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REPORT

Precision and scaling in morphogen gradient read-out

Aitana Morton de Lachapelle1,2and Sven Bergmann1,2,*

1Department of Medical Genetics, University of Lausanne, Lausanne, Switzerland and2Swiss Institute of Bioinformatics, Lausanne, Switzerland

* Corresponding author. Department of Medical Genetics, University of Lausanne, Bugnon 27, Lausanne 1005, Switzerland.

Tel.: þ41 21 692 5452; Fax: þ41 21 692 5455; E-mail: Sven.Bergmann@unil.ch

Received 30.7.09; accepted 16.1.10

Morphogen gradients infer cell fate as a function of cellular position. Experiments in Drosophila

embryos have shown that the Bicoid (Bcd) gradient is precise and exhibits some degree of scaling.

We present experimental results on the precision of Bcd target genes for embryos with a single,

double or quadruple dose of bicoid demonstrating that precision is highest at mid-embryo and

position dependent, rather than gene dependent. This confirms that the major contribution to

precision is achieved already at the Bcd gradient formation. Modeling this dynamic process, we

investigate precision for inter-embryo fluctuations in different parameters affecting gradient

formation. Within our modeling framework, the observed precision can only be achieved by a

transient Bcd profile. Studying different extensions of our modeling framework reveals that scaling

isgenerallypositiondependentanddecreasestowardtheposteriorpole.Ourmeasurementsconfirm

this trend, indicating almost perfect scaling except for anterior most expression domains, which

overcompensate fluctuations in embryo length.

Molecular Systems Biology 6: 351; published online 9 March 2010; doi:10.1038/msb.2010.7

Subject Categories: development

Keywords: bicoid; Drosophila; morphogen gradients; precision; scaling

This is an open-access article distributed under the terms of the Creative Commons Attribution Licence,

which permits distribution and reproduction in any medium, provided the original author and source are

credited.Creationofderivativeworksispermittedbuttheresultingworkmaybedistributedonlyunderthe

same or similar licence to this one. This licence does not permit commercial exploitation without specific

permission.

Introduction

Bicoid (Bcd) is a well studied morphogen involved in the

patterningoftheanterior–posterior(AP)axisoftheDrosophila

embryo (Driever and Nu ¨sslein-Volhard, 1988a,b). Zygotic

downstream genes read out this gradient and their expression

domains determine the basic body plan of the embryo along

this axis. The positions of these domains are remarkably

insensitive to fluctuations in the external environment

(Houchmandzadeh et al, 2002; Crauk and Dostatni, 2005;

Lucchetta et al, 2005) and their relative proportions are

maintainedacrossembryosofdifferentsizes.Thelatterfeature

is referred to as scaling and occurs within a single species

(Houchmandzadeh et al,2002; Lottet al, 2007)and alsoacross

different species (Gregor et al, 2005, 2008; Lott et al, 2007).

Recent experiments also show that the Bcd gradient itself is

rather precise and that its length scale correlates to some

extent to the embryo size (Gregor et al, 2007a; He et al, 2008).

These new findings suggest that the precision and scaling of

Bcd target genes may, at least in part, be attributed to that of

the morphogen gradient itself. Therefore, we focus here on

single morphogen models that aim at explaining the precision

and scaling of Bcd target genes at the level of the morphogen

gradient. However, it is likely that other mechanisms (e.g. Bcd

interactions with the staufen gene (Aegerter-Wilmsen et al,

2005), gap genes interactions (Jaeger et al, 2004a,b, 2007;

Jaeger and Reinitz, 2006; Manu et al, 2009a,b), Bcd interac-

tions with maternal Hunchback and the terminal system

(Ochoa-Espinosa et al, 2009) or bistability (Lopes et al, 2008))

alsocontributetofurtherincreaserobustnessinAPpatterning.

In this work, we assess the spatial precision of expression

domains for the gap genes Kru ¨ppel (Kr), Giant (Gt) and

Hunchback (Hb), as well as the pair-rule gene Even-skipped

(Eve) in more than 150 staining images of embryos with a

single, double and quadruple dose of bcd. Our data indicate

thattheprecisionismaximalatmid-embryoaswellasposition

dependent rather than gene dependent. This provides inde-

pendent support for the conclusions drawn from direct

measurements of Bcd (Gregor et al, 2007a; He et al, 2008)

that the morphogen gradient itself is the main contributor to

the precision of the target genes. It motivates our subsequent

analytical investigation of noise propagation during Bcd

gradient formation within a single morphogen modeling

framework. Our analysis shows that fluctuations in the

parameters affecting morphogen production, degradation,

diffusion as well as nuclear trapping can give rise naturally

Molecular Systems Biology 6; Article number 351; doi:10.1038/msb.2010.7

Citation: Molecular Systems Biology 6:351

& 2010 EMBO and Macmillan Publishers Limited

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& 2010 EMBO and Macmillan Publishers LimitedMolecular Systems Biology 2010 1

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to highest precision at mid-embryo, provided the Bcd gradient

is decoded at pre-steady-state. Finally, we investigate the

scaling of the morphogen gradient in a variety of models that

couple the embryo size to parameters affecting gradient

formation. To single out the most likely scenario, we measure

scaling at the level of the target genes. We find that expression

domain boundaries posterior to B40% embryo length (L)

exhibit almost perfect scaling, whereas in the more anterior

region we observe over-compensation to fluctuations in

embryo size. This effect appears to be position dependent

rather than gene dependent. Interestingly, it is in good

agreement with a model that explicitly includes the nuclear

trapping of Bcd. We conclude that the formation of the Bcd

gradient itself is likely to be a main contributor to robust

patterning along the AP axis and that pre-steady-state

decoding and nuclear trapping are efficient means to increase

robustness.

Results

The gap and pair-rule gene expression domains

are precise

We studied staining images of 154 Drosophila melanogaster

embryos at cleavage cycle 14 (see Methods in Bergmann et al,

2007). Using image processing tools described in ‘Materials

and methods’, we measured the relative positions x/L of the

protein domains of Gt, Hb, Kr and Eve. We screened both wild-

type embryos with two copies of bcd and mutant strains with

one or four bcd copies resulting in shifted expression domains.

We observed that inmid-embryo, thestandard deviation of the

relative domain localizations, s(x/L), is between 1 and 2%

and it tends to be higher (2–4%) toward the anterior and

posterior poles (Figure 1).

Theseresultsareingoodagreementwithpreviousresultson

the Hb domain (Houchmandzadeh et al, 2002). Importantly,

they show that the precision of the target genes has the same

magnitude and positional trend as that of Bcd according to

Gregor et al (2007a). Moreover, our experimental analysis

shows that precision is more position dependent than gene

dependent. Indeed, when expression domains are shifted

because of different bcd mRNA dosages, the precision of the

domain seems to change according to the new domain

position. Overall, all the investigated domains follow similar

precision trends.This suggests that a majorcontribution to the

precision of the Bcd target genes can be attributed to the

morphogen gradient itself. We note that position dependent

precision may also arise as an experimental artifact when

analyzing embryos with different orientations and varying

ages (times classes T5–T8 of nuclear cycle 14) or because of

imperfect scaling (cf. Supplementary Text S3). However,

estimating the maximal contributions of such effects we still

find that higher precision at mid-embryo is statistically signi-

ficant (cf. Supplementary Text S1; Supplementary Figure S4).

Modeling precision in a single morphogen

framework

Given the aforementioned pieces of evidence for achieving

precision at the level of the gradient formation, we consider a

French-flag model (Wolpert, 1969) in which a single morpho-

gen gradient induces expression domain boundaries in a

concentration-dependent manner (see Materials and meth-

ods). During the first stages of embryonic development, 13

nuclear divisions occur about every 10min. Gregor et al

(2007b) have recently shown that nuclear concentrations

stabilizeatcycle10.Fromarobustnessperspective,thismeans

that Bcd read-out is based on a similar number of molecules

and so the associated stochastic noise does not change

significantly from cycle 10 on. Therefore, we focus here on

the noise propagated from the external Bcd gradient and

investigate the fluctuations in domain localization when

perturbing the various parameters characterizing the gradient

formation (i.e. production, degradation, diffusion and nuclear

trapping rates).

We want to assess the variation in domain position Dx,

which is induced by embryo-to-embryo fluctuations of

magnitude Dq in a parameter q affecting the morphogen

gradient formation. Dx can be estimated analytically to first

order in Dq

????

where we used that the threshold concentration is fixed,

yieldingdM¼(qM/qx)dxþ(qM/qq)dq¼0.Thisestimatecanbe

computed foranymorphogen distribution M(x, t) with explicit

dependence on q. Here, we consider the time-dependent

solution of a model in which Bcd is produced at the anterior

pole with production rate s0, diffuses according to a uniform

diffusion constant D, is degraded at uniform rate a and is

trapped and released by the nuclei in the embryo at rates kn,

k?n, respectively (see Materials and methods). In Figure 2, we

plotted the imprecision measure Dx for small fluctuations

Dx ¼

dx

dq

????Dq ¼

qM

qx

???1

?qM

qq

?????

?????Dqð1Þ

x (%L)

2030405060708090

0.5

1

1.5

σ(x/L) (%)

2

2.5

3

3.5

4

4.5

Figure 1

s(x/L) ofthe gap andpair-rule gene expression domains asafunction of position

x along the AP axis (see Supplementary Dataset S1). Errors (bars) were

estimated by computing the standard deviation across 50 independent

(semi-automated) markings of expression domain boundaries. For Gt (filled

triangles) and Kr (empty triangles), we show results for their left (v) and right

(x) boundaries, as well as for the center (n) of their expression domain. The Hb

(&) domain is characterized by the boundary where its concentration drops and

the Eve stripes (J) by the position at which their intensity is maximal. Color

code: 1xbcd (red), 2xbcd¼wild type (green), 4xbcd (blue).

Measuring precision of Bcd target genes. Measure of imprecision

Precision and scaling in morphogen decoding

AM de Lachapelle and S Bergmann

2 Molecular Systems Biology 2010

& 2010 EMBO and Macmillan Publishers Limited

Page 3

(5%) in each of these parameters (Figure 2A–D) as well as a

combination of fluctuations in the production and diffusion

rates (Figure 2E) (numerical simulations gave very similar

results, data not shown). We choose ti¼100min as the time

when patterning is initiated, but qualitatively our results only

depend on the relation between tiand the Bcd decay time

t¼1/a (for ti? t the gradient has not yet reached steady-

state). We find that fluctuations in s0(Figure 2A) give rise to

decreasing Dx toward the posterior pole if the pre-steady-state

gradient is read out (i.e. for small a). In contrast, for

fluctuations in D or a, Dx increases toward the posterior pole

(Figure 2B and C). Thus, in the case of Bcd pre-steady-state

decoding (bluish curves) and fluctuations both in the

production and degradation or diffusion rates (Figure 2E),

maximalprecisionoftheBcdgradientaround mid-embryocan

arise naturally, as was indeed observed directly by Gregor et al

(2007a) and indirectly by our analysis (cf. Figure 1). In

contrast, for high decay rates (reddish curves) the gradient

rapidly equilibrates such that fluctuations in the production

rate give rise to uniform noise in Bcd concentrations,

preventing the possibility of minimal noise in the central

region. Interestingly, fluctuations in the nuclear trapping rate

alone also yield higher precision around mid-embryo

(Figure 2D) if the Bcd gradient is decoded early in pre-

steady-state. However, the position with highest precision is

sensitive to the decoding time, shifting more toward the

posterior pole for later decoding.

Our modeling approach provides a proof of principle that

maximal precision at mid-embryo can arise if the gradient is

decoded before steady-state. However, we cannot rule out that

other mechanisms may yield such a pattern of precision even

for a steady-state Bcd gradient (e.g. in the wing disc, it was

shown that cell-to-cell variability in the production, diffusion

and degradation rates can yield higher precision around mid-

field, Bollenbach et al, 2008).

In our analysis, we assumed the classical French-flag model

(Wolpert, 1969) in which domain boundaries are determined

from critical morphogen concentrations. This is a simplifica-

0.0

0.5

1.0

1.5

2.0

2.5

3.0

s′0 ~ N (s0,0.05s0)

k′n ~ N (kn,0.05kn)

?′ ~ N (?,0.05?)

D′ ~ N (D,0.05D)

20304050

x (%L)

607080902030405060708090

0.0

0.5

1.0

1.5

2.0

2.5

3.0

AC

D

B

Δx (%L)

Δx (%L)

x (%L)

20 3040506070 8090

x (%L)

Δx (%L)

E

= 0

= 0.3 min–1

= 0.1 min–1

= 1 min–1

= 0.03 min–1

Degradation rate

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure 2

corresponding diffusion constants D¼{167, 55, 17, 5, 3}mm2/s, ensuring l¼0.2L at t¼150min (numerical fit); 5% noise was added to the Bcd profile at the level of

(A)theproductionrate,(B)thediffusionconstant,(C)thedegradationrateand(D)thenucleartrappingrate.In(E),acombinationofnoisewasaddedtotheBcdprofile

(17% on the production- and 6% on the diffusion rates; all contributions are assumed to be independent). (L¼1 for all embryos.)

Modeling precision of nuclear Bcd. Predicted positional variability at t¼100min for various Bcd degradation rates a¼{1, 0.3, 0.1, 0.03, 0}min?1and their

Precision and scaling in morphogen decoding

AM de Lachapelle and S Bergmann

& 2010 EMBO and Macmillan Publishers LimitedMolecular Systems Biology 2010 3

Page 4

tion and recent work by the Reinitz group (Jaeger et al,

2004a,b, 2007; Jaeger and Reinitz, 2006; Surkova et al, 2008;

Manu et al, 2009a,b) showed that the gap gene expression

domains are established by a highly dynamic process

characterized bydrifting domain boundaries. In our modeling,

we neglected the gap gene dynamics and rather focused on

how precision can be achieved already at the level of the Bcd

gradient, as suggested by our data (as well as those reported in

Gregor et al, 2007a; He et al, 2008). Thus, we assume that the

positional information is transmitted relativelyearlyto the gap

genes, around cycle 10 (as suggested in Lucchetta et al, 2008).

The precision and proper scaling of this read-out is then

maintained (and possibly refined) by the gap genes indepen-

dently of Bcd, because of their cross- and auto-regulation

(Bergmann et al, 2007). Meanwhile, the Bcd gradient may

continue to evolve and eventually decays at the onset of

gastrulation (e.g. by activation of its PEST domain, Niessing

et al, 1999).

Modeling scaling

We next sought to extend our modeling framework to

accommodate also scaling. To quantify scaling analytically at

the morphogen level, we define a scaling coefficient S(x, t) for

a position x and time t

S ?dx

dL?L

x

ð2Þ

Perfect scaling corresponds to S¼1. In this case, fluctuations in

embryo length, dL/L, are exactly compensated by fluctuations

in position, dx/x, implying perfectly conserved proportions.

We use the terms hypo- and hyper-scaling to refer to So1

and S41, respectively. A position that hypo-scales does not

compensate enough for a change in embryo size, meaning that

in a bigger embryo the absolute position is not shifted enough

posteriorward to keep the correct proportions. In contrast,

hyper-scaling is the tendency to overcompensate for a change

in embryo size.

Scaling is a property of the external gradient, which is then

transmitted to the nuclear concentrations (cf. equation (7)).

Assuming that the threshold concentration is fixed (implying

dM¼(qM/qx)dxþ(qM/qL)dL¼0) it follows from equation (2)

that the scaling coefficient is

?

The abovedefinition ofscalingis genericand canbe computed

for any morphogen distribution M(x, t) with explicit depen-

dence on L. Here, we consider a model in which the embryo

length impacts the generation of the morphogen gradient

through the nuclei density. Specifically, assuming that at

decoding time all embryos have the same number of nuclei

independent of their size (which agrees with the deterministic

doubling of nuclei at each cycle), the nuclei density N depends

on the embryo size like NpL?n, where nA[1;3] (n¼3

corresponds to a uniform distribution of nuclei, whereas

n¼2 is true if nuclei are distributed on a shell with a fixed

width, Gregor et al, 2007b). In this scenario, we find that

scaling is time dependent and position dependent (Figure 3B;

see Supplementary Figure S9 for the dependence of scaling on

S ¼ ?

qM

qx

??1

?qM

qL?L

x

ð3Þ

N/K and n). Specifically, anterior domain boundaries hyper-

scale, in particular if decoding occurs relatively late, whereas

posterior domains show very good scaling for a wide range of

decoding times.

We also investigated three other models that give rise to

scaling by coupling the nuclear degradation, cytoplasmic

degradation or the morphogen production rate to the embryo

length (cf. Supplementary Text S2). We find that their scaling

behavior is qualitatively similar (in the sense that S decreases

toward the posterior pole) but more sensitive to the Bcd

decoding time (Supplementary Figure S8).

The gap and pair-rule gene expression domains

scale with embryo size

The model we investigated above predicts that scaling should

be position dependent rather than gene dependent and close to

perfect except for anterior most domains. To test these two

predictions experimentally on our collection of staining

images, we adapted the continuous definition of the scaling

coefficient S(x, t) in equation (2) for discrete measurements as

follows:

?L

? x¼covðx; LÞ

where^b is the estimated slope from a linear regression

x¼aþbL of the domain positions xi(with mean value x ¯) onto

their respective embryo sizes Li(with mean value L ¯) (see

Figure 3A). In our data, the relative embryo size fluctuations

wereoftheorderof10–15%.Theresultsofourscalinganalysis

for Gt, Hb, Kr and Eve in embryos with single, double and

quadruple bcd dosage are presented in Figure 3B. We observe

that anterior domains indeed tend to hyper-scale, whereas

mid-embryo and posterior domains show good scaling

(P-valueo0.00021,cf.SupplementaryTextS1;Supplementary

Figure S5). Moreover, the magnitude of the scaling coefficient

depends mainly on the position of the respective domain

(boundary) rather than the associated gene. This is in good

agreement with the model discussed above, whereas in the

other models we investigated (cf. Supplementary Text S2) it is

difficult to obtain close to perfect scaling both at mid-embryo

and toward the posterior pole.

S ?^b ?

varðLÞ

?

?L

? x

ð4Þ

Perspectives

Precision and scaling are important to achieve robust pattern

formation. Our new position-dependent measures provide a

unified quantification of scaling for both functional descrip-

tions of gradient profiles derived from models (equation (2))

and experimentally determined expression domains (equation

(4)). These measures will be useful to address scaling also in

different systems like the wing-disk. Importantly, our mea-

sures clearly identify perfect scaling (S¼1) corresponding to

the correct preservation of proportions, which is not necessa-

rily equivalent to perfect correlation (see Supplementary Text

S4 for a detailed discussion).

Interestingly, our experimental analysis using embryos with

different bcd dosages showed that both precision and scaling

are more position dependent than gene dependent, suggesting

that the morphogen gradient itself has an important function

Precision and scaling in morphogen decoding

AM de Lachapelle and S Bergmann

4 Molecular Systems Biology 2010

& 2010 EMBO and Macmillan Publishers Limited

Page 5

to set robust domain boundaries. As our study showed,

gradient formation of a single morphogencan naturallyensure

maximal robustness only in a limited patterning domain. For

Bcd, this appears to be mid-embryo, where most of its targets

are expressed. Outside of this domain, other systems and

mechanisms may have evolved to cooperate in maintaining

and potentially increasing robustness. Considering precision

and scaling as position- (and possibly time-) dependent

features will allow to better characterize and develop models

for these systems.

Materials and methods

Image analysis

We use an interface developed in MATLAB (Supplementary Figure S1)

to extract and analyze expression profiles from the staining images of

different Bcd target genes, yielding quantitative information on the

positionsofeachproteindomain(Bergmannet al,2007).Atotalof154

staining images were analyzed using a semi-automated analysis tool

where the position of the anterior and posterior poles were marked 50

times for each embryo. On the basis of this input, the software

extracted a rectangular region from the image from which it generated

20304050

x (%L)

60708090

0

0.5

Scaling coefficient

1

1.5

2

2.5

3

3.5

Stripe 2

ΔL/L

S=1.34 ± 0.20

−0.2 −0.100.10.2

−0.2

−0.1

0

0.1

0.2

Stripe 3

ΔL/L

S=1.18 ± 0.12

−0.2 −0.1 00.10.2

−0.2

−0.1

0

0.1

0.2

Stripe 1

ΔL/L

−0.2 −0.10 0.10.2

−0.2

−0.1

0

0.1

0.2

S=1.62 ± 0.27

Stripe 4

ΔL/L

S=1.07 ± 0.09

−0.2 −0.100.1 0.2

−0.2

−0.1

0

0.1

0.2

Stripe 5

ΔL/L

S=0.97 ± 0.09

−0.2 −0.10 0.1 0.2

−0.2

−0.1

0

0.1

0.2

Stripe 6

ΔL/L

S=0.87 ± 0.10

−0.2 −0.100.10.2

−0.2

−0.1

0

0.1

0.2

Stripe 7

ΔL/L

S=0.85 ± 0.12

−0.2 −0.100.10.2

−0.2

−0.1

0

0.1

0.2

A

B

30

60

90

120

150

180

Time (min)

Figure 3

(images on top), fluctuations in the domain position Dx/x ¯ are plotted against fluctuations in embryo size DL/L¯. Scaling coefficients are then estimated by linear

regression.Errorsshow68%confidenceintervalsfromtheregressionanalysis.(B)Measuredscalingcoefficientsofthegapandpair-rulegeneexpressiondomainsasa

function of position x along the AP axis (see Supplementary Dataset S2). Errors (bars) represent 68% confidence intervals from the linear regression. For Gt (filled

triangles) and Kr (empty triangles), we show results for their left (v) and right (x) boundaries, as well as for the center (n) of their expression domain. The Hb (&)

domainis characterized bythe boundarywhere itsconcentration dropsandthe Evestripes (J)bythe position at whichtheir intensityis maximal(filledcircles represent

Eve co-stained with Hb, as in (A)). Color code: 1xbcd (red), 2xbcd¼wild type (green), 4xbcd (blue). We also show scaling as predicted by our model (grayscale).

Parameters are chosen such that the profile is closest to an exponential decay with length scale l¼0.2L at t¼150min (see Supplementary Figure S12 for temporal

evolution).

Measuring scaling of Bcd target genes. (A) Scaling of the Eve stripes for wild-type bcd mRNA dosage, co-stained with Hb. For each stripe in each embryo

Precision and scaling in morphogen decoding

AM de Lachapelle and S Bergmann

& 2010 EMBO and Macmillan Publishers Limited Molecular Systems Biology 2010 5

Page 6

protein concentration profiles and automatically determined the

positional information of the gap and pair-rule gene expression

domains. Thus, for each embryo, its total length and the domain

localizations were characterized by a mean value and a standard

deviation computed from the 50 markings. The right boundary of the

anterior Hb expression domain was defined as the position at which

the decline in the Hb concentration is the steepest. Gt and Kr domains

were described similarly by their left and right boundaries, whereas

thepositionwithmaximalstainingintensitycharacterizedtheircenter.

As the Eve stripes are not so broad, we only extracted the position at

which the Eve intensity was maximal. (All the staining images are

available for download at http://www.unil.ch/cbg/morphogen/

Images_paper.zip)

We also developed a second, almost fully automated analysis tool

that only requires human input for selecting which of the two poles is

the anterior one (while their position was detected automatically).

This tool was used to analyze the expression domain positions of a set

of 119 embryos, which included 78 (51%) from the first analysis.

Precision and scaling results turned out to be very similar for the two

analysis approaches, suggesting that measurement errors and inter-

embryo variation (e.g. because of different embryo orientations) can

be safely neglected (Supplementary Figure S6). We also note that part

of the observed fluctuations in embryo size may be induced by the

fixation procedure. However, computing precision and scaling by

considering embryos on different slides separately yielded similar

results (Supplementary Figure S7), suggesting that fixation or other

batcheffectshavenopositionalbiastowardeitheroftheembryopoles.

Modeling

WeconsideramodelinwhichBcdisproducedattheanteriorpolewith

production rate s0, diffuses according to a uniform diffusion constant

D, is degraded at uniform rate a and is trapped and released by the

nuclei in the embryo at rates kn, k?n, respectively. The corresponding

coupled partial differential equations for the free morphogen

concentration M(x, t) and the nuclear concentration Mn(x, t) read

(Bergmann et al, 2007; Coppey et al, 2007; Gregor et al, 2007b):

(

qM

qt¼ Dq2M

qMn

qt¼ ?k?nMnþ knM ? N

qx2þ s0dðxÞ ? aM þ k?nMn? knM ? N

ð5Þ

where N is the nuclei density and s0d(x) the source term localized at

x¼0 (anterior pole). We consider zero-flux (Neumann) boundary

conditions at the posterior pole and that there is no morphogen at the

initial time (t¼0). Assuming that nucleo-cytoplasmic exchanges occur

rapidly (qMn/qt|ME0), as demonstrated by the experiments reported

in Gregoret al (2007b), the following effective diffusion equationholds

for the free morphogen (Bergmann et al, 2007; Coppey et al, 2007;

Gregor et al, 2007b):

qMðx; tÞ

qt

¼~Dq2Mðx; tÞ

qx2

? ~ aMðx; tÞ þ ~ s0dðxÞð6Þ

where D ˜¼D/(1þ(N/K)), ~ a ¼ a=ð1 þ ðN=KÞÞ and s ˜0¼s0/(1þ(N/K))

(with K¼k?n/kn). At time t¼150min (corresponding to nuclear cycle

14), the morphogen profile is numerically fitted to an exponentially

decaying gradientwith length scale l¼0.2L. Thus, given a degradation

rate, the diffusion constant is adjusted accordingly (see caption of

Figure 2 for numerical values). We account for the presence of the

nuclei by setting N/K¼1 (i.e. to be specific, we assume that the

probability that external Bcd is trapped equals the probability that

nuclear Bcd is released; yet our qualitative results for precision are

robust with respect to the exact choice of N/K).

According to the nuclear trapping model (equation (5)), the nuclear

concentration cn, which determines the precision and scaling of Bcd

target genes, is given by (see also Coppey et al, 2007)

cn¼

Mn

N ? vn

¼

M

K ? vn

ð7Þ

where nnis the nuclear volume. Importantly, according to equation (7)

the fluctuations in the nuclear concentrations cnare proportional

to those in the external gradient, provided we can neglect variability

in nnand K.

Supplementary information

Supplementary information is available at the Molecular Systems

Biology website (www.nature.com/msb).

Acknowledgements

We thank Naama Barkai, Ben-Zion Shilo and Zvi Tamari for careful

reading of the manuscript, as well as all the members of the Bergmann

Computational Biology Group and David Morton de Lachapelle for

theirsupportandcommentsduringthewholeproject.Inparticular,we

thank Toby Johnson, Karen Kapur and Sascha Dalessi for their help

with the theoretical analyses and are very grateful to Zolta ´n Kutalik

who implemented the fully automated version of the profile extraction

tool. We are also grateful for financial support from the Giorgi-

Cavaglieri Foundation, the Swiss National Science Foundation (Grant

#3100AO-116323/1),SystemsX.ch(throughtheWingXproject)andthe

European Framework Project 6 (through the AnEuploidy project).

Conflict of interest

The authors declare that they have no conflict of interest.

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