Abstract
An equation for the dynamics of the vesicle supply center model of tip growth infungal hyphae is derived. For this we analytically prove the existence anduniqueness of a traveling wave solution which exhibits the experimentallyobserved behavior. The linearized dynamics around this solution is analyzed andwe conclude that all eigenmodes decay in time. Numerical calculation of thefirst eigenvalue gives a timescale in which small perturbations will die out.
Highlights
Tip growth is a process in which single-celled organisms grow roots or hairs, called hyphae, which lengthen at a constant speed, often achieving lengths much larger than their diameters
Experiments using markers on the cell wall [2] indicate that the wall expands orthogonally to its surface, with growth highly localized in the tip
Vesicles travel from the vesicle supply center (VSC) to the cell wall, producing growth of the cell wall orthogonal to the wall surface
Summary
Tip growth is a process in which single-celled organisms grow roots or hairs, called hyphae, which lengthen at a constant speed, often achieving lengths much larger than their diameters It is a mechanism by which organisms increase the ratio of surface area to volume probably in order to increase nutrient uptake. The concept of a vesicle supply center (VSC), first proposed by BartnickiGarcia et al [1], [3], lies at the basis for a whole hierarchy of mathematical models which attempt to explain tip growth It assumes that there is a point source in the tip which distributes cell wall material for the tip. Vesicles are emitted uniformly in all direction and travel in straight lines from the VSC to the cell wall where they immediately are absorbed resulting in orthogonal growth. Eggen [4] showed that with these assumptions, the dynamics of the model can be expressed in terms of the mean curvature of the surface, we continue with this idea
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