Table 3.

Equations of the growth model

^aThroughout these equations superscript t or t + δt is used to indicate the value of the parameter at time t or t + δt. Equation 1 expresses the Gaussian distribution of exocyst density used in the model, which is parameterized from the experimental observation for σ. In equation 2, the new area of an annular region at time t + δt is determined by calculating the amount of cell wall synthesis in the small time δt (∝S_i^t A_i^t δt) and converting it to a change in area using the parameter γ. The two forms of equation 3 express the method by which the new density of synthase at time t + δt is calculated and differ only by whether endocytosis is included (equation 3b) or is neglected (equation 3a). In both forms, the amount of synthase added to an annular region in time δt is ∝E_i^t A_i^t δt and is converted to units of S_i via the parameter ε. This amount is added to the preexisting amount of synthase and is then scaled down by the new area. In this way, the decrease in synthase density due to an increase in cell wall area is included. In equation 3b, if s is >s_endo, a fraction of synthase (ϕ) is removed per time step due to endocytosis. In equations 4, the new radii and arc lengths of annular regions are calculated assuming an isotropic mode of growth in which the radius and arc length of an annular region grow by the same factor. The parameters ε and γ do not affect the form of the hypha; they affect only the amount of growth per time step (see Materials and Methods). The very apex of the tip is represented by a small circular region. This region grows according to the same growth model as the annular regions except that it is described only by a radius and not by an arc length. After a set number of steps, the region is replaced by a combination of a new annular region and the circular region with its radius reset to its initial value.