Allometry in Physarum plasmodium during free locomotion: size versus shape, speed and rhythm

Physarum plasmodium is a giant unicellular organism whose length can vary by more than three orders of magnitude. Using plasmodia ranging in size from 100 μm to 10 cm, we investigated the size dependency of their thickness distributions and locomotion speeds during free locomotion. (1) In the longitudinal direction, the organism is thickest close to the front, and decreases exponentially in thickness towards the rear. The slenderness ratio varies with body size according to a power law, such that large plasmodia are long and flat, whereas small plasmodia are short and thick. (2) The mean locomotion speed is proportional to the mean maximum thickness of the frontal part. By conducting a dimensional analysis, possible physical models are discussed. (3) The intrinsic period of the thickness oscillation, which is related to shuttle streaming (period 1-2 min), increases logarithmically with body size. (4) Various characteristics exhibit size-independent, long-period (20±10 min) oscillations, including speed, shape and intrinsic thickness oscillation period. These variations are closely coupled to formation of the entire cell shape, including undulation of thickness along the longitudinal axis and timing of branching of the frontal tip. Based on these experimental results and those reported previously, we propose a simple mathematical model for cell locomotion.