SUPPLEMENTARY DATA
Supplemental Text…………………………………………………………….p. 1-5
Supplemental Figure Legends………………………………………………..p. 6-8
Supplemental Figures………………………………………………………....p. 9-13
Supplemental Text
Mathematical argument of why the plateau levels of the amplitude are approximately
independent of L
Consider two embryos with DV axis lengths L1 and L2 and let Tsna be the nuclear concentration
threshold at which Dorsal activates sna. Therefore, at the end of nc 14, the absolute distance
from of the ventral midline of the sna border (denoted by x1 and x2 in each embryo) are
determined by the following equations:
(S1)
where A1, 1 and A2, 2 are the amplitudes and widths of the Dorsal gradient (assumed
Gaussian) in each embryo at the end of nc 14. From Figure 2D and 3C, we know that both the
Dorsal gradient width,  and the border of sna, x, satisfy the following linear relationships with
respect to the length of the DV axis, L:
,
(S2)
.
(S3)
Since both  and x obey a strict scaling relationship with L (Fig. 2D and Fig. 3C), we can assume
that
. Applying equations (2) and (3) to the embryos of DV length L1 and L2, we
get:
(S2’)
.
(S3’)
Finally, since the Dorsal gradient is not too steep at the location that determines the sna border,
we can substitute equations (2’) and then equation (3’) into equation (1) to get:
.
(S4)
Therefore, based on our experimental observations that the width of the Dorsal gradient and the
location of the sna border strictly scale, we can infer that the plateau level of the amplitude at the
end of nc 14 is approximately independent of L.
Details about the mathematical model in Figure 6
Here we discuss the details of the mathematical model described in Fig. 6. The model considers
the concentration of two species of Dorsal: a nuclear species [dln], and a freely diffusible
cytoplasmic species [dlc]. The dynamics of these species is given by the following equations:
(S5)
In these equations, kin and kout are the rates of import and export of Dorsal into a nucleus, D is
the diffusion coefficient of cytoplasmic Dorsal in the embryonic syncitium, and f is a space2
dependent function that bias the import of Dorsal into ventral nuclei. Therefore, f(x) is assumed
to be graded and takes higher values in ventral positions than in dorsal positions in the embryo
(Fig. 6A). The analysis of the model can be done entirely without determining f(x) explicitly, but
for plotting purposes we assumed that f has a Gaussian shape. As discussed below, this choice of
f does not affect any of our conclusions qualitatively.
We assumed that initially Dorsal was entirely cytoplasmic and imposed zero-flux boundary
conditions.
(S6)
where [dl0] is a constant.
The equations (S5) with initial conditions and boundary conditions (S6), can be solved
analytically at the steady state. The steady-state solution can be obtained as follows. First, we set
the left-hand side of the differential equations to zero. Adding the two steady-state equations, it
follows that,
(S7)
where the SS subscript denotes the steady-state concentration. Equation (S7) has the following
general solution:
(S8)
where C0 and C1 are integration constants to be determined. Applying zero-flux boundary
conditions (equation S6), we conclude that C1=0. Therefore, at the steady state, the nuclear
distribution of Dorsal is given by:
(S9)
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Equation (S9) shows that at the steady state, the nuclear gradient of Dorsal is proportional to f(x).
In order to determine C0 explicitly, we need to make additional assumptions. Integrating the sum
of nuclear and cytoplasmic Dorsal, and using the initial condition (equation S6) we have:
.
(S10)
Thus, the initial condition implies that the total levels of maternally-deposited Dorsal are
proportional to L at time t0. Since Dorsal is a maternal gene product, Dorsal is not transcribed in
the embryo. If we also assume that Dorsal degradation is negligible, we have that equation (S10)
applies for all times. This is the Conservation law in Fig. 6A. Now, we can use this Conservation
SS
law to obtain C0. Substituting the expressions of [dlSS
above into the
c ](x) and [dln ](x)
Conservation law (equation S10), we get:
This expression shows that L has an effect on the amplitude of the Dorsal gradient. Substituting
this into equation (S9), we get equation (1) in the text:
Note that f(x) determines the shape of the gradient, but its choice does not affect the
proportionality constant qualitatively since any gradient-like function has a similar integral (area
under the curve).
Finally, assume that f (x) can be written in relative coordinates, i.e., f =f (x/L). Consider
the change of variables, y = x / L. Then,
, and the integrand no longer
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depends on L. Writing this integral in terms of the new variable y and substituting into equation
(S9), we get that C0 is a constant and the Dorsal gradient obeys strict scaling:
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SUPPLEMENTARY FIGURE LEGENDS
Supplementary Fig. 1: The amplitude of the Dorsal gradient is largely uncorrelated with
the length of the DV axis. (A) Dorsal gradient amplitude (pixel intensity) was plotted against
the DV axis length for small 9.3, WT, large 2.1 and large 2.4 embryos. The 68% (dark shading)
and 95% (light shading) confidence intervals are shown.
Supplementary Fig. 2: Scaling behaviors are similar for wild-type embryos alone or when
considered in combination with large 2.1 embryos. Several replicates of wild-type (WT)
alone showed similar results to those achieved by combining WT with large 2.1. The main
difference is that due to a smaller size distribution the WT alone data contained more noise and
was less conclusive. The data shown here are from single experiments that were conducted
independently of the ones in the main Fig.s. (A) Dorsal gradient width was plotted against DV
axis length, L. (B-C) The position of the sna border was plotted against L (B) and Dorsal
gradient width (C). (D-E) The positions of the dorsal and ventral borders of vnd were plotted
against L (D) and Dorsal gradient width (E). (F-G) The positions of the dorsal and ventral
borders of ind were plotted against L (F) and Dorsal gradient width (G). Error bars for each
embryo are displayed although generally small. For all plots n is equal to the number of embryos
in each experiment, the R2 value is the square of the Pearson coefficient and the 68% (dark
shading) and 95% (light shading) confidence intervals are shown.
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Supplementary Fig. 3: The average number of nuclei per cross-section increases with DV
length while the number of ind expressing nuclei remains constant. (A-C) Cross-sections of
embryos stained with anti-H3 antibody were imaged and the number of nuclei was counted in
‘small’ (A), wild-type (B), and ‘large’ (C) embryos. The absolute length and average number of
nuclei of the semi-circumference/DV axis is shown. (D) The number of nuclei was plotted
against the DV axis length. The data is color coded as in Fig. 5: ‘small 9.3’ (blue), wild-type
(red), ‘large 2.4’ (green). (E-F) The width of ind in number of nuclei was calculated and plotted
against DV axis length (E) and Dorsal gradient width (F). Error bars for each embryo are
displayed.
Supplementary Fig. 4: The scaling behavior of the dorsal and ventral borders of vnd is
similar to wild-type in the several mutants that were tested. The scaling behavior of vnd was
analyzed in rho vn, wntD, DN-TKV, and ‘large 2.4’. (A-H) The positions of the dorsal and
ventral borders was plotted against DV axis length (L, left column) and Dorsal gradient width
(right column) for rho vn (A-B), wntD (C-D), DN-TKV (E-F), and large 2.4 (G-H). Error bars
for each embryo are displayed although generally small. For all plots n is equal to the number of
embryos in each experiment, and the R2 value is the square of the Pearson coefficient and the
68% (dark shading) and 95% (light shading) confidence intervals are shown.
Supplementary Fig. 5: The average DV axis length in rho vn and large 2.4 is larger than in
WT embryos. (A) Box plots of the DV axis length for wild-type (WT; n = 18) and various
mutant lines: rho vn (n = 43), wntD ( n = 20), DN-TKV (n = 25) and ‘large 2.4’ ( n = 38). Twosample T-test showed that the mean of the DV axis length for rho vn (p = 2.0 x 10-16) and ‘large
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2.4’ (p = 1.2 x 10-14) is statistically different from WT. The mean of the DV axis length for
wntD (p = 0.05) and DN-TKV (p = 0.06) were not statistically different from WT. (B) Box plots
of the absolute Dorsal gradient width (m) in WT and rho vn. Two-sample T-test show a
significant difference in the means (p = 4.7 x 10-4). (C) Box plots of the relative Dorsal gradient
width normalized to DV axis length in WT and rho vn, no significant difference in the means (p
= 0.37).
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