Continuum Mechanical Model of Leukocytes During Protopod Formation

Abstract

A new continuum mechanical theory for protopod extension in leukocytes is developed. Protopod formation is an active process which is the basis for amoeboid displacement on substrates. Leukocytes may form protopods both when adhering to a substrate and when freely suspended in plasma. Therefore the required energy is derived from the cell itself. Protopods are depleted of granules and other organelles, they have a fine fibrillar ultrastructure, and they are covered by a cell membrane. They grow at about 5 μm/min until they reach a length of 4–5 μm. A period of protopod retraction follows during which granules re-enter via the protopod base by Brownian motion. Micropipette experiments have indicated that the protoplasm in the leukocyte has viscoelastic properties, whereas the protopod is stiffer and shows elastic behavior. We propose a continuum theory based on the polymerization of the actin matrix in the cell which results in gelation with a preferred orientation. It is triggered by influx of Ca++ across local regions of the cell membrane and the polymerization occurs along an interface at the base of the polymerized protopod. As cytoplasm passes through the interface it is subject both to a volumetric strain due to exclusion of granules and a shear strain due to alignment of actin molecules. The polymerization provides an active force leading to projection of the protopod and cell deformation. The base of the protopod rests on the unpolymerized cytoplasm along the interface. As the external plasma medium and the cell membrane, if it is not stretched taut, offer little resistance, the projection of the protopod proceeds outward with simultaneous unfolding of the membrane. On the other hand, in osmotically swollen cells the membrane offers considerable resistance as it is under tension and the actin polymerization proceeds inward. A general set of equations are formulated and some special solutions are discussed.