DERIVATION OF FORMULA OF APPROXIMATE IDEALIZED HYPHAL CONTOUR AS BUILT-IN HYPHAL FITTING PROFILE

Received 13 October 2021 Accepted 15 November 2021 Available online 22 November 2021 Hypha consists of two regions; cap (apex) and cylindrical shaft (subapex and mature combined). The hyphalcap is the most critical part due to its dominant role in the hyphal-wall growth and morphogenesis. Just how the hyphal-wall growth is regulated in order to maintain its tubular shape has been the subject of much research for over 100 years. Here, we derived a formula of approximate idealized hyphal-contour based on gradients of secant lines joining a fixed coordinate at the starting hyphal-shaft to any coordinates on the contour. The formula is capable of being a control for experimental analysis in which it is not limited to one specific shape of the hyphal-like cell. Also, it potentially can play a role as built-in or ready-made hyphalfitting profile that “best fits” microscopic images of various actual hyphallike cells. In other words, given a microscopic image of hyphal-like cell, mycologists and microbiologists would not have to wonder about mathematical representation of its contour since the formula has prepared for it.


INTRODUCTION
Hypha is a single filament in a fungus consisting of two regions; cap (apex) and cylindrical shaft (subapex and mature combined). Hypha grows at its tip and such growth mode is called hyphal tip growth (see Figure 1 (Left)) (Davidson, 2007;Harold, 1997). The cap is the most critical part due to its dominant role in hyphal wall growth and morphogenesis. Penetratingphysical power and powerful tensile strength possessed by hypha is what affords it to exploit physically complex environments. This results in hypha has evolutionary advantage and ecological significance offering an important role in industries useful for human life as well as threaten food security, health and biodiversity (Read and Steinberg, 2008;Boswell, 2007;Flrdh, 2003;Harold, 1997;Davidson, 2007). However, complete details of its complex growth process remains a matter of discussion. Hence, just how hyphal tip growth is regulated in order to maintain its tubular shape has been the subject of much research for over 100 years.
Examining such cell forms has been developed from both biomechanical and geometrical models (Gierz and Bartnicki-Garcia, 2001;Goriely et al., 2005;Trinci and Saunders, 1977;Saunders and Trinci, 1970;Bartnicki-Garcia et al., 1989). The former focuses on balance of forces on the cell wall, where the wall is assumed to be a differentially thin elastic membrance. As for geometrical model, it offers the plausible fundamental description of possible hypha-like shapes. For example, geometrical model proposed by a previous researcher, where its key assumption is tip curvature, κ, is proportional to vesicle-incorporation for wall building process, n (Goriely et al., 2005). Also, tip monotonicity is stated as κp, for p ≥ 1. In following years, a researcher suggested that p > 0 also results in a monotonically increasing function of κ (Jaafar and Davidson, 2013). From what it follows, it serves as a motivation for this study to derive a formula of a two-dimensional approximate idealized hypha as control hypha allowing experimental hypha to be examined from the context of model proposed by (Goriely et al., 2005). The contour generated from the formula evolves naturally from trends of gradients of secant lines along the idealized contour. Tendency of points on idealized cap contour to depart from tangent lines drawn to the contour at those points is represented by exponentiality featured in the formula. The state of idealized hyphal-cap being curved is restricted by a constraint paving the way for tangency-comparison between actual cap contours and idealized cap contour.
The paper is structured as follows. In Section 2, we proposed geometrical setting of two dimensional idealized hypha and associated secant lines While, Section 3 suggests plausible proportionality of gradient-trends. Next is Section 4, which it analyzes range of tip monotonicity. Finally, Section 5 draws a conclusion of this study and its future study.

GEOMETRY OF APPROXIMATE IDEALIZED HYPHAL CON-TOUR
Half the approximate idealized hyphal contour denoted by r(x) shown in Figure 1 (Right). Let (b, r(b)) = (−π/2, π/2) be a fixed coordinate. The secants lines are drawn from the fixed coordinate to any coordinates on r(x) denoted by (x, r(x)), where, x ∈ [−2π, 0]. Hence, gradient is (1) Values of g of the secant lines on r(x) for −2π ≤ x ≤ 0 can be generally summarized by (2)

Case 2:
The radicand is positive.

DISCUSSION
The formula stated in (4) displays the two-dimensional hyphal-contour, where the starting hyphal-shaft contour is at x = −π/2 as proposed by (Goriely et al., 2005). Also, notice that the starting hyphal-shaft contour is actually length of hyphal-cap contour, Lc, here, Lc = π/2. Furthermore, radius of hyphal-cap contour, is denoted by Rc, that is, Rc = π/2 as well. While, the exponentiality, k x (k ∈ R + ), in (4) reflects the state of the hyphalcap contour being curved. In reality, Lc and Rc of actual hyphal contours are not necessarily equal to π/2. Differences in size and in shape depend upon species and other factors influencing the growth. Speaking of species, it was observed that re-placing Lc = π/2 with either Lc > π/2 or Lc < π/2 with a suitable k still ables for r(x) to produce different sizes and shapes of hyphal-like contours. This tells that the formula can also be used as control for experimental results of other hyphal-like cells such as pollen tube, root hair and neurons in animals. Besides that, the formula stated in (4) can also act as a built-in hyphal-fitting profile making it easy for mycologists and microbiologists to quickly find the "best fit" to microscopic images of various actual hyphal-like cells. This means that mycologists and microbiologists should not be concerned about finding a mathematical expression that "best fits' the contour of the given microscopic image. Instead, mycologists just have to adjust Lc and k to smooth the profile and improve its appearance. Therefore, (4) can be rewritten generally as In the light of the above, this profile illustrates relationship between Lc and k. Caution should be taken when plotting (4) in order to study the relationship due to graphical illusion. Such an illusion appears that the profile idealistically obeys its idealistic basic mathematical rules that state 0 ≤ r(x) < π/2 and r(x) = π/2 for −π/2 ≤ x < 0 and −2π ≤ x ≤ -π/2, respectively. The illusion is immediately crushed by examining the ´(x)| −2π≤x≤−π/2 , where r j (x) can never be equal to zero but closer to zero since r´(x) = −(1/2)k −x . Another important observation is just because the hyphal-cap contour satisfies its constraint, wˆ(x), does not necessarily mean the hyphal-shaft contour satisfy its basic mathematical rule. The hyphal-shaft contour violates its idealistic rule when ´(x) ≠ 0| −2π≤x≤−π/2 . This actually boils down to choosing a suitable k, whichs means r´(x) is closely approaching 0 as k increases.

CONCLUSION
The formula derived in this work refers to approximate idealized hyphallike contour, which is based on gradients of secant lines. Its capability of being a control for experimental analysis is not limited to one specific shape of the hyphal-like cell. Also, it potentially can play a role as built-in hyphal-fitting profile that "best fits" microscopic images of various actual hyphal-like cells.