Journal Club September 2010: Modeling the Mechanics of Cellular Membranes

Constitutive relations, 2-D vs. 3-D. The starting point for modeling cellular membranes is the constitutive relations in 2-D space. It is important to set up the corresponding equations directly in two dimensions rather than to consider them as an asymptotic limit of three-dimensional relationships, like it is done in the shell theory. The main reason for the direct 2-D relations is that 3-D continuum approaches are not applicable to membranes whose thickness in on the order of magnitude of the dimension of a single molecule.

Solid-like membranes. Red blood cell membrane. Some cellular membranes (e.g., the membranes of the red blood cell or cochlear outer hair cell), where the plasma membrane is attached to the underlying 2-D cytoskeleton, can be treated as elastic or viscoelastic solids. The model of the red blood cell membrane can be considered as a gold standard in the analysis of cellular membranes, and it is based on nonlinear viscoelastic relationships, including the bending mode. Nonlinearity is critical for the description of the large deformations during cell's squeezing through narrow capillaries, and the bending component is important for the modeling of cell shape inside the blood flow. The red blood cell constitutive relations and their computational implementations developed earlier have broad applications today. One important application (for review, Pozrikidis [1]) is the simulation of the cell movement through the blood flow, including the interactions among red blood cells in the rouleau. Another application is the modeling of red blood cells in disease. As an important example, it has been shown that the mechanical properties change significantly in malaria. Suresh et al. [2] demonstrated how the red blood cell shear modulus, the main parameter of the cell's membrane mechanics, increases with different stages of the P. falciparum parasite maturation in the cell.

Liquid-like membranes. Membrane tethers. Pure plasma membranes have to be considered as special fluids whose primary properties are tension and bending. From the standpoint of general continuum mechanics, it has been shown that the strain energy functional (function) of liquid membranes is determined by the curvatures and local area change. In many cases, liquid membranes are considered as completely unstretchable, and the area preservation condition is introduced via a Lagrange multiplier. In 1973, Herfrich proposed a broadly used energy functional in the form of a function of the membrane curvature after deformation and the membrane original (spontaneous) curvature. In axisymmetric cases, the Euler-Lagrange equation for this functional can be explicitly derived in the form of a nonlinear 4-th order ODE and solved by using asymptotic and finite difference methods (e.g., Derenyi et al. [3]). Alternatively, finite element methods can be directly applied to the minimization of the bending energy functional. Membrane tethers (narrow tubes) are an interesting and general phenomenon in liquid membranes. Such tethers can form naturally for cell-cell communication or for the slowing down of moving cells (such as leukocytes). Pulling membrane tethers is also an effective method to probe the membrane local properties. There has been a significant effort to model this membrane phenomenon. An approach based on the thermodynamic balance of a tether system was previously applied to estimate the membrane bending modulus, membrane-cytoskeleton adhesion energy, and tension. The details of shape of a narrow tether are often unavailable in light microscopy. One approach to resolve this problem is to combine the tether pulling experiment with micropipette aspiration of the membrane on the opposite side of the cell and to compute the radius of the tether based on the length of the aspirated part of the membrane inside the pipette. This approach was successfully applied to the estimation of the membrane-cytoskeleton adhesion and membrane tension in red blood cells. Alternatively, membrane shape in the whole tether region can be computed from the analysis of the equilibrium state or steady state deformation of the tether (Derenyi et al. [3]). Such an approach was recently extended to model tether shape in different cells, taking into account particular arrangement of bonds between the membrane and the cellular cytoskeleton.

Electromechanical coupling in membranes.  Electromechanical coupling occurs naturally in biological membranes because they possess and are surrounded by a system of electric charges. One mode of such coupling bi-directionally relates the membrane curvature to an applied electric field. Physical models of this membrane phenomenon, called flexoelectricity, relate coupling between the membrane curvature and the electric field to a redistribution of the electrical charges associated with the bilayer (Harland el al. [4]). Another mode of electromechanical coupling in cellular membranes is associated with the recently discovered membrane protein prestin. The membranes (cells) containing this protein exhibit dimensional changes in response to the application of a transmembrane electric field, a phenomenon named electromotility. Similar to piezoelectric materials, these membranes generate (transfer) electric charges in response to membrane (cell) deformation. Originally, prestin was found in the cochlear outer hair cell, and it was shown that this protein is critical to the amplification and frequency selectivity of the mammalian ear. Lately, a number of originally nonmotile cells, such as human embryonic kidney cells, were transfected with prestin in order to use genetic manipulations in studies of electromotility. The membranes with prestin can be modeled by using mechanically linear (small deformations) and electrically nonlinear (saturating electric charge) piezoelectric-like constitutive relations. Prestin is probably the fastest known protein, and one of the main challenges for models of membranes with prestin is their applicability to high-frequency regimes (up to tens of kHz) of cell performance (for review, Spector et al. [5]).

References

[1] Pozrikidis, C. (Ed) Computational Hydrodynamics of Capsules and Biological Cells. Taylor&Francis, 2010.

[2] Suresh, S., Spatz, J., Mills, A., Dao, M., Lim, C.T., Beil, M., Seufferlein, T. Connections between single-cell biomechanics and human disease states: gastrointestinal cancer and malaria, Acta Biomater., 1, 15, 2005.

[3] Derenyi, I., Julicher, F., and Prost, J. Formation and interaction of membrane tubes, Phys. Rev. Letts, 88, 238101, 2002.

[4] Harland, B., Brownell, W.E., Spector, A.A., and Sun, S.X. Voltage-induced bending and electromechanical coupling in lipid bilayers, Phys. Rev. E., 81, Art. 031907, 2010.

[5] Spector, A.A., Deo, N., Grosh, K., Ratnanather, J.T., and Raphael, R.M. Electromechanical models of the outer hair cell composite membrane, J. Membr. Biol., 209, 135, 2006.