240 Mechanical Resonances of Biological Cells / 1 3 : 3 



3. Gelatinous-Shell Model 



The static experiments used to measure interfacial tensions of nonmobile 

 or slowly moving cells could be interpreted in other ways. Some 

 involving ultracentrifugation may measure the tensile strength of the 

 cell membrane. Others, depending on the gravitational distortion of 

 cell shape, may actually be measuring the rigidity of an elastic outer 

 layer (or cortex) of the individual cell. In a like fashion, the optimum 

 frequencies, or resonances, observed in the ultrasonic destruction of single 

 cells in acavitating suspension can also be interpreted as due to resonant 

 vibrations of a rigid spherical cell immersed in, and filled with, an in- 

 compressible fluid. 



This rigid-shell model is very different from the interfacial-tension 

 model, in terms of both its mechanical structure and its biochemical 

 make-up. However, its predictions for distortions and resonances of 

 biological cells are very similar to those of the interfacial-tension model. 

 Indeed, there is no simple way to distinguish one from the other. 



The rigidity of the cell cortex is negligible compared to steel, glass, 

 or even wood. Rigidities are described by elastic moduli called 

 coefficients of rigidity or shear moduli, which are about 10 8 -10 10 dynes/cm 2 

 for solid objects. All protein gels have much smaller, but nonetheless 

 measurable, shear moduli in the range of 10 3 -10 5 dynes/cm 2 . Assuming 

 gelatinous properties for the outer layers of the single cell leads to pre- 

 dicted resonant frequencies in the ranges observed for protozoans and 

 erythrocytes. 



The rigid-shell model is considerably more complex than the inter- 

 facial-tension model. The analysis of the resonances of the rigid-shell 

 model is similar to that of closed rigid shells in air. The restriction of a 

 closed shell is important because most analyses of the vibrations of shells 

 and plates assume no extension of the midsurface of the shell, a condition 

 which cannot be met for closed shells. 



For vibrations of rigid shells with extension of the midsurface in air, 

 both the kinetic and potential energies are proportional to the shell 

 thickness h. Most of the modes occur at frequencies independent of h. 

 For the cell cortex, the liquid on both sides may move, as well as the cell 

 cortex. Accordingly, some of the resonant frequencies depend on the 

 effective thickness of the cortex h or, at any rate, on its ratio to the 

 effective cell radius a. This is shown by a detailed derivation. 



Rather than attempting to present the entire derivation, only the 

 results will be described. Two general types of motion of the shell are 

 considered, those which include radial motion as well as tangential 

 motion, and those involving tangential motion only. The latter are 

 simpler and will be described first. 



