A continuum theory of Allen's frontal contraction model of amoeboid pseudopodium extension

A continuum theory is proposed for the chemically controlled cytoplasmic streaming observed in pseudopodium extension in Chaos Carolinensis. Amoeboid cytoplasm is assumed to consist of submicroscopic contractile fibers bathed by viscous fluid. The fiber constituent models the actin-like and myosin-like contractile machinery known to be present in Chaos Carolinensis cytoplasm. A "trigger chemical", produced at the pseudopodium tip, moves by diffusion in, and convection by, the viscous fluid, and causes the contractile fibers to contract in their own length. The contracting fibers, attached at the tip and running continuously back toward the amoeba cell body, pull the fluid constituent of the cytoplasm forward and ultimately crosslink to form the outer gel tube of the advancing pseudopodium. That is, streaming cytoplasm is modeled as a two constituent porous medium, with the fluid constituent free to flow through a porous matrix of oriented (contractile fiber) rods, while the matrix of rods itself moves as the fibers contract, with fiber contraction controlled by a trigger chemical born by the fluid constituent. According to this theory, in the region behind the advancing pseudopodium tip, the contractile fiber rods move forward toward the tip faster than the fluid constituent. The hydrostatic pressure in the fluid therefore increases from the cell body toward the tip (Just the opposite from flow driven by pressure excess generated in the cell body). The excess of hydrostatic pressure above ambient built up at the tip provides the force to roll out the advancing pseudopodium tip. The cell membrane plays no active mechanical role. The mathematical transcription makes a precise theory of R. D. Allen's "frontal (or fountain zone) contraction model". The general system of coupled, non-linear, partial differential equations is solved for its simplest non-trivial special case, that of a steady-state motion, as seen from a coordinate system attached to the advancing tip. Solutions exist, and, for each distinct forward speed (which is left to the discretion of the amoeba) the solution is unique. The theory predicts both upper and lower bounds for possible pseudopodium lengths.