. L., & Kirschner, M. W. (1982). Subunit treadmilling of microtubules or actin in the presence of cellular barriers: possible conversion of chemical free energy into mechanical work. Proc. Nat. Acad. Sci., 79, 490–4.

Howard, J. 2001. Mechanics of Motor Proteins and the Cytoskeleton. Sunderland, MA: Sinauer Associates.

Kovar, D. R., & Pollard, T. D. (2004). Insertional Assembly of Actin Filament Barbed End in Association with Foweins Produces Piconewton Forces. Proc. Nat. Acad. Sci., 101, 14725– 14730.

Knupp, P., & Steinberg, S. 1994. Fundamentals of Grid Generation. Boca Raton, FL: CRC Press. Lee, E., Shelden, E. A., & Knecht, D. A. (1997). Changes in Actin Filament Organization during

Pseudopod Formation. Exp. Cell Res., 235, 295–9.

Masayuki, T., & Tanaka, T. (1991). Friction coefficient of polymer networks of gels. J. Chem. Phys., 95, 4613–9.

Mast, S. O. (1926). Structure, Movement, Locomotion, and Stimulation in Amoeba. J. Morphology and Physiology, 41, 347–425.

Mogilner, A., & Oster, G. (1996). Cell Motility Driven by Actin Polymerization. Biophys. J., 71, 3030–48.

Oster, G. F., & Perelson, A. S. (1987). The Physics of Cell Motility. J. Cell Sci. Suppl., 8, 35– 54.

Peskin, C. S., Odell, G. M., & Oster, G. F. (1993). Cellular Motions and Thermal Fluctuations: the Brownian Ratchet. Biophys. J., 65, 316–24.

Popper, K. 1968. The Logic of Scientific Discovery. New York, NY: Harper & Row.

Rash, P. J., & Williamson, D. L. (1990). On Shape-Preserving Interpolation and Semi-Lagrangian Transport. SIAM J Statist Computation, 11, 656–87.

Raucher, D., & Sheetz, M. P. (1999). Characteristics of a Membrane Reservoir Buffering Membrane Tension. Biophys. J., 77, 1992–2002.

Scheidegger, A. E. 1960. The Physics of Flow through Porous Media. New York: MacMillan. Temam, R. 1979. Navier-Stokes Equations Theory and Numerical Analysis. Studies in Mathematics

and Its Applications. Amsterdam: North-Holland.

Theriot, J. A., & Mitchison, T. J. (1992). Comparison of Actin and Cell Surface Dynamics in Motile Fibroblasts. J. Cell Biol., 118, 367–77.

Tokita, M., & Tanaha, T. (1991). Friction Coefficient of Polymer Networks of Gels. J. Chem. Phys., 95, 4613–19.

224M. Herant and M. Dembo

Watts R. G., & Howard, T. H. (1993). Mechanisms for Actin Reorganization in Chemotactic FactorActivated Polymorphonuclear Leukocytes. Blood, 81, 2750–7.

Zhelev, D. V, Alteraifi, A. M., & Hochmuth, R. M. (1996). F-actin Network Formation in Tethers and in Pseudopods Stimulated by Chemoattractant. Cell Motil. Cytoskeleton, 35, 331– 44.

Zhelev, D. V., Alteraifi, A. M., & Chodniewicz, D. (2004). Controlled Pscudopod Extension of Human Neutrophils Stimulated with Different Chemoatlractants. Biophys. J., 87, 688–95.