Modeling Fusome Formation in Drosophila Oogenesis

In this section, we introduce a theoretical model for fusome formation during Drosophila oogenesis using known biological features of the process, accompanied by a small number of assumptions. This minimal model, in coordination with the measured fusome volumes across stages, seeks a more coarse-grained, less stochastic view of fusome formation during oogenesis.
It was previously shown that the cystoblast contains the spectrosome, a fusomal precursor [25 (link), 36 (link)]. Therefore, if the volume of the spectrosome in the cystoblast after division to form the 2-cell cyst is given by v₀ and the fusome volume added at the first division is v₁, the volume fraction of cell 1 in the 2-cell cyst (f_1,2) is described by:
$$\begin{array}{c}\hfill f_{1,2}=\alpha =\frac{v_0+\beta v_1}{v_0+v_1},\end{array}$$
where α is defined as the volume fraction of fusome in cell 1 after the first division and β is defined as the fraction of the fusome volume added to the already existing cell at each division (Fig 3A). Extending this quantitative description to the next division, from a 2-cell to 4-cell cyst, we have two fusome plugs to add, each contributing v₂ to the total fusome volume (Fig 3B). Under the assumption that fusome plugs added at each division are the same (in this case, v₂), the volume fractions in the cells of the cyst, f_i,4, can be derived:
$$\begin{array}{c}\hfill f_{1,4}=\frac{v_0+\beta v_1+\beta v_2}{v_0+v_1+2v_2},\end{array}$$
$$\begin{array}{c}\hfill f_{2,4}=\frac{(1-\beta )v_1+\beta v_2}{v_0+v_1+2v_2},\end{array}$$
$$\begin{array}{c}\hfill f_{3,4}=f_{4,4}=\frac{(1-\beta )v_2}{v_0+v_1+2v_2}.\end{array}$$
These derivations can be repeated at each subsequent division, allowing for fusome volume fractions in each cell of any cyst size to be defined (Fig 3A–C).
Our experimental measurements revealed that the average volume fractions of cells 1 and 2, cells 3 and 4, cells 5 through 8, and cells 9 through 16 at the 16-cell stage are given in the ratio 2 : 1 : 1 : 1. One can therefore write the following equations for the volume fractions of these groups of cells and, along with Eq 1, solve for v₁, v₂, v₃, and v₄, the average volume of the fusome fragment being added between mother and daughter at each division, in terms of v₀:
$$\begin{array}{c}\hfill f_{1,16}+f_{2,16}=\frac{2}{5}=\frac{v_0+v_1+2\beta v_2+2\beta v_3+2\beta v_4}{v_0+v_1+2v_2+4v_3+8v_4},\end{array}$$
$$\begin{array}{c}\hfill f_{3,16}+f_{4,16}=\frac{1}{5}=\frac{2(1-\beta )v_2+2\beta v_3+2\beta v_4}{v_0+v_1+2v_2+4v_3+8v_4},\end{array}$$
$$\begin{array}{c}\hfill \sum _{i=5}^8{f}_{i,16}=\frac{1}{5}=\frac{4(1-\beta )v_3+4\beta v_4}{v_0+v_1+2v_2+4v_3+8v_4},\end{array}$$
$$\begin{array}{c}\hfill \sum _{i=9}^{16}{f}_{i,16}=\frac{1}{5}=\frac{8(1-\beta )v_4}{v_0+v_1+2v_2+4v_3+8v_4}.\end{array}$$
Solving this system of equations, along with Eq 1, allows for one to solve for v₁, v₂, v₃, and v₄ in terms of v₀. Inserting the derived relationships into equations for the volume fractions for each cell in the 2-, 4-, 8-, and 16-cell cysts yields volume fractions that can be compared with experimental measurements at the parameter values α = 0.7 and β = 0.5 heuristic values close to the parameter values found to minimize the total error, as shown above (Fig 3D and S1 Fig).