Experimental approach and parameters for analysis

We explored a modified staining procedure to detect Bcd intensities in wt embryos (see

Materials and Methods for details). For our analysis, we used anti-Bcd and anti-Hb antibodies to simultaneously detect Bcd and Hb proteins in individual embryos (). High resolution digital images of double-stained embryos at early nuclear cycle 14 were captured, Bcd and Hb intensities measured and plotted against the anterior-posterior (A-P) position

*x* (). Throughout this work, Bcd intensities were captured within a linear range and expressed as raw data (unless otherwise noted), without any normalization or adjustment at either imaging or data processing steps. To minimize measurement errors, all images for a group of embryos were captured with identical settings in a single imaging cycle. In addition, we "spiked" embryos with those lacking Bcd (from

*bcd*^{E1} females) to specifically measure background intensities under identical experimental conditions. To facilitate our analysis, we measured the following values for each embryo: embryo length (

*L*), the Hb expression boundary (

*x*_{Hb}) which is the A-P position where Hb intensity is ½ maximal, the highest Bcd intensity (

*B*_{max}), Bcd intensity at the anterior (

*B*_{0}), and Bcd intensity at the Hb boundary position (

*B*_{xHb}). When describing a group of embryos, any value given refers to the mean unless noted otherwise; for example,

*x*_{Hb} of a group of embryos refers to the group's mean Hb expression boundary position.

Native Bcd profiles are reproducible in wt embryos

shows that Bcd intensity profiles in wt embryos are highly reproducible (see for Hb intensity profiles). This analysis represents the measurement of 28 double-stained wt embryos at early nuclear cycle 14. Bcd intensities were measured by scanning the nuclear layer of the dorsal part of individual embryos and plotted against normalized A-P position (

*x*/

*L*). The reproducibility of the Bcd profiles in these wt embryos is qualitatively evident. First, these profiles are visually less variable than those reported previously using stained wt embryos (

Gregor et al., 2007a;

Holloway et al., 2006;

Houchmandzadeh et al., 2002;

Spirov and Holloway, 2002). In addition, Bcd intensities from these embryos do not appear to have a higher variability than the data measured from individual nuclei of a single embryo (see

Supplemental Fig. 1C), suggesting that Bcd intensity variations among individual embryos at early nuclear cycle 14 are comparable to those between neighboring nuclei in single embryos. As shown in , Bcd intensities of random pairs of wt embryos or random pairs between wt and 1×-

*bcd* embryos both exhibit a linear relationship, further supporting the suggestion that the measured Bcd profiles in wt embryos are reproducible and intensities are linear to the

*bcd* gene dose (also see (

Gregor et al., 2007a)).

To quantify reproducibility of native Bcd profiles in wt embryos, we calculated the mean and standard deviation of Bcd intensities along the A-P position (also see

Supplemental Fig. 2 and additional discussions in

Supplemental Data). We plotted Bcd intensity noise (standard deviation divided by the mean) as a function of fractional embryo length (

*x*/

*L*). The raw Bcd intensity noise (red line, ) is ~10–20% for almost the entire A-P length of the embryos. The noise of background-subtracted Bcd intensities (blue line, ) remains low in the anterior half of the embryos (generally ~15–20%). These results are in contrast with previously reported studies using stained embryos (

Holloway et al., 2006;

Houchmandzadeh et al., 2002;

Spirov and Holloway, 2002) but are in agreement with the recently reported live-imaging study (

Gregor et al., 2007a). As shown in , Bcd intensity noise (blue line) exhibits a gradual increase toward the posterior, but even in this part of the embryo it remains <60%, a noise level lower than that detected by the live-imaging approach (

Gregor et al., 2007a). Bcd intensity variations among individual embryos and between neighboring nuclei of single embryos exhibit overall similar profiles (

Supplemental Fig. 1D). shows the effect of correcting background and measurement noise on Bcd intensity variations (see

Materials and Methods for details).

Bcd profiles are scaled with embryo length

The analysis described above indicates that Bcd intensity profiles in wt embryo as a function of normalized A-P position (

*x*/

*L*) are highly reproducible. It is well established that the Hb expression boundary

*x*_{Hb} is scaled with embryo length, i.e.,

*x*_{Hb} is correlated with

*L* (

*r* = 0.52,

*P* = 0.005; also see (

Houchmandzadeh et al., 2002)). However, how embryo length variations may affect Bcd gradient precision in embryos has not been fully characterized (

Gregor et al., 2007a). When discussing precision and scaling of the Bcd gradient, we pay particular attention to Bcd profile behaviors at and around

*x*_{Hb} because they will directly affect our interpretation of how precision and scaling of Hb expression is achieved. We reasoned that, if the Bcd gradient itself is scaled with embryo length, Bcd intensity variations as a function of normalized A-P position

*x*/

*L* should be lower than those measured as a function of absolute distance from the anterior (

*x* in µm), particularly at and around

*x*_{Hb}. In other words, scaling of the Bcd gradient is expected to "make" Bcd profiles in this region more precise when A-P position is normalized than without normalization. Our results shown in clearly demonstrate such a reduction in Bcd intensity variations at and around

*x*_{Hb}, supporting the notion that the Bcd gradient is scaled with embryo length.

We conducted a second test to further analyze the effect of embryo length variations on Bcd gradient precision. In this test, we divided the wt embryos into two groups according to their embryo length and analyzed their average Bcd intensity profiles. But before we discuss our Bcd data, we will use the Hb expression boundary *x*_{Hb} that is known to be scaled with embryo length to help illustrate how our analysis works. As shown in , the two average Hb intensity profiles (for large and small embryos) are different from each other when plotted as a function of absolute distance from the anterior (*x* in µm), with small embryos as a group having a shorter *x*_{Hb} than large embryos (*P* = 0.016, student's *t*-test). However, when these two Hb intensity profiles are expressed as a function of fractional embryo length *x*/*L* (), they converge and effectively eliminate *x*_{Hb} differences (*P* = 0.42). The observed convergence of the two average Hb intensity profiles is simply another way of illustrating the well-documented embryo length scaling of *x*_{Hb} in wt embryos. In a similar analysis for Bcd, inset shows that the average Bcd intensities for large and small embryos are significantly different from each other at and around *x*_{Hb} when plotted as a function of absolute A-P position *x* (*P* = 0.01 at *x*_{Hb}; see legend for *P*-values at surrounding locations). However, when these two curves are plotted as a function of normalized A-P position *x*/*L*, they converge near *x*_{Hb} ( inset) effectively eliminating Bcd intensity differences (*P* = 0.20 at *x*_{Hb}; see legend for *P*-values at surrounding locations). Together, these analyses demonstrate that the Bcd intensity profiles are scaled with embryo length to provide precise and scaled activator information for Hb activation in wt embryos.

Probing mechanisms of embryo length scaling of Bcd gradient

Our analysis of Bcd intensity profiles of large and small embryos () also provides critical insights into mechanisms of embryo length scaling. In particular, our results show that Bcd intensity at the anterior (*B*_{0}) is significantly higher in large embryos as a group than in small embryos (*P* = 0.024, student's *t*-test). To investigate the propagation of this positive correlation along the A-P length, we plotted the correlation coefficient between *B* and *L* (*r*_{B–L}) as a function of either *x* or *x*/*L*. As shown in , *r*_{B–L} peaks at *x* = ~200 µm (*r*_{B–L} = 0.66; *P* = 10^{−4}; see inset for a scatter plot between *B* and *L* at this position) and then drops gradually toward the posterior as a function of *x*. In the *x*/*L* plot (), however, *r*_{B–L} begins to drop almost immediately from the anterior, effectively attenuating the propagation of this anteriorly-originated positive correlation toward *x*_{Hb} and beyond (profiles of Spearman's rank correlation coefficient exhibit behaviors similar to those shown in , data not shown). inset shows a scatter plot between *B* and *L* at the peak position as identified from the *x*/*L* plot.

To test whether a positive

*B*_{0}-

*L* correlation might be sufficient to explain embryo length scaling of Bcd profiles, we performed simulation studies using the exponential decay Bcd profile

*B* =

*B*_{0} exp(−

*x*/λ), where λ is the length constant (

Houchmandzadeh et al., 2002). Our simulation results demonstrate two features expected of scaling when

*B*_{0} is correlated with

*L*, namely, 1) a reduced Bcd noise in

*x*/

*L* plot compared with that in

*x* plot (), and 2) the convergence of the average Bcd intensity curves for large and small embryos in the middle of the embryo (data not shown). Both of these properties are absent when

*B*_{0} and

*L* are uncorrelated ( and data not shown), demonstrating that a

*B*_{0}-

*L* correlation is sufficient to explain our observed scaling properties of the Bcd gradient.

Reduced Bcd reproducibility in *stau* embryos directly contributes to Hb variations

To further understand the molecular mechanisms of developmental precision and size scaling, we analyzed

*Drosophila* mutants that exhibit variable Hb expression patterns. The only

*Drosophila* mutant that has been reported to affect Hb precision is the maternal gene

*staufen* (

*stau*; (

Crauk and Dostatni, 2005;

Houchmandzadeh et al., 2002)). Hb boundary in embryos from homozygous

*stau*^{HL} females (referred to as

*stau* embryos) is almost twice as variable as in wt embryos (σ = 1.43% and 2.78% embryo length for wt and

*stau* embryos, respectively; see and inset for Hb intensity profiles in wt and

*stau* embryos, respectively). Unlike in wt embryos,

*x*_{Hb} and

*L* are no longer correlated in

*stau* embryos (

*r* = 0.043,

*P* = 0.83), suggesting a loss of Hb boundary scaling. To understand defects of

*stau* embryos, we measured Bcd gradient intensities in these embryos (). To ensure a direct comparison,

*stau* embryos were stained side-by-side with wt embryos and images taken in the same imaging cycle as wt embryos. Overall, Bcd intensities in these embryos exhibit a greater variability (green line, ) than in wt embryos (blue line). Bcd intensity noise is over 20% in the anterior half of

*stau* embryos and is increased dramatically toward the posterior. In addition, unlike in wt embryos, Bcd intensity variations in

*stau* embryos are not lower around the

*x*_{Hb} region when expressed as a normalized A-P position than without normalization ( inset), suggesting a loss of scaling of the Bcd gradient (see

Supplemental Fig. 3 and additional discussions in

Supplemental Data).

To directly determine whether increased Bcd variability is responsible for increased *x*_{Hb} variability in *stau* embryos, we grouped the embryos according to their normalized Hb boundary positions. We reasoned that, if Hb boundary variations in *stau* embryos are caused by Bcd intensity variations, embryos that have an anteriorly-shifted *x*_{Hb} as a group should cross Bcd thresholds at a more anterior position than embryos with a posteriorly-shifted *x*_{Hb}. and D show, respectively, the average Hb and Bcd intensity profiles of the two groups of *stau* embryos that have either an anteriorly- or posteriorly-shifted *x*_{Hb}. As shown in inset, the two average Bcd intensity curves cross thresholds at different A-P positions, with the anteriorly-shifted group crossing at more anterior positions (see legend for *P*-values). In a similar test for wt embryos (data not shown), the average Bcd intensity curve for embryos with a smaller (than mean) normalized *x*_{Hb} as a group does not cross thresholds at more anterior positions than the other group (see legend for *P*-values). These results suggest that increased Bcd intensity variability in *stau* embryos directly contributes to increased *x*_{Hb} variations.

Positional errors of the Bcd gradient and Hb precision

Our studies described thus far suggest that Bcd gradient precision is necessary for precise Hb expression. To determine whether the observed Bcd profile reproducibility is sufficient to account for Hb precision in wt embryos, we converted the measured Bcd intensity errors to positional errors σ

_{x}, i.e., errors in

*x* at which individual Bcd profiles cross given thresholds (

Gregor et al., 2007a). Our results show that σ

_{x} is ~3.5–4% embryo length around

*x*_{Hb}, which is more than twice the observed Hb boundary variations (1.4% embryo length; see for positional errors of Bcd and Hb). Even after Bcd intensity errors are corrected for measurement and background noise, positional errors remain higher for Bcd (~3–3.5% embryo length) than for Hb (generally <2% embryo length). These results suggest that a precise Bcd gradient, though necessary, is insufficient on its own to account for Hb precision (also see (

Reinitz, 2007)). We must stress here again that our Bcd intensities are not adjusted or normalized in any way except background subtraction (also see

Supplemental Data for additional discussions). Previous models proposed to explain Hb precision--assuming that Bcd profiles are noisy--all suggest the operation of additional factors (

Aegerter-Wilmsen et al., 2005;

Bergmann et al., 2007;

Houchmandzadeh et al., 2005;

Howard and Rein ten Wolde, 2005), but most of these efforts remain at a theoretical level, underscoring a need for experimental investigations.

To further understand mechanisms controlling Hb expression and precision, we focused our analysis on two parameters that directly describe Hb activation by Bcd in embryos. The first parameter is the Hill coefficient

*n*, which has a best fit of 5.1 ± 2.7 for wt embryos, a value that is in agreement with a recent estimate in embryos (

Gregor et al., 2007a) and with our biochemical studies (

Ma et al., 1996). The Hill coefficient

*n* depicts the steepness of the Hb boundary but

*n* variations have little or no effect on Hb boundary position. The second parameter is the Bcd level at the Hb boundary position

*B*_{xHb}, which is the measured Bcd threshold for Hb activation in individual embryos.

*B*_{xHb} has a variability of 24% (5.0 ± 1.2) in wt embryos, a variability higher than that of Bcd profiles. If our observed

*B*_{xHb} variations represent meaningful differences that are indicative of the properties of embryos (as opposed to measurement errors), then such variations must be incorporated into our analysis of the Bcd-Hb relationship.

We entertained the possibility that

*B*_{xHb} variations might actually be reflective of a correction mechanism(s) that further reduces positional errors of an already precise Bcd gradient. Our analysis reveled a positive correlation between

*B*_{xHb} and the length constant λ (

*r* = 0.44,

*P* = 0.018; see for a scatter plot of

*B*_{xHb} and λ), a correlation that is further improved for embryos at a more uniform developmental stage (

*r* = 0.57,

*P* = 0.014; Spearman's rank correlation coefficient

*r*_{S} = 0.49,

*P* = 0.038; see

Materials and Methods for details on developmental stage definitions). To determine whether the observed

*B*_{xHb}-λ correlation could reduce positional errors of the Bcd gradient, we conducted simulation studies using the exponential decay Bcd profile

*B* =

*B*_{0} exp(−

*x*/λ) and the Hill equation for Hb expression

*H* =

*B*^{n} / (

*B*^{n} +

*B*_{xHb}^{n}). All parameters in our simulations were based on experimentally-determined values, resulting in a mean

*x*_{Hb}/

*L* of 0.44 under all simulation conditions (see

Materials and Methods for details). In the absence of any correlations, normalized

*x*_{Hb} variability σ

_{x} was 6.1% embryo length (), which was reduced to 1.1% embryo length when both

*B*_{0}-

*L* and

*B*_{xHb}-λ correlations were applied (). Simulations with either

*B*_{0}-

*L* or

*B*_{xHb}-λ correlation alone () resulted in an

*x*_{Hb} variability of 5.4% and 2.3% embryo length, respectively. Furthermore, unlike in panel D, where Hb expression is both precise and scaled (i.e.,

*x*_{Hb} is correlated with

*L*;

*r* = 0.93), Hb expression is not scaled with embryo length in panel F (i.e.,

*x*_{Hb} is uncorrelated with

*L*;

*r* = −0.006). These simulation results show that a scaled and precise Hb boundary requires both

*B*_{0}-

*L* and

*B*_{xHb}-λ correlations. Further simulation studies of altering the parameters

*L* and

*B*_{xHb} revealed that a Bcd gradient based on the two observed properties is robust. In particular, target gene precision is insensitive to embryo length variations () or movements of the target boundary position (), features in full agreement with experimental data (

Crauk and Dostatni, 2005;

Houchmandzadeh et al., 2002).