Journal of General Microbiology (1977), 103, 343-248. Printed in Great Britain Tip Growth of Fungal Hyphae By A. P. J. TRINCI Department of Microbiology, Queen Elizabeth College, Campden Hill Road, London W8 7AH AND P. T. SAUNDERS Department of Mathematics, Queen Elizabeth College (Received 24 June 1977) 243 The shapes of tips of rapidly growing hyphae approximate more closely to half ellipsoids of revolution than to hemispheres. It is concluded that provided growth is isotropic the specific rate of wall area expansion will be more nearly proportional to the cotangent of the position angle than to its cosine. Certainly better qualitative fits to the observed data are given by cotangent rather than by cosine curves. INTRODUCTION The portion of a hyphal tip involved in extension is called the extension zone. In a hypha growing at a linear rate the shape of this zone, its length and the radius of the hypha at the base of the zone are all constant. These observations imply that both longitudinal and circumferential aspects of hyphal extension are patterned in some stable fashion. If a tip-growing filament has a hemispherical extension zone in which wall expansion is isotropic and extension zone ‘activities’ show radial symmetry, then the specific rate of area expansion will be proportional to the cosine of the angle (angle AOP in Fig. I) between the longitudinal axis of the filament and the point on the tip wall (De Wolff & Houwink, 1954; Green & King, 1966; Green, 1974). Although Green (1969, 1974), Bartnicki-Garcia (1973) and Burnett (1976) have suggested that it may be possible to apply the cosine hypothesis to hyphal extension, no critical analysis has been made of this po- tential application. This is surprising since a mathematical model which is valid for tip growth would be of considerable value, particularly in studies of hyphal morphogenesis (Da Riva Ricci & Kendrick, 1972). In this paper we consider the extent to which hyphae conform to the assumptions implicit in the cosine relationship and, in particular, we assess the importance of tip shape. We have measured the extension zones of several different hyphae and found that they are not in fact hemispherical but are closer to half ellipsoids of revolution. We then con- sidered what effect this has on Green & King’s result developed for the alga, Nitella, and found that for hyphae it implies that the specific rate of area expansion is approximately proportional to the cotangent of the position angle, rather than to the cosine. METHODS Spores on sterile microscope slides were overlaid with sterile cellophane (P.T. 300, British Cellophane) which in turn was overlaid with a thin (about I mm thick) layer of malt extract agar medium (Difco). The slide cultures were incubated in Petri dishes at 25 “C for 36 h. Hyphae were fixed in Helly’s fluid [5 % (w/v) mercuric chloride, 3 % (w/v) potassium dichromate and 0.18 % (w/v) formaldehyde], the tips of leading hyphae were photographed ( x 400 magnification)and measurements of their extension zones were made on enlargements (x 10) of the negatives using the method of Trinci & Halford (1975).