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superseded ours.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:

- Meredith Goldsmith, Yale '90
- Ted Levin, University of Massachusetts '90
- Julia Long, Harvey Mudd College '90
- Sally Relova, Mount Holyoke College '91
- Sebastian Schreiber, Boston University, '89
- Latha Venkataraman, Mount Holyoke College -> MIT '91

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