Mathematical modelling of morphogenesis in fungi: spatial organization of the gravitropic response in the mushroom stem of Coprinus cinereus

Abstract

The purpose of this work was to establish how the distribution of local curvatures changed during the mushroom stem gravitropic reaction and to suggest a suitable mathematical model based on these new data.

The gravitropic bending of base- and apex-pinned Coprinus cinereus (Fries) S. F. Gray stems was recorded on videotapes. The images were captured from the tapes after each 10 min, rotated by 45° and transformed into tables of changing co-ordinates of points for each stem. The non-linear regression of these points was performed using Legendre polynomials. From the resulting equations the patterns of changing local curvature for 50 subsections per stem during 400 min of gravitropic reaction were calculated.

It was observed that base-pinned stems first bent from the apex, but later the curvature of this part decreased, and in the late stages the apex became nearly completely straight again. Subsections, located about one third of stem length from the base determined the main part of the final curvature. The free basal part of the apex-pinned stems bent upward and after a certain bending time also began to straighten. However, this process started significantly later and was weaker. Bending of the subsections close to the pinned apex did not stop when they reached the vertical position, and the final angle of gravitropic curvature could exceed 180°.

Plotting various functions of local bending speed and its derivatives against each other and against local angle indicated that, if the hypothetical signal about reorientation arises in the apex, its propagation towards the base did not follow simple wave or simple diffusion laws. The importance of the local angle of all subsections both for signal origin and transmission was established and a signal transmission equation, involving local angle of each subsection, was derived. After creating a suitable program this partial differential equation was solved numerically. The generated shapes of the bending stem coincided in high degree with experimentally observed images.