REU 1989: The Red Blood Cell Shape (M. Peterson)

The REU project for 1989 directed by Mark Peterson investigated the equilibrium shapes of fluid vesicles, that is, membrane bounded fluid cells. The red blood cell is a naturally occurring vesicle, and artificial vesicles can be made to form spontaneously in lipid-water mixtures. Their shapes are believed to minimize curvature energy at fixed volume V and area A. As first pointed out by Helfrich, this leads to the problem of minimizing, by choice of membrane shape M,

where lambda and mu are Lagrange multipliers, H is the mean curvature, kc is the bending modulus, c0 the spontaneous curvature (biasing H), and the integral goes over M, the membrane surface. c0 can also be interpreted as a Lagrange multiplier, fixing the average of H over the surface M.

If M is a surface of revolution, the Euler-Lagrange equation for this problem is a system of ODE's. Solutions include shapes which actually occur for red blood cells: the "normal" discocytic shape, and stomatocyte shapes (i.e., "cup" shapes). Three such equilibrium shapes are shown below (the rotational symmetry axis is horizontal). The left shape is the normal shape, the other two are stomatocytes.

We computed families of equilibrium shapes, which lie along hypersurfaces in the space of parameters (Lagrange multipliers). There are bifurcations. We also checked the infinitesimal stability of the shapes we found (using the method described by Peterson in J. Appl. Phys 57 (1985), 1739). Below is a typical picture, corresponding to c0*rho=-2, A=4*pi*rho^2, (rho arbitrary).

Note that as mu increases along the stable normal shape, one reaches a bifurcation beyond which no infinitesimally stable shape seems to exist. The unstable mode has the symmetry of the stomatocyte. This is the stomatocyte transition (actually observed in red blood cells). The instability initiates a shape change which ultimately leads to the stable stomatocyte shape, but not continuously through stable shapes. (Stable stomatocytes lie off the graph.)

The "phase diagram" for this problem turns out to be surprisingly complex. The paper by Udo Seifert, K. Berndl and R. Lipowsky, "Shape transformations of vesicles: Phase diagrams for spontaneous- curvature and bilayer-coupling models," Phys Rev A44 (1991), 1182-1202, which appeared soon after our work, largely superseded ours.

The student participants were:

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