Probing the limits to positional information

Thomas Gregor,a−d David W. Tank,a−c Eric F. Wieschaus, b,d and William Bialeka,c

aJoseph Henry Laboratories of Physics, bDepartment of Molecular Biology,cLewis-Sigler Institute for Integrative Genomics, and dHoward Hughes Medical Institute,Princeton University, Princeton, New Jersey 08544 USA

(Dated: October 14, 2006)

The reproducibility and precision of biological patterning is limited by the accuracy with whichconcentration profiles of morphogen molecules can be established and read out by their targets.We consider four measures of precision for the Bicoid morphogen in the Drosophila embryo: Theconcentration differences that distinguish neighboring cells, the limits set by the random arrivalof Bcd molecules at their targets (which depends on the absolute concentration), the noise inreadout of Bcd by the activation of Hunchback, and the reproducibility of [Bcd] at correspondingpositions in multiple embryos. We show, through a combination of different experiments, that allof these quantities are ∼ 10%. This agreement among different measures of accuracy indicatesthat the embryo is not faced with noisy input signals and readout mechanisms; rather the systemexerts precise control over absolute concentrations and responds reliably to small concentrationdifferences, approaching the limits set by basic physical principles.

INTRODUCTION

The macroscopic structural patterns of multicellularorganisms have their origins in spatial patterns of mor-phogen molecules (Wolpert 1969, Lawrence 1992, Ger-hart & Kirschner 1997). Translating the qualitativepicture of morphogen gradients into quantitative termsraises several difficulties. First is the problem of preci-sion (Figure 1): Neighboring cells often adopt distinctfates, but the signals that drive these decisions involvevery small differences in morphogen concentration, andthese must be discriminated against the inevitable back-ground of random noise. The second problem is repro-ducibility: If cells “know” their location based on theconcentration of particular morphogens, then generatingreproducible final patterns requires either that the abso-lute concentrations of these molecules be reproduciblefrom embryo to embryo or that there exist mechanismswhich achieve a robust output despite variable input sig-nals.

These problems of precision and reproducibility arepotentially relevant to all biochemical and genetic net-works, and analogous problems of noise (de Vries 1957;Barlow 1981; Bialek 1987, 2002) and robustness (LeMas-son et al 1993; Abbott & LeMasson 1993; Goldman etal 2001) have long been explored in neural systems. Onemight even hope that maximizing reliability in the pres-ence of noise or maximizing robustness to variation inprotein copy number could provide “design principles”with which to understand or predict the structure anddynamics of these networks. Here we address these issuesin the context of the initial events in fruit fly develop-ment.

The primary determinant of patterning along theanterior-posterior axis in the fly Drosophila melanogasteris the gradient of Bicoid (Bcd), which is established bymaternal placement of bcd mRNA at the anterior end ofthe embryo (Driever & Nusslein-Volhard 1988a,b); Bcd

acts as a transcription factor, regulating the expressionof hunchback (hb) and other downstream genes (Struhlet al 1989; Rivera-Pomar et al 1995; Gao & Finkelstein1998). This cascade of events generates a spatial pat-tern so precise that neighboring nuclei have readily dis-tinguishable levels of expression for several genes [as re-viewed by Gergen et al (1986)], and these patterns arereproducible from embryo to embryo (Crauk & Dostatni2005; Holloway et al 2006).

In trying to quantify the precision and reproducibilityof the initial events in morphogenesis, we can ask fourconceptually distinct questions:

• If cells make decisions based on the concentrationof Bcd alone, how accurately must they “measure”this concentration to be sure that neighboring cellsreach reliably distinguishable decisions?

• What is the smallest concentration difference thatcan be measured reliably, given the inevitable noisethat results from random arrival of individual Bcdmolecules at their target sites along the genome?

• What level of precision does the system actuallyachieve, for example in the transformation fromBcd to Hb?

• How reproducible are the absolute Bcd concentra-tions at corresponding locations in different em-bryos?

The answer to the first question just depends on the spa-tial profile of Bicoid concentration (Houchmandzadeh etal 2002). To answer the second question about the phys-ical limits to precision we need to know the absolute con-centration of Bcd in nuclei, and we measure this usingthe Bcd-GFP fusion constructs described in a compan-ion paper (Gregor et al 2006). To answer the third ques-tion we characterize directly the input/output relationbetween Bcd and Hb protein levels in each nucleus of

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FIG. 1 Schematic of the readout problem for the Bicoid gradient. At left, the conventional picture. A smooth gradient ofBcd concentration is translated into a sharp boundary of Hb expression because Bcd acts as a cooperative activator of thehb gene. Although intended as a sketch, the different curves have been drawn to reflect what is known about the scales onwhich both the Bcd and Hb concentrations vary. Note that neighboring cells along the anterior-posterior axis experience Bcdconcentrations that are very similar (differing by ∼ 10%, as explained in the text), yet the resulting levels of Hb expressionare very different. At right, we consider a larger number of cells in the mid-embryo region where Hb expression switches fromhigh to low values. From direct experiments on simpler systems we know that, even when the concentrations of transcriptionfactors are fixed, the resulting levels of gene expression will fluctuate (Elowitz et al 2002; Raser & O’Shea 2004), and thereare physical limits to how much this noise can be reduced (Bialek & Setayeshgar 2005, 2006). If the noise is low, such thata scatter plot of Hb expression vs Bcd concentration is relatively tight, then the qualitative picture of a sharp Hb expressionboundary is perturbed only slightly. If the noise is large, so that there is considerable scatter in the relationship betweenBcd and Hb measured for individual cells, then the sharp Hb expression boundary will exist only on average, and not alongindividual rows in individual embryos.

individual embryos. Finally, to answer the fourth ques-tion we make absolute concentration measurements onmany embryos, as well as using more classical antibodystaining methods. In the end, we find that all of thesequestions have the same answer: ∼ 10% accuracy in theBcd concentration. While this number is interesting, itis the agreement among the four different notions of pre-cision that we find most striking.

A number of previous groups have argued that thecentral problem in thinking quantitatively about theearly events in development is to understand how an or-ganism makes precise patterns given sloppy initial dataand noisy readout mechanisms (von Dassow et al 2000;Houchmandzadeh et al 2002; Spirov et al 2003; Jaegeret al 2004; Eldar et al 2004; Martinez Arias & Hayward,2006; Holloway et al 2006). In contrast, our results leadus to the problem of understanding how extreme preci-sion and reproducibility–down to the limits set by basicphysical principles–are achieved in the very first steps ofpattern formation.

RESULTS

Setting the scale

Two to three hours after fertilization of the egg, ad-jacent cells have adopted distinct fates, as reflected intheir patterns of gene expression. At this stage, the em-bryo is ∼ 500 µm long and neighboring nuclei are sepa-rated by ∆x ∼ 10 µm. Distinct fates in neighboring cellstherefore means that they acquire positional informationwith an accuracy of ∼ 2% along the anterior-posterioraxis. Measurements of Bcd concentration by immunos-taining reveal an approximately exponential decay alongthis axis, c(x) = c0 exp(−x/λ), with a length constantλ ∼ 100 µm (Houchmandzadeh et al 2002). Neighboringnuclei, at locations x and x + ∆x, thus experience Bcdconcentrations which differ by

∆c(x) =∣∣∣∣dc(x)

dx

∣∣∣∣∆x =1λ

c(x)∆x. (1)

The exponential form of the Bcd profile implies that thefractional concentration difference between neighbors isconstant along the length of the embryo,

∆c(x)c(x)

=∆x

λ∼ 0.1. (2)

2

To make decisions that reliably distinguish individualnuclei from their neighbors using the Bcd morphogenalone therefore would require each nucleus to “measure”the Bcd concentration with an accuracy of ∼ 10%.

Absolute concentrations

The difficulty of achieving precise and reproduciblyfunctioning biochemical networks is determined in partby the absolute concentration of the relevant molecules:for sufficiently small concentrations, the randomness ofindividual molecular events must set a limit to precision(Berg & Purcell 1977; Bialek & Setayeshgar 2005, 2006).For a quantitative discussion, we thus would like to knowthe absolute concentration of Bcd molecules. Since Bcdis a transcription factor, what matters is the concentra-tion in the nuclei of the forming cells. A variety of exper-iments on Bcd (Ma et al 1996; Burz et al 1998; Zhao etal 2002) and other transcription factors (Ptashne 1992;Pedone et al 1996; Winston et al 1999) suggest that theyare functional in the nanoMolar range, but to our knowl-edge there exist no direct in vivo measurements in theDrosophila embryo.

In Figure 2A we show an optical section through a liveDrosophila embryo that expresses a fusion of the Bcdprotein with the green fluorescent protein (GFP), as de-scribed in a companion to this paper (Gregor et al 2006).There we established that this Bcd-GFP fusion quanti-tatively reproduces the functions of the native Bcd. Inscanning two-photon microscope images we identify in-dividual nuclei to measure the mean fluorescence inten-sity in each nucleus, which should be proportional to theprotein concentration. To establish the constant of pro-portionality we bathe the embryo in a solution of purifiedGFP with known concentration and thus compare fluo-rescence levels of the same moiety along the same opticalpath (see Methods).

Some of the observed fluorescence is contributed bymolecules other than the Bcd-GFP, and we test for thisbackground by imaging wild type embryos under exactlythe same conditions. As shown in Figure 2B, this back-ground is almost spatially constant and essentially equalto the level seen in the Bcd-GFP flies at the posteriorpole, consistent with the idea that the Bcd concentrationis nearly zero at this point.

Figure 2B shows the concentration of Bcd-GFP in nu-clei as a function of their position along the anterior-posterior axis. The maximal concentration near the an-terior pole, corrected for background, is cmax = 55 ±3 nM, while the concentration in nuclei near the mid-point of the embryo, near the threshold for activation ofhb expression (at a position x/L ∼ 48% from the anteriorpole), is c = 8 ± 1 nM. This is remarkably close to thedisassociation constants measured in vitro for binding ofBcd to its target sequences in the hb enhancer (Ma et al1996; Burz et al 1998; Zhao et al 2002).

FIG. 2 Absolute concentration of Bcd. A: Scanning two-photon microscope image of a Drosophila embryo express-ing a Bcd-GFP fusion protein (Gregor et al 2006); scale bar50µm. The embryo is bathed in a solution of GFP with con-centration 36 nM. We identify individual nuclei and estimatethe mean Bcd-GFP concentration by the ratio of fluorescenceintensity to this standard. B: Apparent Bcd-GFP concentra-tions in each visible nucleus plotted vs. anterior-posteriorposition x (reference line in A) in units of the egg length L;red and blue points are dorsal and ventral, respectively. Re-peating the same experiments on wild type flies which do notexpress GFP, we find a background fluorescence level shownby the black points with error bars (standard deviation acrossfour embryos). In the inset we subtract the mean backgroundlevel to give our best estimate of the actual Bcd-GFP concen-tration in the nuclei near the midpoint of the embryo. Pointswith error bars show the nominal background, now at zeroon average.

Physical limits to precision

Our interest in the precision of the readout mecha-nism for the Bcd gradient is heightened by the theo-retical difficulty of achieving precision on the ∼ 10%level. To begin, we note that 1 nM corresponds to0.6 molecules/µm3. Thus the observed concentration ofBcd in nuclei near the midpoint of the embryo is c =4.8 ± 0.6 molecules/µm3, which corresponds to 690 to-tal molecules in the nucleus [with diameter d ∼ 6.25 µm(Gregor et al 2006)] during nuclear cycle 14. A ten per-cent difference in concentration thus amounts to changesof ∼ 70 molecules.

Berg and Purcell (1977) emphasized, in the context of

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bacterial chemotaxis, that concentration measurementsare limited by fundamental physical noise sources thatderive from the random character of diffusion and bind-ing at the level of single molecules. This noise is deter-mined not by the total number of available molecules,but by the dynamics of the their random arrival at theirtarget locations. Consider a receptor of linear size a, andassume that the receptor occupancy is integrated for atime T . Berg and Purcell argued that the precision ofconcentration measurements is limited to

δc

c∼ 1√

DacT, (3)

where c is the concentration of the molecule to whichthe system is responding and D is its diffusion constantin the solution surrounding the receptor. Recent workshows that the Berg-Purcell result really is a lower limitto the noise level (Bialek & Setayeshgar 2005, 2006): thecomplexities of the kinetics describing the interaction ofthe receptor with the signaling molecule just add extranoise, but cannot reduce the effective noise level belowthat in Equation (3). These theoretical results encourageus to apply this formula to understand the sensitivity ofcells not just to external chemical signals (as in chemo-taxis) but also to internal signals, including morphogenssuch as Bcd.

The measurements above show that the total concen-tration of Bcd in nuclei is c = 4.8 ± 0.6 molecules/µm3

near the point where the “decision” is made to activateHb. In a companion to this paper we show that Bcddiffuses slowly through the dense cytoplasm surround-ing the nuclei with a diffusion constant D < 1 µm2/s(Gregor et al 2006), which is similar to that observed inbacterial cells (Elowitz et al 1999), and we take this asa reasonable estimate of the effective diffusion constantfor Bcd in the nucleus (see Methods for more details onthe estimates of c and D).

Receptor sites for eukaryotic transcription factors are∼ 10 base pair segments of DNA with linear dimensionsa ∼ 3 nm. Multiple sites might make the effective sizeslightly larger, but wrapping of the DNA around thenucleosome could effectively convert a longer sequence ofsites into a two dimensional patch of sites with the samelinear dimensions, and we know from the Berg-Purcellanalysis and its generalization that such a patch acts asone effective site in setting the limiting noise level.

The remaining parameter, which is unknown, is theamount of time T over which the system averages indetermining the response to the Bcd gradient; the longerthe averaging time the lower the effective noise level.One way to express our results, then, is to ask for theminimum averaging time required to reach a precision ofδc/c ∼ 10%. With the parameters listed above,

δc

c∼ [DacT ]−1/2

=[(1 µm2/s)(3 nm)(4.8 /µm3)T

]−1/2

FIG. 3 Hb vs. Bcd concentrations from fixed and stainedembryos. A: Scanning confocal microscope image of aDrosophila embryo in early nuclear cycle 14, stained for DNA(blue), Hb (red) and Bcd (green); scale bar 50µm. Inset(28× 28µm2) shows how DNA staining allows for automaticdetection of nuclei (see Methods). B: Scatter plot of Hbvs Bcd immunofluorescent staining levels from 1299 identi-fied nuclei in a single embryo. C: Scatter plot of Hb vs Bcdconcentration from a total of 13,366 nuclei in 9 embryos, nor-malized (see Methods). Data from the single embryo in B arehighlighted.

∼ 0.1, (4)

implies that this minimum time is T ∼ 115 min, or nearlytwo hours. This is almost the entire time available fordevelopment from fertilization up to gastrulation, andit seems implausible that downstream gene expressionlevels reflect an average of local Bcd concentrations overthis long time; given the enormous changes in local Bcdconcentration during the course of each nuclear cycle(Gregor et al 2006), it is not even clear that there aremechanisms which could achieve this integration.

We emphasize that our discussion ignores all noisesources other than the fundamental physical process ofrandom molecular arrivals at the relevant binding sites;additional noise sources would necessitate even longeraveraging times. Already the minimum time requiredto push the physical limits down to the ∼ 10% level isimplausible given the pace of developmental events.

Input/output relations and noise

The fact that neighboring cells can generate distinctpatterns of gene expression does not mean that any single

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step in the readout of the primary morphogen gradientsachieves this level of precision. Indeed, the discussionin the previous section makes clear that any simple pic-ture in which the expression levels of downstream genesprovides a “measurement” of the Bcd concentration insingle nuclei will run up against fundamental physicallimits if we ask for precision sufficient to distinguish re-liably among neighboring nuclei. To test the plausibilityof these ideas more directly, we focus on the transfor-mation from Bcd to Hb, one of the first steps in thegeneration of anterior-posterior pattern.

In Figure 3A we show confocal microscope images ofa Drosophila embryo fixed during nuclear cycle 14 andimmunostained for DNA, Bcd and Hb; the fluorescencepeaks of the different labels are sufficiently distinct thatwe can obtain independent images of the three stains.The DNA images allow us to locate automatically thecenters and outlines of the nuclei (see Methods). In atypical image of this sort we identify ∼ 1200 nuclei; man-ual inspection shows that misidentifications occur at lessthan the 1% level, and these are easily corrected. Giventhese outlines we can measure the average intensity ofBcd and Hb staining in each nucleus (Figure 3B).

It is conventional to assume that immunofluorescentstaining intensity is proportional to protein concentra-tion; more precisely, that in each pixel the observed in-tensity I is proportional to the concentration c plus somenon-specific background, so that I = Ac + B, where Aand B are constant in a single image. Recently we havebeen able to test this assumption quantitatively usingflies that express a Bcd-GFP fusion protein (Gregor etal 2006), showing that the patterns of fluorescence fromGFP, anti-GFP and anti-Bcd antibodies all are consis-tent with one another as expected if staining is relatedto concentration through this linear model. With thislinearity, a single image provides more than 1000 pointson the scatter plot of Hb expression level vs Bcd concen-tration, as in Figure 3C.

Scatter plots as in Figure 3 contain information bothabout the mean “input/output” relation between Bcdand Hb and about the precision or reliability of this re-sponse. Indeed, we can think of these data as the gener-alization to multicellular, eukaryotic systems of the in-put/output scatter plots measured for engineered regu-latory elements in bacteria [e.g. Figure 3B of Rosenfeldet al (2005)]. To analyze these data we discretize theBcd axis into bins, grouping together nuclei which havevery similar levels of staining for Bcd; within each binwe compute the mean and variance of the Hb intensity.We normalize the input/output relation by measuringthe Hb level in units of its maximal mean response, andthe Bcd level in units of the level which generates (onaverage) half maximal Hb.

Mean input/output relations between Bcd and Hb areshown for nine individual embryos in Figure 4A. We seethat the results from different embryos are very similar;differences among embryos are smaller than the outputnoise (see Methods for discussion of normalization across

embryos). Pooling the results from all embryos yields aninput/output relation that fits well to the Hill relation,

Hb = HbmaxBcdn

Bcdn + Bcdn1/2

. (5)

We find that the best fit is with n = 5. This is con-sistent with the idea that Hb transcription is activatedby cooperative binding of effectively five Bcd molecules,as expected from the identification of seven Bcd bindingsites in the hb promoter (Struhl et al 1989; Driever &Nusslein-Volhard 1989).

In Figure 4B we show the standard deviation in Hblevels as a function of the apparent Bcd concentration.We see that the fractional output fluctuations are below10% when the activator Bcd is at high concentration,similar to what has been found in direct measurementson artificially constructed regulatory modules (Elowitzet al 2002; Raser & O’Shea 2004). If we think of the Hbexpression level as a readout of the Bcd gradient, then wecan convert the output noise in Hb levels into an equiva-lent level of input noise in the Bcd concentration. This isthe same transformation as ‘referring noise to the input’in specifying the performance of an amplifier or sensor,and the mathematical transformation is the same as forthe propagation of errors: we ask what level of error δcin Bcd concentration would generate the observed levelof variance in Hb expression,

σ2Hb(Bcd) =

∣∣∣∣ d[Hb]d[Bcd]

∣∣∣∣2 (δc)2, (6)

or

δc

c= σHb(Bcd)

∣∣∣∣ d[Hb]d ln[Bcd]

∣∣∣∣−1

. (7)

It is this “equivalent fractional noise level,” shown inFigure 4C, that cannot fall below the physical limit setby Equations (3) and (4). For individual embryos wefind a minimum value of δc/c ∼ 0.1 near c = c1/2.

It should be emphasized that all of the noise we ob-serve could in principle result from our measurements,and not from any intrinsic unreliability of the underly-ing molecular mechanisms. In particular, because theinput/output relation is very steep, small errors in mea-suring the Bcd concentration will lead to a large apparentvariance of the Hb output. In separate experiments (seeMethods) we estimate the component of measurementnoise which arises in the imaging process. Subtractingthis instrumental variance from the inferred concentra-tion noise results in values of δc/c ∼ 0.1 on average (cir-cle with error bars in Figure 4C). The true noise levelcould be even lower, since we have no way of correctingfor nucleus-to-nucleus variability in the staining process.

Before proceeding it is important to emphasize thelimitations of our analysis. We have treated the relation-ship between Bcd and Hb as if there were no other factorsinvolved. In the extreme one could imagine (although

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FIG. 4 Input/output relations and noise. A: Mean input/output relations for 9 embryos. Curves show the mean level of Hbexpression as a function of the Bcd concentration, where we use a logarithmic axis to provide a clearer view of the steep,sigmoidal nonlinearity. Points and error bars show, respectively, the mean Hb level and standard deviation of the output noisefor one of the embryos. Inset shows mean Hb output (points) and standard errors of the mean (error bars) when data from allembryos are pooled. The mean response is consistent with the Hill relationship, Equation (5), with n = 5, corresponding to amodel in which five Bcd molecules bind cooperatively to activate Hb expression (red line). In comparison, Hill relations withn = 3 or n = 7 provide substantially poorer fits to the data (green lines). B: Standard deviations of Hb levels for nuclei withgiven Bcd levels. C: Translating the output noise of (B) into an equivalent input noise, following Equation (7). Blue dots aredata from 9 embryos, green line with error bars is an estimate of the noise in our measurements (see Methods), and red circleswith error bars are results after correcting for measurement noise. D: Correlation function of Hb output noise, normalized byoutput noise variance, as a function of distance r measured in units of the mean spacing ` between neighboring nuclei. Linesare results for four individual embryos, points and error bars are the mean and standard deviation of these curves. We havechecked that the dominant sources of measurement noise are uncorrelated between neighboring nuclei. The large differencebetween r = 0 and r = ` arises largely from this measurement noise. Inset shows the same data on a logarithmic scale, witha fit to an exponential decay C ∝ exp(−r/ξ); the correlation length ξ/` = 5± 1.

we know this is not true) that both Bcd and Hb con-centrations vary with anterior-posterior position in theembryo, but are not related causally. In fact, if we lookalong the dorsal-ventral axis, there are systematic varia-tions in Bcd concentration (as can be seen in Figure 3),and the Hb concentrations are correlated with these vari-ations, suggesting that Bcd and Hb really are linked toeach other rather than to some other anterior-posteriorposition signal. It has been suggested, however, that hbexpression may be responding to signals in addition toBcd (Howard & ten Wolde 2005; Houchmandzadeh etal 2005; McHale et al 2006). If these signals ultimatelyare driven by the local Bcd concentration itself, then

it remains sensible to say that the Hunchback concen-tration provides a readout of Bcd concentration with anaccuracy of ∼ 10%. If, on the other hand, additional sig-nals are not correlated with the local Bcd concentration,then one might expect that collapsing our descriptiononto just the Bcd and Hb concentrations would treatother variables as an extrinsic source of noise; in thiscase the true reliability of the transformation from Bcdto Hb could be even better than what we observe.

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Noise reduction by spatial averaging?

The observed precision of ∼ 10% is difficult to rec-oncile with the physical limits [Equation (4)] given theavailable averaging time. If the precision cannot be in-creased to the observed levels by averaging over time,perhaps the embryo can achieve some averaging overspace: If the Hb level in one nucleus reflects the averageBcd levels in its N neighbors, the limiting noise level inEquation (4) should decrease by a factor of

√N .

If we imagine that communication among nuclei is me-diated by diffusion of a protein with diffusion constantcomparable to that of Bcd itself, then in a time T it willcover a radius r ∼

√4DT and hence an area A ∼ 4πDT .

But at cycle 14 the nuclei form an approximately regularlattice of triangles with side ` ∼ 8.5 µm, so the area Acontains

N ∼ 8π√3

DT

`2(8)

nuclei. Putting all the factors together (with D ∼1 µm2/s from above), we find that in just four min-utes it should be possible to average over roughly 50nuclei. Since averaging over time and averaging over nu-clei have the same effect on the noise level–reducing thenoise in proportion to the square root of time or number–averaging over fifty nuclei for four minutes is the same aseach nucleus acting independently but averaging for twohundred minutes. More generally, with communicationamong nuclei the physical limit becomes

δc

c∼ 1√

DacTN=

[ √3

8πac

]1/2`

DT(9)

∼ 20 sT

. (10)

Thus 10% precision is possible with mechanisms that in-tegrate for only ∼ 200 s, or ∼ 3 minutes, rather thanhours. In particular, it would be possible to make mea-surements with this precision over the course of a singlenuclear cycle.

If each nucleus makes independent decisions in re-sponse to the local Bcd concentration, then noise in theHb levels of individual nuclei should be independent. Onthe other hand, if Hb expression reflects an average overthe nuclei in a neighborhood, then noise levels necessar-ily become correlated within this neighborhood. Goingback to our original images of Bcd and Hb levels, wecan ask how the Hb level in each nucleus differs fromthe average (along the input/output relation of Figure4A) given its Bcd level, and we can compute the cor-relation function for this array of Hb noise fluctuations(see Methods). The results, shown in Figure 4D, reveala component with a correlation length ξ = 5± 1 nuclei,as predicted if averaging occurs on the scale required tosuppress noise.

Reproducibility in live embryos

The results of Figure 4 indicate that individual em-bryos can “read” the profile of Bcd concentration withan accuracy of ∼ 10%. A particular value of the Bcdconcentration thus has a precise meaning within eachembryo, raising the question of whether this meaning isinvariant from embryo to embryo. Such a scenario wouldrequire control mechanisms to insure reproducibility ofthe absolute copy numbers of Bcd and other relevantgene products. The alternative is that the spatial profilesof Bcd vary from embryo to embryo, but other mecha-nisms allow for a robust response to this variable input.A number of groups have argued for the latter scenario(von Dassow et al 2000; Houchmandzadeh et al 2002;Spirov et al 2003; Jaeger et al 2004; Howard & ten Wolde2005; Holloway et al 2006). In contrast, the similarityof Bcd/Hb input/output relations across embryos (Fig-ure 4A) suggests that reproducible outputs result fromreproducible inputs.

To measure the reproducibility of the Bcd gradient,we use live imaging of the Bcd-GFP fusion construct, asin Figure 2. To minimize variations in imaging condi-tions, we collect several embryos that are approximatelysynchronized and mount them together in a scanningtwo-photon microscope. Nucleus-by-nucleus profiles ofthe Bcd concentration during the first minutes of nu-clear cycle 14 are shown for 15 embryos in Figure 5A(see Methods). Qualitatively it is clear that these pro-files are very similar across all embryos. We emphasizethat these comparisons require no scaling or separate cal-ibration of images for each embryo; one can compare rawdata, or with one global calibration (as in Figure 2) wecan report these data in absolute concentration units.

We quantify the variability of Bcd levels across em-bryos by measuring the standard deviation of concen-tration across nuclei at similar locations in different em-bryos, and expressing this as a fraction of the mean. Theresults, shown in Figure 5B, are consistent with repro-ducibility at the 10−20% level across the entire anteriorhalf of the embryo, with variability gradually rising inthe posterior half where Bcd concentrations are muchlower. Thus the reproducibility of the Bcd profile acrossembryos is close to precision with which it can be readout within individual embryos.

The average concentration profile c(x) defines a map-ping from position to concentration; the basic idea ofpositional information is that this mapping can be in-verted so that we (and the embryo!) can “read” theposition by measuring concentration. But then if a cellat position x “sees” a Bcd concentration that deviatesslightly from the average appropriate to that location,c(x) = c(x) + δc(x), then the readout will produce anerror δx in the position signal defined by

c(x + δx) = c(x) + δc(x) (11)

⇒ δx ≈ δc(x)[dc(x)dx

]−1

. (12)

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] nuc(n

M)

FIG. 5 Reproducibility of the Bcd profile in live embryos.A: Bcd-GFP profiles of 15 embryos. Each dot represents theaverage concentration in a single nucleus at the mid-sagittalplane of the embryo (on average 70 nuclei per embryo). Allnuclei from all embryos are binned in 50 bins, over which themean and standard deviation were computed (black pointswith error bars). Scale at left shows raw fluorescence inten-sity, and at right we show concentration in nM, with back-ground subtracted, as in Figure 2. B: For each bin fromA, standard deviations divided by the mean as a function offractional egg length (blue); errorbars are computed by boot-strapping with 8 embryos. Grey and black lines show esti-mated contributions to measurement noise (see Methods). C:Variability of Bcd profiles translated into an effective rms er-ror σ(x) in positional readout, as in Equation (13); error barsfrom bootstrapping. Green circles are obtained by correctingfor measurement noise.

Through this relation we can convert the measured stan-dard deviations or rms errors δc(x) in the concentrationprofiles into an effective rms error σ(x) in positional in-formation,

σ(x) = δc(x)

∣∣∣∣∣dc(x)dx

∣∣∣∣∣−1

. (13)

This is equivalent to drawing a threshold concentrationθ and marking the locations xθ at which the individualBcd profiles cross this threshold; σ(x) is the standarddeviation of xθ when the threshold is chosen so that themean of xθ is equal to x. We find (Figure 5C) that theBcd profiles are sufficiently reproducible that near themiddle of the embryo it should be possible to read out

positional information with an accuracy of ∼ 2% of theembryo length, close to the level required to specify thelocation of individual cell nuclei.

We emphasize once again that what we characterizehere as variability still could result from imperfectionsin our measurements (see Methods for details). In Fig-ure 5B we show the summed contributions from noise inimaging and from errors in locating the nuclei, as well aseparate contribution from variations in the location ofthe nuclei relative to our plane of focus. Another sourceof variation is imperfect synchronization of the embryos;during the course of nuclear cycle 14 the Bcd concen-tration in nuclei is dropping systematically at a rate of∼ 2%/min as a result of nuclear swelling (Gregor et al2006), so if we compare embryos that differ by just a fewminutes in the time since initiation of this cycle we makea significant contribution to the apparent variability ofthe concentration. There also is a significant variationof Bcd concentration along the dorsal-ventral axis (ascan be seen in Figure 3, for example), and so variationsin the orientation of the embryo will lead to apparentvariability of the Bcd concentration measured in nucleiat the rim of the mid-saggital plane. The conservativeconclusion is that nuclear Bcd concentration profiles areat least as reproducible as our measurements, which arein the range of 10− 20%. In Fig 5C we correct for thosesources of measurement error that we have been ableto quantify, and we find that the resulting reproducibil-ity translates into specifying position with a reliability∼ 1− 2% of the embryo length.

It is interesting to note that the absolute variabilityin the concentration profile is largest near the anteriorpole. Although some of this effect could result from vari-ations in the imaging geometry near the embryo poles,it also could arise if the degree of localization of theBcd mRNA varies in addition to variations in the to-tal mRNA copy number (see Methods, where we explorethis idea in a simple model). Functionally, this meansthat measurements of Bcd alone cannot provide repro-ducible positional information to cells within the anterior10− 20% of the embryo. Consistent with this view, pat-terning genes with boundaries in the anterior head regiondepend substantially on the torso terminal system anddo not appear to be as dependent on Bcd (Furriols &Casanova 2003).

Quantifying reproducibility via antibody staining

Previous work, which quantified the Bcd profiles us-ing fluorescent antibody staining (Houchmandzadeh etal 2002), concluded that these profiles are quite variablefrom embryo to embryo, in contrast to our results inFigure 5. We argue here that the discrepancy arises be-cause of the specific normalization procedure adopted inthe earlier work, and that with a different approach tothe data analysis the two experiments (along with a newset of data on immunostained embryos) are completely

8

0 0.5 1

0

0.5

1[B

cd] re

lativ

e

x/L

A

0 0.5 1

0

0.5

1

[Bcd

] rela

tive

x/L

B

0 0.2 0.4 0.6 0.8

0.1

0.2

0.3

0.4

0.5

x/L

δ[B

cd]/[

Bcd

]

C

0.45 0.5 0.550

0.05

0.1

σ(x)

/L

x/L

D

FIG. 6 Reproducibility of the Bcd profile in fixed and stainedembryos. Upper panels: Data on Bcd concentration fromHouchmandzadeh et al (2002). A: Normalization based onmaximum and minimum values of staining intensity. B: Nor-malization to minimize χ2, as in Equation (14). Note thatthese are the same raw data, but with the normalization tominimize χ2 the profiles appear much more reproducible, es-pecially near the midpoint of the anterior-posterior axis. Thisis quantified in C, where we show the standard deviation ofthe concentration divided by the mean, as in Figure 5B, forthe min/max normalization (blue) and the normalization tominimize χ2 (green). D: Equivalent root-mean-square errorin translating morphogen profiles into positional information,from Equation (13). Data on Bcd from Houchmandzadeh etal (2002), with min/max normalization (blue) and normal-ization to minimize χ2 (green); data on Bcd (red) and Hb(cyan) from our experiments (see Methods).

consistent.

As discussed above, the fluorescence intensity at eachpoint in an immunofluorescence image is related to theconcentration through I(x) = Anc(x) + Bn, where An

and Bn are scale factors and backgrounds that are dif-ferent in each embryo n. There is no independent mea-surement of these parameters, and so we have to chosethem in such as way as to make meaningful comparisonsacross images of different embryos. Houchmandzadehet al (2002) set these parameters for each embryo sothat the mean concentration of the 20 points with high-est staining intensity would be equal to one, and simi-larly the mean concentration of the 20 points with loweststaining intensity would be equal to zero. This is equiv-alent to the hypothesis that the peak concentration ofBcd is perfectly reproducible from embryo to embryo.

If we suspect that profiles in fact are reproducible, wecan assign to each embryo the values of An and Bn whichresult in each profile being as similar as possible to themean. We will measure similarity by the mean square

deviation between profiles, and so we want to minimize

χ2 =N∑

n=1

∫dx

∣∣∣∣∣In(x)− [Anc(x) + Bn]

∣∣∣∣∣2

, (14)

where c(x) is the average concentration profile. Thechoice of scaling and background for each embryo de-fines an estimate of the concentration profile cn(x) =[In(x)−Bn]/An, and then c(x) is the mean of these pro-files across all the N embryos in our ensemble. Reana-lyzing the data of Houchmandzadeh et al (2002) in thisway produces Bcd profiles that are substantially more re-producible (Figure 6B vs 6A), down to the ∼ 10% levelfound in the live imaging experiments (Figure 6C).

It should be emphasized that the difference betweenFigures 6A and B is not just a mathematical issue. In onecase (A) we interpret the data assuming that the peakconcentration is fixed, and this ‘anchoring’ of the peakdrives us to the conclusion that the overall profile is quitevariable, especially near the mid-point of the embryo. Inthe other case (B) we allow that the peak concentrationcan fluctuate, or more precisely that there is nothingspecial about the peak relative to any other point alongthe profile; this allows us to find an interpretation of thedata in which the overall profile is more reproducible.

We have collected a new set of data from 47 embryoswhich were fixed during early cycle 14 and stained forboth Bcd and Hb; processing a large group of embryos atthe same time and imaging them side-by-side, we min-imized spurious sources of variability. We confirm the10% reproducibility of the Bcd profiles, and find thatσ(x) is even slightly smaller than in the earlier data,consistent with the live imaging results. Using the samemethods to analyze the Hb profiles, we find that thereproducibility σ(x) inferred from Bcd and Hb are al-most identical in the mid-embryo region, as shown bythe red and cyan curves in Figure 6D. We conclude that,properly analyzed, the measurements of Bcd in fixed andstained embryos give results consistent with imaging ofBcd-GFP in live embryos. Further, the reproducibility ofthe Hb profiles is explained by the reproducibility of theBcd input profile, at least across the range of conditionsconsidered here.

DISCUSSION

The development of multicellular organisms such asDrosophila is both precise and reproducible. Under-standing the origin of precise and reproducible behavior,in development and in other biological processes, is fun-damentally a quantitative question. We can distinguishtwo broad classes of ideas (Schrodinger 1944). In oneview, each of the individual steps involved in the processis noisy and variable, and this biological variability issuppressed only through averaging over many elementsor through some collective property of the whole net-work of elements. In the other view, each step in the

9

process has been tuned to enhance its reliability, per-haps down to some fundamental physical limits. Thesevery different views lead to different questions and toentirely different languages for discussing the results ofexperiments.

Our goal in this paper has been to locate the initialstages of Drosophila development on the continuum be-tween the ‘precisionist’ view and the ‘noisy input, ro-bust output’ view. To this end we have measured theabsolute concentration of Bcd proteins, and used thesemeasurements to estimate the physical limits to preci-sion that arise from random arrival of these molecules attheir targets. We then measured the input/output re-lation between Bcd and Hb, and found that Hb expres-sion provides a readout of the Bcd concentration withbetter than 10% accuracy, very close to the physicallimit. The mean input/output relation is reproduciblefrom embryo to embryo, and direct measurements of theBcd concentration profiles demonstrates that these tooare reproducible from embryo to embryo at the ∼ 10%level. Thus, the primary morphogen gradient is estab-lished with high precision, and it is transduced with highprecision.

Several points which arise along the path to our mainresults are worth noting. First, our analysis of theBcd/Hb input/output relations is similar in spirit tomeasurements of noise in gene expression that have beendone in unicellular organisms (Elowitz et al 2002; Raser& O’Shea 2004; Rosenfeld et al 2005). The morphogengradients in early embryos provide a naturally occur-ring range of transcription factor concentrations to whichcells respond, and the embryo itself provides an exper-imental “chamber” in which many factors that wouldbe considered extrinsic to the regulatory process in uni-cellular organisms are controlled. Perhaps surprisingly,coupling classical antibody staining methods with quan-titative image analysis makes it possible for us to makeprogress on the characterization of noise in the poten-tially more complex metazoan context. This approachshould be more widely applicable.

A second point concerns the analysis of immunoflu-orescence data. Although there are many reasons whyantibody staining might not provide a quantitative indi-cator of protein concentration, our results [see also Gre-gor et al (2006)] indicate that the fluorescence intensityfrom fixed and stained embryos indeed is proportionalto the concentration, and that with proper normaliza-tion such experiments can reveal precision at the 10%level or better. Different approaches to the analysis ofthese data can lead to very different conclusions, how-ever, and it is important to link the assumptions of suchanalyses to their corresponding hypotheses about the un-derlying mechanisms. Thus, although one might haveexpected the peaks of morphogen profiles to be highlyreproducible, in fact these peaks vary in absolute am-plitude; in terms of fractional accuracy, the peak con-centration is not privileged relative to other features ofthe profile. As discussed in the Methods, this could re-

sult from variability in the degree of localization of thematernal mRNA.

A third point concerns the matching of the differentmeasures of precision and reproducibility. We have seenthat, near its point of half-maximal activation, the ex-pression level of hb provides a readout of Bcd concen-tration with better than 10% accuracy. At the sametime, the reproducibility of the Bcd profile from em-bryo to embryo, and from one cycle of nuclear divisionto the next within one embryo (Gregor et al 2006), isalso at the ∼ 10% level. Importantly, these differentmeasures of precision and reproducibility must be deter-mined by very different mechanisms. Our results suggestthat these mechanisms are matched so that no singlestep dominates the overall noise level. For the readout,there is a clear physical limit which may set the scalefor all steps. This limiting noise level is sufficient to pro-vide reliable discrimination between neighboring nuclei,in effect providing sufficient positional information forthe system to specify each “pixel” of the final pattern.

A fourth point concerns the problem of scaling. Pre-vious work has shown that the Bcd profile scales to com-pensate for the large changes in embryo length across re-lated species of flies (Gregor et al 2005), but evidence forscaling across individuals within a species has been elu-sive, perhaps because the relevant differences are small.We find that the Bcd profile is sufficiently reproduciblethat it can specify position along the anterior-posterioraxis within 1 − 2% when we express position in unitsrelative to the length of the embryo (Figures 5C & 6D).But embryos, for example in our ensemble of 15 thatprovide the data for Figure 5, have a standard devia-tion of lengths δLrms/L = 4.1%. Even if the Bcd profilewere perfectly reproducible as concentration vs positionin microns, this would mean that knowledge of relativeposition would be uncertain by 4%, more than what wesee. This suggests that the Bcd profile exhibits some de-gree of scaling to compensate for length differences. Newexperiments will be required to test this more directly.

A fifth point concerns the possibility of spatial aver-aging in order to increase the readout precision to theobserved levels. Our results suggest that if nearby nucleicommunicate through a diffusible “messenger,” so thatthe Hb concentration in one nucleus reflects an averageover Bcd concentrations in the neighborhood that canbe spanned by the messenger, then we can reconcile theobserved precision with the theoretically derived phys-ical limit. In the simplest model, the messenger couldbe Hb itself, since in the blastoderm stages the proteinis free to diffuse between nuclei and hence the Hb pro-tein concentration in one nucleus could reflect the Bcd-dependent mRNA translation levels of many neighboringnuclei. This model predicts that precision will depend onthe local density of nuclei, and hence will be degradedin earlier nuclear cycles unless there are compensatingchanges in integration time. Such averaging mechanismsmight be expected to smooth the spatial patterns of geneexpression, which seems opposite to the goal of morpho-

10

genesis; the fact that Hb can activate its own expres-sion (Margolis et al 1995) may provide a compensatingsharpening of the output profile. There is a theoreti-cally interesting tradeoff between suppressing noise andblurring of the pattern, with self-activation shifting thebalance.

At a conceptual level our results on Drosophila devel-opment have much in common with a stream of resultson the precision of signaling and processing in other bi-ological systems. There is a direct analogy between theapproach to the physical limits in the Bcd/Hb readoutand the ability of the visual system to count single pho-tons (Rieke and Baylor 1998; Bialek 2002). In both casesthe reliability of the whole process is such that the ran-domness of individual molecular events dominates thereliability of the macroscopic output. An even betteranalogy may be to bacterial chemotaxis, where the re-sponse to small concentration gradients is so sensitivethat it approaches the limits set by random arrival ofmolecules at the cell’s surface receptors (Berg & Purcell1977). There are several examples in which the relia-bility of neural processing reaches such limits (de Vries1957, Barlow 1981, Bialek 1987), and it is attractive tothink that developmental decision making operates witha comparable degree of reliability. More concretely, theapproach to physical limits has led to successful predic-tions regarding the mechanisms of neural signal process-ing, and parallel considerations should generate inter-esting predictions regarding the dynamics of signal pro-cessing in morphogenetic systems. In this direction, weneed to understand how signals can be integrated for therequired time without adding significant amounts of ad-ditional noise, and we need independent experiments totest for the spatial averaging which we have proposed isessential in reconciling the observed precision with thephysical limits.

The reproducibility of absolute Bcd concentration pro-files from embryo to embryo literally means that thenumber of copies of the protein is reproducible at the∼ 10% level. As noted above, this fits with the factthat Hb output varies significantly in response to 10%changes in the Bcd input. Understanding how the em-bryo achieves reproducibility in Bcd copy number is asignificant challenge. Again there is a conceptual con-nection to ideas in neuroscience. Models for the elec-trical dynamics of neurons–generalizations of the classi-cal Hodgkin-Huxley model (1952)–predict that even thequalitative behavior of the system depends sensitively onthe number of copies of the different ion channel proteins,suggesting that there must be mechanisms which regu-late these copy numbers, or at least their ratios (LeMas-son et al 1993; Abbott & LeMasson 1993; Goldman etal 2001); such mechanisms have since been discoveredexperimentally (Turrigiano et al 1994). The analogousmechanisms which might insure the reproducibility ofBcd copy number remain to be explored.

Finally, we note that the precision and reproducibilitywhich we have observed in the embryo is disturbingly

close to the resolution afforded by our measuring instru-ments.

METHODS

Bcd-GFP imaging in live embryos

Bcd-GFP lines are from Gregor et al (2006). Live imag-ing was performed in a custom built two-photon microscope(Denk et al 1990) similar in design to that of Svoboda etal (1997). Microscope control routines (Pologruto et al 2003)and all our image analysis routines were implemented in Mat-lab software (MATLAB, MathWorks, Natick, MA). Imageswere taken with a Zeiss 25x (NA 0.8) oil/water-immersionobjective and an excitation wavelength of 900−920 nm. Av-erage laser power at the specimen was 15−35 mW. For eachembryo, three high-resolution images (512 × 512 nm pixels,with 16 bits and at 6.4µs per pixel) were taken along theanterior-posterior axis (focussed at the mid-sagittal plane) atmagnified zoom and were subsequently stitched together insoftware; each image is an average of 6 sequentially acquiredframes (Figures 2 and 5). With these settings, the linear pixeldimension corresponds to 0.44± 0.01 µm. At Bcd-GFP con-centrations of ∼ 60 nM and a raw intensity value of ∼ 400,we counted 36± 5 photons/pixel in a single image.

Calibrating absolute concentrations

GFP variant S65T, a gift of HS Rye (Princeton), was over-produced in Escherichia coli (BL21) from a trc promoter andpurified essentially as described by Rye et al (1997). Absoluteprotein concentration was determined spectroscopically. TheS65T variant of GFP has optical absorption properties nearlyidentical to the eGFP variant used to generate transgenicBcd-GFP fly (Patterson et al 1999; Gregor et al 2006). Liv-ing Drosophila embryos expressing Bcd-GFP were immersedin 7.15± 0.05 pH Schneider’s insect culture medium contain-ing 36 nM GFP. Embryos were imaged 15 min after entry intomitosis 13, focusing at the mid-sagittal plane. Nuclear Bcd-GFP fluorescence intensities were extracted along the edge ofthe embryo as described below (see next section). At each nu-clear location a reference GFP intensity was measured at thecorresponding mirror image of the respective nucleus in theexternal immersion solution, using the vitelline membrane asa mirror.

Identification of nuclei in live images

For each embryo, nuclear centers were hand selected andthe average nuclear fluorescence intensity was computed overa circular window of fixed size (50 pixels). Embryos imagedat the mid-sagittal plane contained on average 70 nuclei alongeach edge. In our high-resolution images nuclei have, on av-erage, a diameter of 150 pixels. Towards the posterior end,where nuclei merge into the background intensity, virtual nu-clei were selected by keeping the same approximate periodic-ity of the anterior end.

11

Which diffusion constant?

In the Berg-Purcell analysis and its generalization, the con-centration c in Figure 3 refers to molecules that are free insolution and hence can diffuse up to the receptor site. Inpractice, when we measure concentrations using fluorescence,as in Figure 2, we are sensitive to a total concentration thatincludes both free molecules and those bound to other sites.Similarly, the diffusion constant D refers to the diffusion offree molecules, but when we measure a diffusion constant us-ing optical methods we see an effective diffusion constant Deff

that includes the effects of binding and unbinding to less mo-bile sites. What we would really like know, then, are cfree andDfree, but what we measure is ctotal and Deff . To understandthe connections among these quantities, consider a model inwhich Bcd molecules are free in solution at concentrationcfree(~x, t) and diffuse with diffusion constant Dfree. Let therebe binding sites for Bcd that are fixed (non-diffusing) at den-sity ρ; and let the second order binding rate be k+ and theunbinding rate be k−. Then

∂cfree(~x, t)

∂t= Dfree∇2cfree(~x, t)− ρ

∂f(~x, t)

∂t(15)

∂f(~x, t)

∂t= k+cfree(~x, t)[1− f(~x, t)]− k−f(~x, t),

(16)

where f(~x, t) is the fraction of binding sites occupied byBcd at position ~x and time t; the classical discussion ofthis problem is for calcium diffusion in neurons (Hodgkin& Keynes 1957). In steady state, the total concentration ofBcd molecule, bound and free, is given by

ctotal = cfree + ρcfree

cfree +K, (17)

where the dissociation constant K = k−/k+. If we consider

a perturbation around the steady state, c→ c+a(t) sin(~k·~x),where the wavelength of the perturbation λ = 2π/|~k|, thenfor long wavelengths it can be shown from Equations (15) and(16) that the amplitude of the concentration profile decays

as a(t) ∝ exp(−t/τ), where τ = 1/(Deff |~k|2). This depen-dence on wavelength is the same as for the ordinary diffusionequation, so the binding sites change the effective diffusionconstant to

Deff =Dfree

1 + ρK/(cfree +K)2. (18)

In the limit that the binding sites are weak, cfree � K, wehave ctotal = cfree(1 + ρ/K) and Deff = Dfree/(1 + ρ/K), sothat ctotalDeff = cfreeDfree. Thus we can compute the Berg-Purcell limit [Equation 3], which depends only on the productDc, from measured quantities alone, without corrections.

Antibody staining and confocal microscopy

All embryos were collected at 25◦C, heat fixed, and sub-sequently labeled with fluorescent probes. We used rat anti-Bcd and rabbit anti-Hb antibodies (Reinitz at al 1998), giftsof J Reinitz (Stony Brook). Secondary antibodies were con-jugated with Alexa-488, Alexa-546 and Toto3 (MolecularProbes), respectively. Embryos were mounted in AquaPoly-mount (Polysciences, Inc.). High-resolution digital images(1024 × 1024, 12 bits per pixel) of fixed eggs were obtained

on a Zeiss LSM 510 confocal microscope with a Zeiss 20x (NA0.45) A-plan objective. Embryos were placed under a coverslip and the image focal plane of the flattened embryo waschosen at top surface for nuclear staining intensities (Figures3 and 4) and at the mid-sagittal plane for protein profile ex-traction (Figure 6D). All embryos were prepared, and imageswere taken, under the same conditions: (i) all embryos wereheat fixed, (ii) embryos were stained and washed together inthe same tube, and (iii) all images were taken with the samemicroscope settings in a single acquisition cycle.

Automatic identification of nuclei in fixed embryos

Images of DNA stainings (Toto3, Molecular Probes, Eu-gene, OR) were used to automatically identify nuclei. Topre-process the images, we examine each pixel x in the con-text of its 11 × 11 pixel neighborhood. Let the mean inten-sity in this neighborhood be I(x) and the variance be σ2(x);we then transform the intensity I(x) into a normalized formψ(x) = [I(x) − I(x)]/σ(x). These normalized images aresmoothed with a Gaussian filter (standard deviation 2 pixels)and thresholded, with the threshold chosen by eye to optimizethe capture of the nuclei and minimize spurious detection.Locations of nuclei were assigned as the center of mass in theconnected regions above threshold. For each embryo a regionof interest was hand selected to avoid misidentification due togeometric distortion at the embryo edge, yielding an averageof 1300–1500 nuclei per embryo. Residual misidentificationwas at the 1% level.

Analysis of input/output relations

The raw data from images such as Figure 3 consist of pairs{IBcd(n; k), IHb(n; k)}, where IBcd and IHb refer to anti-Bcdand anti-Hb fluorescence intensities, respectively; n labelsthe nuclei in a single embryo, and k labels the embryo. Weexpect that IBcd(n; k) = ABcd(k)cnBcd(k)+BBcd(k), and sim-ilarly for the Hb data, where ABcd(k) and BBcd(k) are thescaling and background that relate Bcd concentration to fluo-rescence intensity in the kth embryo. Note that if we analyzeinput/output relations in single embryos, the choice of scalefactors ABcd and AHb is entirely a matter of convention. Toask if all of the input/output relations can be placed consis-tently on the same axes, however, we need to choose thesescale factors more carefully.

Initial guesses for A and B are made by assuming that thesmallest concentration we measure is zero and that the meanconcentration of each species is equal to one in all embryos.If we know the values of these parameters, we can turn all ofthe intensities into concentrations, and we merge all of thesedata into pairs {cnBcd, c

nHb}, where the index n runs over all

the nuclei in all the embryos. We characterize this mergeddata set by computing the mutual information I(cBcd; cHb)between cBcd and cHb, being careful to correct for errors dueto the finite size of the data set; see, for example, Slonimet al (2005). If the shapes of the input/output relations arevery different in different embryos, or if we choose incorrectvalues for the parameters A and B, then I(cBcd; cHb) will bereduced. We use an iterative algorithm to adjust all fourparameters for each embryo until we maximize the mutualinformation. Once this process has converged we can use the

12

merged data to compute the mean input/output relation byquantizing the cBcd axis and estimating the mean value of cHb

associated with each bin along this axis; we then normalizecHb so that the minimum (maximum) of this mean output isequal to 0 (1), and we normalize cBcd relative to the valuewhich produces half-maximal mean output. Computing thestandard deviation of cHb values in each bin gives the outputHb noise. We can then compute input/output relations forthe individual embryos and verify that they are the samewithin the error bars defined by the output noise (Figure 4).

Measurement noise in the input/output relations

Four fixed and antibody-stained embryos were imaged 5times in sequence using confocal microscopy as above. Foreach identified nucleus the mean and standard deviationacross the 5 images was computed. All embryos were nor-malized as above and their data sets merged to generate aquantized cBcd axis. For each bin along this axis we computedaverage cBcd measurement standard deviations by averagingthe measurement variances of all the nuclei in the given bin.The same procedure was used to estimate measurement noisein the Hb images, although this was found to be much lesssignificant.

Correlation function of Hb noise

Let cnHb be the observed concentration of Hb in nucleus n,and similarly for the Bcd concentration cnBcd; these are thecoordinates for each point in the scatter plot of Figure 3C.For individual embryos the contribution to the correlationcoefficient from two nuclei i and j was computed as

Cnm =cnHb − cHb(cnBcd)

σHb(cnBcd)· c

mHb − cHb(cmBcd)

σHb(cmBcd), (19)

where cHb(Bcd) is the mean input/output relation andσHb(Bcd) is the standard deviation of the output, as in Fig-ures 4A and 4B, respectively; we use the same binning of theBcd concentration as in Figure 4 to approximate these func-tions. The correlation function is the ensemble average overthese coefficients, C(r) = 〈Cnm〉, where 〈· · ·〉 averages overall nuclei that are distance r apart, with r quantized into binsof size equal to the spacing between neighboring nuclei.

Measurement noise in live images

We identified 4 different sources of measurement noise: 1.Imaging noise. Small regions of individual embryos were im-aged 5 times in sequence, 3 s per image, with the same pixelacquisition time as in actual data. The variances across those5 images for identified nuclei constitute the instrumental orimaging noise. 2. Nuclear identification noise. To estimatethe error due to mis-centering of the averaging region overthe individual nuclei, we computed the variances across 9averaging regions centered in a 3 × 3 pixel matrix aroundthe originally chosen center. Gray line in Figure 5B stemsfrom the sum of Imaging noise and Nuclear identificationnoise, which are uncorrelated. 3. Focal plane adjustmentnoise. For each individual embryo the focal plane has to be

hand adjusted before image acquisition. We adjusted the fo-cal plane to be at the mid-sagittal plane of the embryo butestimate our accuracy to be 6µm, or one nuclear diameter.Due to power attenuation over that distance (∼ 10% on aver-age) we estimate our error by computing the variances of nu-clear intensities across 7 images taken at consecutive heightsspaced by 1µm in a single embryo (black line in Figure 5B).4. Rotational asymmetry around the anterior-posterior axis.Embryos are not rotationally symmetric around the anterior-posterior axis, and Bcd profiles are significantly differentalong the dorsal vs the ventral side of a laterally orientedembryo. An obvious error source arises from our inability tomount all embryos at the same azimuthal angle. We are un-able to measure this potentially large error source accurately,but we estimate an upper bound by the difference of dorsalvs. ventral gradients. Between 10−50% egg length the upperbound for this contribution of the fractional error is ∼ 13.5%.

Sources of variability

Consider the simplest model for the Bcd profile in whichBcd molecules diffuse with diffusion constant D and aredegraded through a first order reaction with lifetime τ .Restricting our attention to the one dimension along theanterior-posterior axis, the equation for the concentrationprofile is

∂c(x, t)

∂t= D

∂2c(x, t)

∂x2− 1

τc(x, t) + S(x), (20)

where S(x) describes the source of protein which resultsfrom translation of the maternal mRNA. The usual assump-tion is that this source is confined to the anterior pole,x = 0. A slightly more realistic assumption is that thesource is localized within some distance ∆, perhaps as S(x) =(R/∆) exp(−x/∆), where R is the total rate at which Bcdmolecules are synthesized. Then the steady state profile is

c(x) =Rτλ

λ2 −∆2exp(−x/λ)− Rτ∆

λ2 −∆2exp(−x/∆), (21)

where λ =√Dτ and we assume that the embryo is long,

L � λ,∆. We see that if the typical value of ∆ is smallerthan λ, then for x� ∆ the dependence of the concentrationon ∆ is ∼ (∆/λ)2. On the other hand, near x = 0 thereis a term ∼ ∆/λ. Thus the concentration near the anteriorpole will be more sensitive to variations in ∆ from embryo toembryo.

Spatial profiles of Bcd and Hb in fixed and stainedembryos

Bcd and Hb protein profiles were extracted from confocalimages of stained embryos by using software routines that al-lowed a circular window of the size of a nucleus to be system-atically moved along the outer edge of the embryo (Houch-mandzadeh et al 2002). At each position, the average pixelintensity within the window was plotted versus the projec-tion of the window center along the anterior-posterior axis ofthe embryo. Protein concentration measurements were madeseparately along the dorsal and ventral sides of the embryo;for consistency, we compared only dorsal profiles.

13

Minimizing χ2

As written, the minimization of χ2 in Equation (14) sug-gests an iterative process: compute the mean concentrationprofile, adjust the parameters {An, Bn} so that the data fromeach embryo matches the mean as well as possible, recomputethe mean, and iterate. In fact χ2 is quadratic in both the pa-rameters {An, Bn} and in the mean profile c(x), so one canmake analytic progress. To begin, χ2 is minimized when eachBn is chosen so that all of the profiles cn(x) have the samemean value when averaged over x; a convenient first step is tochose the Bn so that this mean is zero. The hypothesis thatmost of the remaining variance can be eliminated by properchoice of the An is equivalent to the statement that the sin-gular value decomposition of the unnormalized, zero meanprofiles is dominated by a single mode, and this mode willbe the mean profile c(x). Thus we perform a singular valuedecomposition of the profiles and choose An so that the pro-jection of each profile onto the dominant mode is normalizedto unity, and this provides the minimum χ2. Once the pa-rameters have been set in this way we still have the freedomto add a constant background (so that the concentration fallsto zero on average at the posterior of the egg) and an overallscale (so that the mean concentration profile has a maximumvalue of one).

Acknowledgements

We thank B. Houchmandzadeh, M. Kaschube, J. Kinney,K. Krantz, D. Madan, R. deRuyter van Steveninck, S. Se-tayeshgar and G. Tkacik. This work was supported in partby the MRSEC Program of the National Science Foundationunder Award Number DMR-0213706, and by NIH grants P50GM071508 and R01 GM077599.

References

Abbott, LF & LeMasson, G (1993). Analysis of neuron mod-els with dynamically regulated conductances, NeuralComp 5, 823–842.

Barlow, HB (1981). Critical limiting factors in the designof the eye and visual cortex, Proc R Soc Lond Ser B212, 1–34.

Berg, HC & Purcell, EM (1977). Physics of chemoreception,Biohys J 20, 193–219.

Bialek, W (1987). Physical limits to sensation and percep-tion, Ann Rev Biophys Biophys Chem 16, 455–478.

Bialek, W (2002). Thinking about the brain, in Physics ofBiomolecules and Cells: Les Houches Session LXXV,H Flyvbjerg, F Julicher, P Ormos & F David, eds,pp 485–577 (EDP Sciences, Les Ulis; Springer-Verlag,Berlin)

Bialek, W & Setayeshgar, S (2005). Physical limits to bio-chemical signaling, Proc Nat’l Acad Sci (USA) 102,10040–10045

Bialek, W & Setayeshgar, S (2006). Cooperativ-ity, sensitivity and noise in biochemical signaling,http://arXiv.org/q-bio.MN/0601001.

Burz, DS, Pivera-Pomar, R, Jackle, H & Hanes, SD (1998).Cooperative DNA-binding by Bicoid provides a mech-anism for threshold dependent gene activation in theDrosophila embryo. EMBO J 17, 5998–6009.

Crauk, O, & Dostatni, N (2005). Bicoid determines sharpand precise target gene expression in the Drosophilaembryo. Curr Biol 15, 1888–1898.

von Dassow, G, Meir, E, Munro, EM & Odell, GM (2000).The segment polarity network is a robust developmen-tal module, Nature 406, 188–192.

Denk, W, Strickler, JH & Webb, WW (1990). Two-photonlaser scanning fluorescence microscopy, Science 248,73–76.

Driever, W & Nusslein-Volhard, C (1988a). A gradient ofBicoid protein in Drosophila embryos, Cell 54, 83–93.

Driever, W & Nusslein-Volhard, C (1988b). The Bicoid pro-tein determines position in the Drosophila embryo, Cell54, 95–104.

Driever, W & Nusslein-Volhard, C (1989). The Bicoid pro-tein is a positive regulator of hunchback transcriptionin the early Drosophila embryo, Nature 337, 138–143.

Eldar, A, Dorfman, R, Weiss, D, Ashe, H, Shilo, BZ &Barkai, N (2002). Robustness of the BMP morphogengradient in Drosophila embryonic patterning, Nature419, 304–308.

Eldar, A, Rosin, D, Shilo, BZ & Barkai, N (2003). Self-enhanced ligand degradation underlies robustness ofmorphogen gradients, Dev Cell 5, 635–646.

Eldar, A, Shilo, BZ & Barkai, N (2004). Elucidating mech-anisms underlying robustness of morphogen gradients.Curr Opin Genet Dev 14, 435–439.

Elowitz, MB, Levine, AJ, Siggia ED & Swain, PD (2002).Stochastic gene expression in a single cell, Science 207,1183–1186.

Elowitz, MB, Surette, MG, Wolf, PE, Stock, JB & Leibler,S (1999). Protein mobility in the cytoplasm of Es-cherichia coli, J Bacteriol 181, 197–203.

Furriols, M & Casanova, J (2003). In and out of Torso RTKsignalling, EMBO J 22, 1947–1952.

Gao, Q & Finkelstein, R (1998). Targeting gene expressionto the head: the Drosophila orthodenticle gene is adirect target of the Bicoid morphogen, Development125, 4185–4193.

Gergen, JP, Coulter, D & Wieschaus, EF (1986). Segmentalpattern and blastoderm cell identities. In Gametoge-nesis and The Early Embryo, JG Gall, ed. (Liss Inc.,New York).

Goldman, MS, Golowasch, J, Marder, E & Abbott, LF(2001). Global structure, robustness, and modulationof neural models, J Neurosci 21, 5229–5238.

Gregor, T, Bialek, W, de Ruyter van Steveninck, RR, Tank,DW & Wieschaus, EF (2005). Diffusion and scalingduring early embryonic pattern formation, Proc Nat’lAcad Sci (USA) 102, 18403–18407.

Gregor, T, Wieschaus, EF, McGregor, AP, Bialek, W &Tank, DW (2006). Stability and nuclear dynamics ofthe Bicoid morphogen gradient, submitted.

Hodgkin, AL & Huxley, AF (1952). A quantitative descrip-tion of membrane current and its application to con-duction and excitation in nerve, J Physiol (Lond) 117,500–544.

14

Hodgkin, AL & Keynes, RD (1957). Movements of labelledcalcium in giant axons, J Physiol (Lond) 138, 253–281.

Holloway, DM, Harrison, LG, Kosman, D, Vanario-Alonso,CE, & Spirov, AV (2006). Analysis of pattern preci-sion shows that Drosophila segmentation develops sub-stantial independence from gradients of maternal geneproducts. Developmental Dynamics online publicationDOI 0.1002/dvdy.2094.

Houchmandzadeh, B, Wieschaus, E & Leibler, S (2002).Establishment of developmental precision and propor-tions in the early Drosophila embryo, Nature 415, 798–802.

Houchmandzadeh, B, Wieschaus, E & Leibler, S (2005).Precise domain specification in the developingDrosophila embryo. Phys Rev E 72, 061920 (2005).

Howard, M & PR ten Wolde (2005). Finding the centerreliably: Robust patterns of developmental gene ex-pression. Phys Rev Lett 95, 208103.

Jaeger, J, Surkova, S, Blagov, M, Janssens, H, Kosman, D,Kozlov, KN, Manu, Myasnikova, E, Vanario-Alonso,CE, Samsonova, M, Sharp, DH & Reinitz, J (2004).Dynamic control of positional information in the earlyDrosophila embryo. Nature 430, 368–371.

Kirschner, M & Gerhart, J (1997). Cells, Embryos and Evo-lution (Blackwell Science, Malden MA).

Kossman, D, Small, S & Reinitz, J (1998). Rapid prepara-tion of a panel of polyclonal antibodies to Drosophilasegmentation proteins. Dev. Genes Evol. 208, 290–298.

Lawrence, PA (1992). The Making of a Fly: The Geneticsof Animal Design (Blackwell Scientific, Oxford).

LeMasson, G, Marder, E, & Abbott, LF (1993). Activity-dependent regulation of conductances in model neu-rons, Science 259, 1915–1917.

Ma X, Yuan D, Diepold K, Scarborough T, & Ma J (1996).The Drosophila morphogenetic protein Bicoid bindsDNA cooperatively. Development 122, 1195–1206.

Margolis JS, Borowsky ML, Steingrimsson E, Shim CW,Lengyel JA, Posakony JW (1995) Posterior stripe ex-pression of hunchback is driven from two promoters bya common enhancer element. Development 12, 3067–3077.

Martinez Arias, A & Hayward, P (2006). Filtering tran-scriptional noise during development: concepts andmechanisms, Nat. Rev. Genet. 7, 34–44.

McHale, P, Rappel, W-J & Levine, H (2006). Embryonicpattern scaling achieved by oppositely directed mor-phogen gradients. http//arXiv.org/q-bio.SC/0601022.

Pedone, PV, Ghirlando, R, Clore, GM, Gronenbron, AM,Felsenfeld, G & Omchinski, JG (1996). The singleCys2-His2 zinc finger domain of the GAGA proteinflanked by basic residues is sufficient for high-affinityspecific DNA binding, Proc Nat’l Acad Sci (USA) 93,2822–2826.

Pologruto, TA, Sabatini, BL, & Svoboda, K (2003). Scan-Image: Flexible software for operating laser scanningmicroscopes. Biomedical Engineering Online 2, 13.

Ptashne, M (1992). A Genetic Switch. Second edition:Phage λ and Higher Organisms (Cell Press, Cam-bridge).

Raser, JM & O’Shea, EK (2004). Control of stochasticity ineukaryotic gene expression, Science 304, 1811–1814.

Rieke, F & Baylor, DA (1998). Single photon detection byrod cells of the retina, Rev Mod Phys 70, 1027–1036.

Rivera-Pomar, R, Lu, X, Perrimon, N, Taubert, H & Jackle,H (1995). Activation of posterior gap gene expressionin the Drosophila blastoderm, Nature 376, 253–256.

Rosenfeld, N, Young, JW, Alon, U, Swain, PS & Elowitz,MB (2005). Gene regulation at the single cell level.Science 307, 1962–1965.

Rye, HS, Burston, SG, Fenton, WA, Beechem, JM, Xu, Z,Sigler, PB, & Horwich, AL (1997). Distinct actionsof cis and trans ATP within the double ring of thechaperonin GroEL, Nature 388, 792–798.

Schrodinger, E (1944). What is Life? (Cambridge Univer-sity Press).

Slonim, N, Atwal, GS, Tkacik, G & Bialek, W (2005). Es-timating mutual information and multi-information inlarge networks, http://arxiv.org/cs.IT/0502017.

Spirov, AV & Holloway, DM (2003). Making the body plan:precision in the genetic hierarchy of Drosophila embryosegmentation. In Silico Biol 3, 89–100.

Struhl, G, Struhl, K & Macdonald, PM (1989). The gra-dient morphogen Bicoid is a concentration-dependenttranscriptional activator, Cell 57, 1259–1273.

Svoboda, K, Denk, W, Kleinfeld, D & Tank, DW (1997). Invivo dendritic calcium dynamics in neocortical pyrami-dal neurons, Nature 385, 161–165.

Turrigiano, G, Abbott, LF & Marder, (1994). Activity-dependent changes in the intrinsic properties of cul-tured neurons, Science 264, 974–977.

de Vries, H (1956). Physical aspects of the sense organs,Prog Biophys 6, 207–264.

Winston, RL, Millar, DP, Gottesfeld, JM & Kent, SB(1999). Characterization of the DNA binding prop-erties of the bHLH domain of Deadpan to single andtandem sites, Biochemistry 38, 5138–5146.

Wolpert, L (1969). Positional information and the spatialpattern of cellular differentiation, J Theor Biol 25, 1–47 (1969).

Zhao, C, York, A, Yang, F, Forsthoefel, DJ, Dave, V, Fu, D,Zhang, D, Corado, MS, Small, S, Seeger, MA & Ma, J(2002). The activity of the Drosophila morphogeneticprotein Bicoid is inhibited by a domain located outsideits homeodomain, Development 129, 1669–1680.

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