SUPPORTING MATERIAL. Mathematical Study of the Role of Delta/Notch Lateral Inhibition during Primary Branching of Drosophila Trachea Development

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1 SUPPORTING MATERIAL Mathematical Study of the Role of Delta/Notch Lateral Inhibition during Primary Branching of Drosophila Trachea Development by Yoshiki Koizumi, Yoh Iwasa and Tsuyoshi Hirashima Supporting Material includes a supporting text and 4 figures. SUPPORTING TEXT Numerical Analysis of Comparison between Lateral Inhibition and Self-Inhibition To confirm the results of Eq.9 we calculated the nonlinear system, Eqs. 1 and 2. For the calculation each parameter set was randomly taken from the range (Methods for the parameter range). We here focus on the frequency when become larger than 1. The lateral inhibition amplifies the output differences between neighboring cells compared with self-inhibition (Fig. S1). This result supports the results of Eq.9 that 1 for j=0,1 and 2, and <1 for j = 3, 4. Also, these do not change even for the variance of each parameter c, d, and g (see Fig. S1(b)-(d)). For the reason that increasing the input gradient g decreases the frequency of 1 at j=2 (see Fig. S1(b)), Eq.9 is based on an assumption that there is a weak gradient of input signal between neighboring cells; thus, the numerical results at j=2 conflicts with the inequality of Eq. 9 when there is a strong gradient of input signal. Nevertheless, our numerical results are compatible with the inequality of Eq. 9 that the average frequency of 1 in the center of input (j=0 and1) is almost 1 for any parameter; in the distant region from the center (j=3 and 4), the frequency becomes very small. 1

2 Explicit Expression of Eq. 5 Eq. D2 Δδ j c is expressed as: From this Δδ j c < 0 because cos 2 jπ N ( ) cos 2 j +1 ( ( )π N) > 0 for j=0,, N/2. These results were performed by Mathematica 8.0 (Wolfram Research). Explicit Expression of Eqs. 6 Eq. E1a xˆ! d j = ˆx d is expressed as:.. d From this and Eq. D3, we obtained ˆ" c = which is negative., Eq. D1b xˆ! g j = δ j g is expressed as:. From this and Eq. D3, we obtained ˆ" x g j c =. Since tip cell selection is suppose to occur in the region j=0,, N/4, cos( 2π j N) 0, which results in ˆ" x g j c < 0. These results were performed by Mathematica

3 Comparison between Linear Gradient and Nonlinear Gradient of Input Numerical simulations were conducted with a linear input gradient to examine sensitivity of the robustness of tip cell selection to a different input profile. The linear input profile is as shown below (Fig.S2a): ( ){ exp( 2g) 1} L j = d j N, (S1) where d and g are the basal level of the input signal and the magnitude of input signal gradient, for j = N 2, N 2. We consider N = 8. Parameter range we examined was the same as used in the main text (See section F of the Methods). We calculated in the linear input gradient (Fig.S2b-d). At j=0, figures S2b-d show that the increases for the increase of the intensity of inhibitory regulation c. Thus, the lateral inhibitory regulation amplifies more the signal difference between neighboring cells that receive the highest input signal than self-inhibitory regulation does. At j=1, however, becomes less than 1 and gradually smaller for the increase of the intensity of inhibition (Fig. S2b-d). Note that for the case of nonlinear input gradient, becomes less than 1 at j=3 (Fig.4a-c). This is due to the fact that the input differences between any neighboring cells are uniform in the case of linear input, while the differences are non-uniform in the case of nonlinear input. For example, the input difference between the cell 0 and the cell 1 in the linear case is larger than that in the nonlinear case; in contrast, the input difference between the cell 1 and the cell 2 in the linear case is smaller than that in the nonlinear case (Fig. S2a). This causes the distinct intensity of lateral inhibition between the two different input profiles. In particular, the tip cell signal at j=1 x! is more suppressed by neighboring cells in the case of linear input compared with the case of nonlinear input. Thus, the tip cell signal difference between cell 1 and cell 2 in the case of linear input becomes smaller than that in the case of nonlinear input. As a result, the linear input gradient produces <, while the nonlinear input does not. For both the linear and the nonlinear gradient cases can perform more robust regulation than (Fig.4d-f and Fig.S2e-f). We also examined the robustness indexes against the fluctuation of d and that of g in the linear input gradient. Using the same condition as calculated in the Fig.3c-h of the main text, numerical simulations were performed (Fig.S3). We found that the robustness of tip cell selection in the lateral inhibitory regulation was always better than that in the self-inhibitory regulation on the criterion 1 Ω (Fig.S3c and f), although this was not the case at some c on the 3cells-threshold and 5cells-threshold (Fig.3a, b, d, and e). The reason for the distinction between the case of linear input and that of 3

4 nonlinear input comes from the fact that the tip cell signal difference between cell 1 and cell 2 in the lateral inhibitory regulation is not amplified compared with that in the self-inhibition as mentioned earlier (Fig.S2b-d). The amplification of the tip cell signal difference between neighboring cells is a key mechanism for the robust tip cell selection. Therefore, the robustness by the lateral inhibition sometimes becomes less than that by the self-inhibition in terms of the 3cells-threshold or the 5cells-threshold. Using a different criterion 1cell-threshold, we found that the robustness for the tip cell selection in the lateral inhibitory regulation with the linear and the nonlinear input gradient profile increases for the increase of c than that in the self-inhibitory regulation (Fig.S4). This means that the amplification of the input signal difference between neighboring cells in the case of linear input gradient contributes to choose one tip cell against the fluctuation of input parameters. 4

5 Figure S1 Figure S1: Lateral inhibitory regulation amplifies the between-cell difference of tip cell signal compared with self-inhibitory regulation. (a) Tip cell signal in the becomes larger than that in the in the center of high input signal (j=0 and 1) (: red; : blue). Such tendency becomes opposite in the region away from the center (j=3 and 4). (b)-(d) Frequency of 1 at each cell j for parameters g (b), c (c), and d (d). Note that the average frequency of 1 in the center region of input (j=0 and 1) is almost 1 for any parameter. In the distant region of the center (j=3 and 4), however, the frequency becomes very small. 5

6 Figure S2 Figure S2: Lateral inhibitory regulation enhances the robustness for the tip cell selection compared with the case of self-inhibitory regulation even when the input gradient has a linear profile. (a) Gradient profiles of linear (black) and nonlinear (white). (b)-(d) Dependence of on the parameters, c, d, and g. Note that becomes never less than 1 except at j=1. (e)-(g) Robustness for the tip cell selection against the fluctuation of c. Black bars show the case of and white bars show that of. The results for (e): 3cells-threshold and (f): 5cells-threshold are shown separately. For any variance the leads to more robust regulation compared with the case of. The sample number is about 2500 for each value. 6

7 Figure S3 Figure S3: Robustness indexes of tip cell selection to the intensity of inhibition in the case of the linear input. (a)-(c) Cases for the fluctuation of parameter d. For the increase of the intensity of inhibition c, the robustness for the tip cell selection increase in the (solid) than that in the (dotted) except at c = 1. (d)-(f) Cases for the fluctuation of parameter g. (d) The robustness index in the becomes less than that in the at some c. (c, f) The robustness of tip cell selection in the was always better than that in the. The number of sample is about 2500 for each value of c. 7

8 Figure S4 Figure S4: Evaluation of robustness of tip cell selection by 1cell-threshold. (a, b) Fluctuation of parameter d and (c, d) fluctuation of parameter g. (a, c): Linear profile; (b, d): nonlinear profile. The robustness for the tip cell selection increases in the for the increase of the intensity of inhibition c (solid), whereas it does not increase in the (dotted). 8