Revision of mathematical basis for the hyphoid curve

Filamentous microorganisms, for example, fungi, experience polarized growth that the elongated filamentous shape stems from intense growth activity occurred at the tip apex. Subsequently, examination on the geometry of the tip apex associated with its growth reveals relationship between physiological and geometrical parameters theoretically. Evidently, such revelation can be seen through contributions of the hyphoid model where it is one of insightful breakthroughs for community of mathematical mycology. The model proposes that geometry of the tip shape is determined by interplay between amount of vesicles and speed of moving Spitzenkorper expressed mathematically as hyphoid equation. Here, we provide additional theoretical relationship focusing on small-angle approximation and Sandwich theorem as an attempt to revise some steps of the derivation of the hyphoid equation. Choosing the small-angle approximation is plausible as the model deals with forms with size in micrometer. While, geometrical setting for proving Sandwich theorem fits geometrical setting for hyphal growth. We further examined the hyphal growth by considering outline shape of Spitzenkorper resembling either circle or horizontal ellipse located within the tip apex. This examination involves solving systems of nonlinear equations where we sought to find center of Spitzenkorper that fits within the tip apex maximally. Next, we proposed a hyphal growth model from the wall-elastic in which it was inspired by previous studies of elastic-wall profile of filamentous microorganisms. Also, we proposed actual coordinate where the wall elasticity collapses completely. Finally, we modeled an ideal filamentous microorganism excluding its growth mechanism and its tip range served as predictive tool for cell-profiling based on microscopic images for laboratory.

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