advanced review mathematical models of morphogen gradients ...shvartsmanlab.com/pdf/wires...

Advanced Review

Mathematical models ofmorphogen gradients and theireffects on gene expressionStanislav Y. Shvartsman1∗ and Ruth E. Baker2

An introduction to mathematical models of pattern formation by morphogengradients is presented, using the early embryo of the fruit fly Drosophila as themain experimental example. Analysis of morphogen gradient formation is basedon the source–diffusion–degradation models and a formalism of local accumulationtimes. Transcriptional control by morphogens is discussed within the frameworkof thermodynamic site occupancy models of gene regulatory regions. © 2012 WileyPeriodicals, Inc.

How to cite this article:WIREs Dev Biol 2012. doi: 10.1002/wdev.55

INTRODUCTION

Mechanisms of developmental pattern formationcan be assigned to two broad classes,

inductive and self-organized. In inductive mechanisms,the emergence of pattern formation depends onnonuniformities established during previous stages ofdevelopment. In self-organized mechanisms, patternsemerge through spontaneous symmetry breaking, ina way that is largely independent of preexistingasymmetries. Molecular studies of development haveprovided examples of both inductive and self-organized mechanisms. For instance, the overall planof our hand depends on an intricate cascade ofinductive events. At the same time, the quasiperiodicarray of the hair follicles on the dorsal side of thehand is likely to be generated by a mechanism that isself-organized.

In this review, we focus on a specific classof inductive pattern formation mechanisms, whichdepend on morphogen gradients, defined as theconcentration fields of chemicals that act as dose-dependent regulators of cell signaling and geneexpression.1–4 The term morphogen was introducedin a 1952 paper by Alan Turing to describe chemical

∗Correspondence to: [email protected] of Chemical and Biological Engineering and Lewis-Sigler Institute for Integrative Genomics, Princeton University, NJ,USA2Centre for Mathematical Biology, Mathematical Institute,University of Oxford, Oxford, USA

substances that ‘persuade’ cells to acquire a specificidentity.5 The molecular identities of morphogensremained a mystery until the late 1980s, when the firstmorphogens were identified by in the early Drosophilaembryo,6–11 an experimental system that has a sim-ple anatomy and can be studied by powerful genetictechniques.

During the first two hours of Drosophiladevelopment, a spatially uniform arrangement ofidentical cells is patterned by graded distributionsof transcription factors, which play a key role inestablishing the body plan of the adult fly (Figure 1).The anterior–posterior concentration gradient ofBicoid (Bcd), a transcriptional activator, is essentialfor specifying the anterior segments of the body.12

The ventral-to-dorsal nuclear localization gradient ofDorsal (Dl), a transcription factor that can act bothas an activator and as a repressor, specifies the spatialarrangement of the muscle, nerve, and skin tissues.13

Graded distribution of a transcriptional repressorCapicua (Cic), with minima at both the anteriorand posterior poles, is important for specifyingthe terminal structures of the embryo.14 Gradeddistributions of Bcd, Cic, and nuclear Dl provideinputs to the regulatory regions of genes involved inmultiple aspects of embryogenesis (Figures 2 and 3).

Each of these gradients arises from asymmetriesin the unfertilized egg. The origin of the Bcdgradient can be traced to the localized spatialpattern of bcd mRNA, which is established duringoogenesis. Following egg fertilization, the bcd mRNA

© 2012 Wiley Per iodica ls, Inc.

Advanced Review wires.wiley.com/devbio

70

DAPI

αBcd

αDI

αCic

D

V

A P

(b)(a)

(c)

(e)

(g)

(d)

(f)

(h)

60

50

40

30Inte

nsity

(au

)In

tens

ity (

au)

Inte

nsity

(au

)

Anterior Posterior

Anterior Posterior

Dorsal DorsalVentral

20

150

100

50

90

70

50

30

FIGURE 1 | Graded distributions of transcription factors in the early Drosophila embryo. (a) Spatially uniform arrangement of nuclei in a fixedembryo, ∼2 h after fertilization. No morphological asymmetries are apparent at this point of development. Nuclei are visualized by DAPI staining.(b) Adult Drosophila, about to take off. Schematic on the right provides the orientation of the anteroposterior (AP) and dorsoventral (DV) body axes.(c) Concentration gradient of Bicoid (Bcd), a transcription factor that organizes anterior development of the embryo. The Bcd distribution is visualizedby antibody staining in a fixed embryo. (d) Fluorescence intensity profile of the signal from an embryo stained with a-Bcd antibody. (e) Nuclearlocalization gradient of Dorsal (Dl), a transcription factor that organizes DV patterning of the embryo. In this image, the nuclear distribution of Dl isvisualized in a vertically oriented embryo. (f) Quantified fluorescence intensity of nuclear Dl, based on a-Dl antibody staining. (g) Fixed embryostained with an antibody that recognizes Capicua (Cic), a transcriptional repressor downregulated at the terminal regions of the embryo. (h) Thegraded pattern of Cic concentration has minima at both anterior and posterior poles.

is translated into a protein that generates a long-range concentration profile. The gradients thatcontrol dorsoventral (DV) and terminal patterningarise through a different mechanism, which involveslocalized activation of receptors on the surface of theembryo. In both cases, the spatially uniform receptorsare activated by ligands generated in a spatial patternthat is also predefined during egg development. Thus,early development of the Drosophila embryo providesa clear example of multiple concurrent inductiveevents.

Studies of pattern formation in the earlyDrosophila embryo provide insights into generalprinciples of inductive patterning mechanisms. Forexample, some aspects of pattern formation by the Dlgradient are essentially identical to those discoveredduring pattern formation in the vertebrate neuraltube.15,16 Over the past decade, analyses of patternformation in the Drosophila embryo and other

systems that depend on morphogens have begun toundergo a transformation to a stage in which inductivesignals and their effects are studied at a quantitativelevel.17–29 The results of these studies enable theformulation of mathematical models that can be usedto organize existing results and lead to a deeperunderstanding of pattern formation mechanisms.30–40

Our review provides an introduction to thebasic mathematical models of gradient formationand interpretation, using the Drosophila embryo asthe main example. When studying these models, thereader should keep in mind a quote from Turing5:‘. . .a mathematical model of the growing embryo willbe described. This model will be a simplification andan idealization, and consequently a falsification. It isto be hoped that the features retained for discussionare those of greatest importance in the present state ofknowledge.’ Over the five decades since these wordswere written, the state of knowledge has changed


WIREs Developmental Biology Mathematical models of morphogen gradients

(a) (b)

FIGURE 2 | Fluorescent in situ hybridization images showing the expression patterns of some of the genes regulated by Bicoid (Bcd), Capicua(Cic) and Dorsal (Dl). (a) Expression domains of Bcd and Cic targets along the anteroposterior (AP) axis of the embryo. Note that hunchback (hb;yellow), Kruppel (Kr; green), knirps (kni; red), giant (gt ; blue), tailless (tll; magenta) and huckebein (hkb; orange) are expressed in distinct domainsalong the AP axis. In anterior region of the embryo, the expression domains of these genes dependend on the activating functions of the Bcd gradientand the repressive functions of Cic. Anterior: left. (b) Expression domains of Dl targets. snail (sna; blue) is only detected in the ventral regions of theembryo, while rhomboid (rho; green) and short gastrulation (sog; yellow) are expressed in more lateral domains and zerknullt (zen; red) is confinedto the dorsal region. Shown are sections through the embryo, ventral: bottom, dorsal: top.

dramatically. With this has come a change in thestatus of mathematical models, which can be testednow using an ever increasing number of experimentaltechniques. However, striking a balance between therealism of mathematical models and their practicalvalue remains an art. As a rule, it is best to startwith the simplest possible model, which captures theessence of observed patterns and can be used to do aback-of-the-envelope calculation. This is the approachwe take in our review, which is addressed to studentsof developmental and quantitative biology.

FORMATION OF MORPHOGENGRADIENTS

One of the simplest mechanisms for morphogen gra-dient formation relies on the diffusion and degra-dation of locally produced chemical signals.4,41,42

This mechanism can operate in both intracellularand extracellular compartments, establishing chem-ical gradients with characteristic length scales rangingfrom microns to millimetres. Importantly, each of thethree ingredients of this mechanism can be realized in



(a)

(c)

(b)

FIGURE 3 | Fragments of regulatory networks that convert maternalmorphogen gradients into patterns of zygotic gene expression.(a) Anterior expression pattern of tailless (tll) depends on its directactivation by Bicoid (Bcd) and repression by Capicua (Cic). (b) Dorsal(Dl) expression of zerknullt (zen) is generated by a combination ofgraded Dl, which acts a direct repressor, and spatially uniform Zelda(Zld), an early activator of zygotic transcription. Dl-dependentrepression of zen also depends on Cic (not shown). (c) Lateral stripes ofrhomboid (rho) expression along the dorsoventral axis of the embryoare generated by a network that combines a coherent feedforward loop,an incoherent feedforward loop, and a positive feedback autoregulationmotif. Dl regulates rho both directly and through Twist (twi, anactivator) and Snail (sna, a repressor). In addition to sna and rho, Twialso regulates its own expression.

multiple ways. Signal production can be mediated bylocalized protein synthesis, secretion, or phosphory-lation. Degradation can be realized by proteolysis orreceptor-mediated endocytosis. Diffusion can resultfrom any form of nondirected transport, such asmolecular diffusion or repeated rounds of ligand inter-nalization and recycling.

In most examined experimental systems to date,the basic gradient mechanism is augmented by mul-tiple additional effects, such as directed transportor complex chemical interactions. However, a rela-tively simple class of linear reaction–diffusion modelsfrequently provides a very insightful first step inthe analysis of biological systems. Below we reviewtwo essential characteristics of gradient formationpredicted by these models and illustrate how theycan be used to address some of the key questionsarising in studies of morphogen gradients. First, it isimportant to determine what controls the length scale

of the gradient. Second, it is important to establishwhether gradients reach steady states on time scalesrelevant for developmental patterning. Given a mathe-matical model of gradient formation, answers to thesequestions can be obtained by evaluating an analyti-cal expression for a quantity that we call the localaccumulation time, defined below.

Steady State Gradient and Definitionof Local Accumulation TimeMuch model-based analysis of morphogen gradientsrequires access to time-dependent solutions of theunderlying model equations. For most existingmodels, time-dependent solutions can be obtainedonly numerically. However, given the high uncertaintyin model parameters, detailed numerical solutionsare not always useful. Instead, it is more practicalto have simple analytical expressions that dependexplicitly on model parameters (to analyze the effectsof their variations) and can be used for back-of-the-envelope calculations. We now outline the approachthat leads to such analytical expressions based onthe steady state solution of the model, and give adefinition for the local accumulation time, the latter ofwhich provides a compact summary of concentrationdynamics at a given position within the patternedfield. The approach we present can be readily appliedto mathematical models that describe different formsof signal release, tissues of different sizes, and cascadesof reaction–diffusion processes.43–45

We focus our discussion on models formulatedin one spatial dimension and assume that signalproduction starts at time zero, when morphogen con-centration is zero throughout the system (Figure 4). Atlong times, the time-dependent concentration profileapproaches a steady state that is determined by thebalance of localized synthesis, diffusion, and degrada-tion. The steady state concentration field is denotedby Cs(x), where x is the spatial coordinate within thetissue. In the rest of this presentation, we will assumethat, for all points within the patterned domain, themorphogen concentration approaches its steady statemonotonically, i.e., without overshoots. These casesare clearly important, but we will neglect them fornow.

Cells (or nuclei) at different positions withinthe tissue will reach their approximate steady stateconcentrations on different time scales, which dependon the parameters of the problem and on the distancefrom the morphogen source (Figure 4(c)). As a firststep in dealing with these dynamics, let us considerthe following function, which can be interpreted asthe fractional deviation of the concentration from its



0.8

1

0.6

0.4

C/Cs (0)

F(x,t)C(x,t)/Cs (0)

0.2

0

0.8

1

0.6

0.4

0.2

0

0.8

0.6

0.4

0.2

0

0 2x/λ

4

0 2 40 2tk tk

4

Distance

(a) (b)

(c) (d)

Source

Degradation

Con

cent

ratio

n

FIGURE 4 | Local accumulation time. (a) Schematic of morphogen gradient formation by synthesis, diffusion, and spatially uniform degradation.The source of morphogen production is distributed exponentially. (b) Concentration profile at four different times: t = 1/2k , 1/k , 2/k , ∞. Theconcentration profile has been plotted for λs = 0, and scaled by the maximal concentration at x = 0, Cs (0) = Q/

√Dk . The spatial coordinate has

been rescaled by the dynamic length scale, λ =√

D/k . (c) Concentration dynamics at x = λ/2 and x = 2λ. (d) Local kinetics of the fractionaldeviation from the steady state concentration, plotted at two different positions, x = λ/2 and x = 2λ.

steady value at a given time (Figure 4(d)):

F(x, t) = 1 − C(x, t)Cs(x)

.

Clearly, 1 − F(x, t) = C(x, t)/Cs(x) is the frac-tion of the steady state concentration that has beenaccumulated at position x by time t. Furthermore, thefraction of the steady state value accumulated betweentimes t and t + dt is equal to F(x, t) − F(x, t + dt). Asthe difference between these two times tends to zero,F(x, t) − F(x, t + dt) can be approximated by a timederivative:

F(x, t) − F(x, t + dt) ≈ −∂F(x, t)∂t

dt.

Hence, −[

∂F(x,t)∂t

]dt can be interpreted as the

fraction of the steady state concentration accumulatedbetween times t and t + dt. We can use this to definethe local accumulation time, τ (x), as the mean timeneeded to form a steady gradient at a given pointin space. In this definition, we add the contributionsof all time intervals from zero to infinity, weighedby the corresponding fraction of the steady stateconcentration that has been accumulated during this

interval. The result is the following integral43:

τ (x) =∞∫

0

t[−∂F(x, t)

∂t

]dt.

Importantly, this integral can be evaluated with-out knowledge of the time-dependent concentrationprofile. Details of relevant computations are describedin recent papers, which provide expressions for τ (x)for several canonical versions of the source–diffu-sion–degradation (SDD) model.43–45

Possible configurations of a regulatory regionwith two binding sites for activator A (oval) andB (pentagon).

Using the Local Accumulation Timeto Analyze Gradient DynamicsCalculation of τ (x) can be used to decide whetherthe gradient reaches a steady state on a time scalerelevant for developmental patterning. To do this,τ (x) must be compared to the time allotted for aspecific developmental event. We denote this time byTdev. For instance, the time by which the Bcd gradientmust be established, in order to perform its patterning



function, is less than 90 minutes, from the onset ofBcd synthesis to cellularization.46

When τ (x) is significantly less than Tdev for allx (as a rule of thumb, one can use τ (x) < Tdev/3),the tissue is patterned by a steady state gradient. Thelength scale of this gradient can be extracted fromthe steady state concentration profile, given by Cs(x).On the other hand, when τ (x) is not very differentfrom Tdev or exceeds it, the tissue is patterned by atime-dependent signal.

The length scale of the gradient at a given timecan be extracted from an approximate time-dependentsolution, which can be constructed based only onthe steady profile and the local relaxation time. Toconstruct this approximation, the approach to steadystate at a given point is approximated by a singleexponential, with the decay time equal to τ (x):

Capprox(x, t) = Cs(x) × (1 − exp(−t/τ (x))).

Clearly, the exact and approximate solutionsmatch both when t = 0 and when t = ∞. Theapproximation also works quite well at intermediatetimes that are not much smaller than τ (x).43

Importantly, this approximation allows us to getaround the problem of finding the exact expressionfor the time-dependent solution. Up until this point,we have not specified the model or its parameters. Wedo this in the next section, for a commonly used andexperimentally validated model of the Bcd gradient.First, we provide the analytical expressions for Cs(x)and τ (x) and then supplement these expressions withnumerical estimates of model parameters.

Steady State and Local Accumulation Timein a Model of the Bcd GradientExisting models of Bcd gradient formation havedifferent levels of complexity.47 All of them accountfor the localized synthesis of Bcd protein in theanterior region of the embryo, together with diffusion,and degradation. A commonly used model takes theform of the following reaction–diffusion problem(a recent review provides a good introduction toreaction–diffusion models4):

∂C(x, t)∂t

= D∂2C(x, t)

∂x2 − kC(x, t) + Q(x)

D∂C(0, t)

∂x= 0, D

∂C(L, t)∂x

= 0

C(x, 0) = 0.

C(x, t) represents the time-dependent concentra-tion of Bcd protein, t is the time from the onset of

Bcd synthesis; 0 < x < L is the distance from the ante-rior pole of the embryo with the size of the embryodenoted by L. The first of these equations accountsfor the rate of change of Bcd concentration over time,with contributions from diffusion of the Bcd pro-tein, first order degradation, and spatially distributedsynthesis. The spatial distribution of Bcd synthesisis approximated by a decaying exponential function:Q(x) = (Q0/λs) exp(−x/λs), which represents the gra-dient of bcd mRNA.46 The second and third equationsare the no-flux boundary conditions, reflecting the factthat the Bcd protein cannot leave the embryo. The lastequation states that the protein concentration at t = 0is equal to zero.

This model has only a handful of parameters:the size of the system (L), the signal diffusivity and thedegradation rate constant (D and k, respectively), andtwo parameters that characterize the pattern of signalproduction (Q0 and λs). The experimentally observedgradient decays to zero before the posterior pole17

(x = L), which implies that the model operates in aregime where the solution is independent of the sizeof the system. This reduces the number of parametersto only four. Furthermore, when Bcd synthesis islocalized right at the anterior pole (corresponding toλs ≈ 0, an assumption that will be relaxed in thenext section), the steady is given by a single decayingexponential:

Cs(x) = Q0

Dλexp(−x/λ)

where λ =√

D/k is the dynamic length scale, and canbe interpreted as the average distance to which thesignal molecule diffuses before it is degraded.

One can show that local accumulation time is alinear function position43:

τ (x) = 12k

(1 + x

λ

).

Unlike the steady state profile, the localaccumulation time does not depend on the strength ofthe source of morphogen production. This is alwaystrue for linear degradation models.

Combining Analytical Results withNumerical Estimates of Model ParametersIn order to investigate the implications of ourcalculations upon the Bcd gradient, we will nowcombine the expressions for the steady state profileand local accumulation time with numerical valuesof Bcd diffusivity, lifetime, and the spatial patternof synthesis. We will use D ≈ 4 μm2/s, an estimate



based on a combination of fluorescence recovery afterphotobleaching (FRAP) and fluorescence correlationspectroscopy (FCS) studies with a eGFP-tagged Bcdprotein.18 The lifetime of the Bcd protein, equal to1/k for the first order degradation kinetics assumedhere, is ≈50 minutes, based on optical pulse–chaseexperiments with a photoswitchable version of thefluorescently tagged Bcd protein.48

With these choices of D and k, we get λ ≈110 μm, which leads to the spatial profile of τ (x)shown in Figure 5(a). The minimal value of τ (x)is 33 minutes, reached at x = 0 (the ‘anterior poleof the embryo’). The most distant gene known tobe directly controlled by Bcd is located two thirdsalong the embryo, at x ≈ 375 μm.12 At this location,τ (x) = 110 minutes. Comparing these numbers to 90minutes, the upper limit of the time of the Bcd gradientformation, we see that, for a significant fraction of theanteroposterior (AP) axis, concentrations do not reachtheir steady values. In this regime, the spatial profile ofBcd can be estimated using the analytical expression

140(a)

(b)

120

100

80

60

τ (x), min

40

20

1

0.8

0.6

0.4

0.2

0

200

x, μm

C(x,t)/Cs (0)

4000

200

x, μm

4000

FIGURE 5 | Dynamics in the model of Bicoid (Bcd) gradientformation. (a) Local accumulation time as a function of position,calculated for D = 4 μm2/s, 1/k = 50 min, λs = 0 μm (black) andλs = 50 μm (red). (b) Morphogen concentration profiles, evaluated at30 min, 60 min, and at steady state, based on exact and approximatesolutions (black and red curves, respectively).

for Capprox(x, t), and it is remarkably close to the exactprofile,C(x, t) (Figure 5(b)).

We now ask what happens when the spatialpattern of Bcd synthesis is itself distributed in a gra-dient, as suggested by the quantitative analysis ofbcd mRNA? In this case (where λs > 0), an expres-sion for the local accumulation time can also bederived:

τ (x) = 12k

[(1 + x

λ

) λ exp(−x/λ)λ exp(−x/λ) − λs exp(−x/λs)

+ 2λ2s

λ2s − λ2

].

To evaluate this expression, we need λs, ameasure of the spatial extent of signal production.Direct measurements of the spatial pattern of Bcdprotein synthesis are not available. As a proxy for thispattern, we can use the results of fluorescence in situhybridization (FISH) experiments that measured thedistribution of bcd mRNA.46 Quantitative analysisbcd FISH data reveals that 90% of bcd RNA particlesare located in the anterior tenth of the embryo. Basedon these measurements, λs ≈ 50 μm. Combining thisnumber with the previous estimates of λ = 110 μmand 1/k = 50 minutes, we get the red curve inFigure 5(a). We see that the effect of varying λs isquite modest. We stress that these conclusions arebased on parameters that have been estimated veryrecently and are yet to be confirmed by independentmeasurements and experimental techniques.

Signal TransductionFormation of the morphogen gradients that con-trol DV and terminal patterning of the Drosophilaembryo is considerably more complex than formationof the Bcd gradient. In both of these systems, patternformation is initiated by locally produced extracellu-lar ligands that activate uniformly expressed plasmamembrane receptors. The ligand-bound receptor trig-gers a signal transduction cascade that changes theabundance, phosphorylation state, or nuclear local-ization of a transcription factor, which thereby actsas a spatial regulator of gene expression. Thus, pat-tern formation is based on the localized activation ofreceptor-dependent signaling cascade.

For the DV patterning system, the receptoris Toll, which signals through the NF-κb pathwayand culminates in nuclear accumulation of Dl. Tollreceptor signaling is involved in numerous aspects ofcellular responses across species, but its tissue pat-terning function appears to be restricted to insects.For the terminal system, the receptor is Torso, a



tyrosine kinase, which signals through a highly con-served MAPK signaling cassette. Graded activationof receptor tyrosine kinases and MAPK is usedubiquitously throughout the development of multi-ple species. Below, we provide a very brief overviewof the steps between localized activation of Torsoto the generation of a gradient of transcriptionalactivity that organizes terminal patterning of theembryo.49

The spatial pattern of Torso occupancy dependson the localized production of its ligand Trunk, whichis generated in its active form only at the embry-onic poles.50 Active ligand diffuses in the extracellularspace above the plasma membrane. Ligand diffusionis interrupted by its binding to Torso, activating theenzymatic activity of this receptor. Active receptor isinternalized, thus terminating the process of liganddiffusion. Lateral mobility of Torso along the plasmamembrane is very low.51 Thus, Torso plays a dualrole in the formation of the terminal gradient: it trans-duces the signal provided by the extracellular ligandand limits its spatial range.52,53 This situation is com-monly encountered in the formation of morphogengradients in cellular tissues.54

A large number of experimentally characterizedpatterns that depend on locally produced extracellularligands can be described by models based on two-statereaction–diffusion systems, where a diffusible particlecan be in two states: diffusible or immobile.55 In theterminal system, the diffusible particle correspondsto free active Trunk and the immobile particlecorresponds to Trunk bound to Torso. In general, thedegradation rate constants in the two different statesare different. For example, in the terminal system, it isbelieved that ligand is degraded only in the immobilestate (when bound to Torso). Detailed analysis ofgradient formation in two-state reaction–diffusionsystems and cascades of such systems is describedin a number of recent papers.45,55,56

A key step in the terminal patterning processis the formation of the two-peaked pattern of activ-ity pattern of MAPK. MAPK phosphorylates severalintracellular substrates, including Cic, which is essen-tial for the terminal patterning of the embryo. In theabsence of phosphorylation by MAPK, Cic repressesgene expression throughout the embryo. Two of thegenes repressed by Cic are tailless (tll) and huckebein(hkb). In response to phosphorylation by MAPK, Cicis degraded. The Cic degradation pattern is gradedand negatively correlated with the pattern of MAPKactivation.57 This graded pattern gives rise to thelocalized derepression of tll and hkb. Thus, simi-lar to the system that forms the Bcd gradient, theresult of Torso signaling is graded distribution of a

transcription factor. In the rest of the review, we dis-cuss models that can be used to describe how a gradeddistribution of transcription factors is converted intospatial patterns of gene expression.

INTERPRETATION OF MORPHOGENGRADIENTS

Local Regulation of Gene ExpressionA key issue in the context of morphogen-dependentpatterning is to understand how and why differentgenes are expressed in different positions, correspond-ing to different concentrations of a morphogen.3 Asa rule, the expression of a single gene is regulatedby multiple transcription factors. Thus, to understandhow morphogens control gene expression, we needto consider the joint effects of multiple regulators,which may be either dependent or independent on themorphogen gradient.

To simplify things, we will assume that theexpression rate of a given gene, i.e., the number of newtranscripts generated per unit time, at a given positionwithin the tissue, can be written as an algebraicfunction of local concentrations of transcriptionfactors. The form of this function depends on theregulatory region of the gene. Amongst other things,it depends on the binding sites of specific transcriptionfactors and the distances between these sites.

Morphogen gradients can control gene ex-pression both directly and indirectly, throughtranscriptional cascades. An archetypal example ofa transcriptional cascade is regulation of the gap andpair rule genes during Drosophila segmentation. Bcddirectly regulates expression of the gap genes, in theform of broad domains; the products of these geneswork with Bcd to establish striped expression of thepair rule genes.58

The expression levels of both direct and indirecttargets of a morphogen are affected by quantitativemanipulations of the morphogen level. The strength ofthe effect depends on the number and strength of thebinding sites for the morphogen within the regulatorysequences of its specific targets. However, this is onlypart of the story, because each step within sucha cascade can be regulated by additional, auxiliaryfactors that do not depend on the graded signal. Thesefactors can be either spatially uniform or graded. Forinstance, gene expression along the DV axis dependsboth on Dl, which is graded, and Zelda (Zld), auniformly distributed transcriptional activator.59,60

Direct Effects of Morphogen GradientsSome of the key aspects of gene regulation bya combination of graded and uniform signals can



1

0.8

0.6

Act

ivity

0.4

0.2

B A

(a)

(c)

(b)

(d)

0

1

0.8

0.6

Act

ivity

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0

0 0.5CA

CB

CA

KB

KB

1

0 0.5x

10 0.5x

1

FIGURE 6 | Gene regulation by combination of graded and uniformsignals. (a) Possible configurations of a regulatory region with twobinding sites for activator A (oval) and B (pentagon). (b) Input/outputmap of the regulatory region as a function of concentration of factor A,at different concentrations of factor B, or at different values of thecooperativity parameter (see text for details). Model parameters are asfollows (see text for details):KA = 1, KB = 8, 4, 200, and ω = 200.(c) Spatial distributions of two transcription factors. (d) Normalizedspatial pattern of the activity of the regulatory region.

be illustrated by the following model, which canbe viewed as a simplified description of the firststep within a transcriptional cascade. The modelassumes that the expression rate of a gene dependson the probability of finding its regulatory regionbound by a specific complement of transcriptionfactors. This type of mathematical description oftranscriptional regulator is called a site occupancymodel.28,34,61–63

Consider a gene controlled by two factors, Aand B, each of which has a single binding site withina hypothetical regulatory region (Figure 6(a)). Theconcentrations of these factors are denoted by CA andCB. Let us think of CA as the graded concentration of amorphogen; CB reflects the concentration of auxiliarysignal and it can be either uniform or graded.

The regulatory region can be found in fourdifferent states: one state with both sites empty, twostates with a single site occupied, and one state withboth sites bound by their corresponding transcriptionfactors. We assume that binding of the morphogen isa necessary condition for activation of the regulatoryregion. This can be realized in two different ways,when A is bound, either by itself or together with B.When the binding reactions are in equilibrium, theprobability of finding the regulatory region in eitherone of the states necessary for gene activation is

given by

P{A bound} = P{A} + P{A & B}

= CA + CACBω/KB

KA + CA + KACB/KB + CACBω/KB

In this expression, KA and KB are thecorresponding equilibrium-binding constants, whichcharacterize the binding affinities of A and B to theirrespective sites, when taken in isolation. ω is a measureof cooperativity, which reflects the ability of factorB to change the binding affinity of A to its site.Clearly, when ω = 1, P{A bound} = CA/(KA + CA),which corresponds to the case of no cooperativity.When ω > 1, the auxiliary factor B can be viewed asa coactivator that ‘recruits’ factor A to its bindingsite.

Let us consider the implications of this modelfor the spatial control of gene expression whenA is graded and B is uniform. To make thingsspecific, the concentration field of CA is describedby a single exponential (Figure 6(b)), which could beestablished by processes of reaction and diffusion.A representative pattern of transcriptional activity isshown in Figure 6(c). According to the model, thispattern is affected by changes in the level of themorphogen (CA) as well as by changes in the strengthof its binding site (KA). Note that the spatial patternof transcriptional activity can also be affected by theuniform factor, through changes in CB, KB, and ω.Predictions of the model can be used to interpretthe results of recent experiments, which demonstratedthat uniform factors play an important role in generegulation, by morphogen gradients.

Our illustrative model can be generalized tothe case when factor B is a repressor and/or whenits spatial distribution is non-uniform. The mainconclusion from analyzing of such models is thatgene regulation by a morphogen can be tuned in avariety of ways by combinatorial interactions. As aconsequence, several alternative models can explainthe fact that different genes are expressed in dif-ferent patterns. Observed variation in expressionpatterns can stem from differences in the number orstrength of binding factors for the morphogen itself.Alternatively, they can reflect differential responseto auxiliary signals. This point has been illustratedby recent studies of gene regulation by the Bcdgradient.64,65

Dynamic Models of Gene ExpressionBefore moving on to transcriptional cascades, we needto set-up a simple dynamic model that describes



the expression level of one direct target of a spa-tially graded signal. This model provides a connec-tion between the dynamics of patterning signals andspatiotemporal patterns of gene and protein expres-sion. Solution of this model will be used to discusspattern formation by a combination of direct andindirect effects. We use perhaps the simplest possi-ble dynamic model of transcriptional regulation. Thismodel has only two variables, denoted by Cm and Cp,which correspond to the concentration of mRNA andprotein, respectively, of a gene controlled by a gradedchemical signal.

Most importantly, production of mRNA isdetermined by the state of a gene regulatory region. Inthe simplest case, it is proportional to the probabilityof finding this region in a state favorable fortranscription. For illustrative purposes, we will use amodel with a single site for an activator, whose time-dependent concentration is denoted by CA(x, t) andcan be found from the models of morphogen gradientformation. Protein production rate is proportional tothe amount of mRNA. The degradation rates for bothmRNA and protein are assumed to follow first orderkinetics. With these assumptions, the model takes thefollowing form:

dCm

dt= G1

CA

KA + CA− km

degCm

dCp

dt= G2Cm − kp

degCp.

In these equations, G1 and G2 are the propor-tionality constants in the simplified descriptions oftranscription and translation, respectively, and KA

is the affinity of a binding site for an activator. Ifthe rate constant for transcript degradation (km

deg)greatly exceeds the rate constant for protein degrada-tion (kp

deg), we can use a steady state approximationfor the transcript level, which leads to a single variablemodel for protein concentration:

dCp

dt= G2G1

kmdeg

CA

KA + CA− kp

degCp.

Because morphogen concentration, CA, is afunction of space and time, solution of this equationwill depend on both x and t. In the rest of this section,we limit our discussion to steady state solutions,which correspond to the case when the time scaleof protein degradation is faster than the time scaleof morphogen gradient formation. This leads to thefollowing expression for the spatial distribution ofprotein product of a gene regulated by a morphogen

gradient:

Cp(x, t) ≈ G2G1

kmdegkp

deg

CA(x, t)KA + CA(x, t)

= Cmaxp

CA(x, t)KA + CA(x, t)

.

Here Cmaxp ≡ G2G1/(km

degkpdeg) is the maximal

possible protein concentration, which is attainedwhen activator concentration greatly exceeds theequilibrium-binding constant (CAKA). In this regime,Cp(x, t) is directly proportional to the probability offinding the gene regulatory region in a state favorablefor transcription. Under the assumptions above, thesame will hold true for models of more complexregulatory regions, e.g., those controlled by multipleactivators and repressors.

Formation of Nonmonotonic Patternsby Incoherent Feedforward LoopsOne of the simplest and most commonly encounteredmotifs in transcriptional cascades is an incoherentfeedforward loop, a three-node network where aninput (A) activates a gene (B) and its repressor (C).66

This network can convert a morphogen gradient intoa spatially nonmonotonic pattern of gene expression.To illustrate this point, let us consider a hypotheticalgene regulatory region with two binding sites: one for

1

(a)

(c)

(b)

(d)

B A

B C

A

0.75

0.5

Act

ivity

0.25

0

1

0.75

0.5

0.25

00 0.25

CA

CACC

0.5 0.75 1 0 0.25x

0.5 0.75 1

FIGURE 7 | Spatial signal processing by the incoherent feedforwardloop. (a) Possible configurations of a regulatory region controlled by anactivator and repressor. (b) Structure of an incoherent feedforward loop.(c) Normalized activity of the regulatory region as a function ofmorphogen concentration. Model parameters are as follows (see text fordetails): K C

A = 0.05,K CB = 0.01, K B

A = 50, CmaxB = 35. (d) Normalized

spatial distribution of the morphogen (black) and normalized spatialdistribution of the activity of the regulatory region of gene C (blue).



an activator, distributed in a gradient, and one fora repressor, which is induced by the same activator(Figure 7(a)). We assume that this module is activewhen it is bound by an activator and not boundby a repressor. Furthermore, we assume that bindingreactions are in equilibrium and that binding events ofrepressor and activator are independent. With theseassumptions, the probability of finding the region in astate bound by an activator and no repressor:

P{A bound, B not bound} = P{A}(1 − P{B})

= CA

KCA + CA

KCB

KCB + CB

.

KCA and KC

B are the equilibrium-binding constantsfor repressor and activator within the regulatoryregion of gene C. As before, let us assume thatthe spatial distribution of CA is established byan independent process of morphogen gradientformation, and leads to a given profile, CA(x).

To evaluate the spatial profile of the activity ofthe gene regulatory region in our model we need tofind the spatial profile of repressor. This can be doneusing a model from the previous subsection, in whichrepressor is directly activated through a regulatoryregion with a single binding site, whose affinity for anactivator is denoted by KB

A:

CB(x) = CmaxB

CA

KBA + CA

.

Substituting this into the expression forP{A bound, B not bound}, we get a single variablefunction of activator concentration. Depending on thevalues of model parameters, this function can havea maximum. For instance, when KB

A is much largerthan the maximal concentration of the activator,CB(x) ≈ Cmax

B CA/KBA, this leads to

P{A bound, B not bound}

= CA

KCA + CA

KCB

KCB + Cmax

B CA/KBA

.

This function has a maximum at CA =√KC

AKCBKB

A/CmaxB . If this concentration falls within

the range of the morphogen gradient, transcriptionalactivity of the regulatory region of gene C has amaximum as a function of position. A representativeexample is shown in Figure 7(b). Thus, an incoherentfeedforward loop can convert a spatially monotonicmorphogen gradient into a nonmonotonic pattern ofgene expression. This mechanism is used during the

DV patterning of the embryo, when the expression ofgenes needed for neural development is excluded fromthe presumptive mesoderm region. In this case, Dlactivates both neuroectoderm-specific genes, such assog, and Sna, which represses them in the presumptivemesoderm (Figure 2(c)).

Formation of Patterns with SharpBoundaries: CooperativityRegulatory regions of genes commonly have closelyspaced sites for the same transcriptional regulator.For example, clusters of Bcd and Dl binding sitesare a common feature of multiple genes expressedin the early fly embryo. This structural feature ofgene regulatory regions can lead to highly cooperativebinding of transcription factors, which can resultfrom attractive interactions between proteins boundto adjacent sites. Below, we will use a model withtwo adjacent binding sites for a single factor B. Whencooperative effects are strong, the probability to findthis cluster is in a state with at least one molecule of Bis closely approximated by the following function:

P{B bound} = C2B

K2B + C2

B

.

This is a particular case of a more generalHill-type nonlinearity, f (c) = cn/(kn + cn), whichis commonly used in models of transcriptionalregulation. This function is more switch-like thanthe hyperbolic dependences used in the models of theprevious sections. A measure of a switch-like behavioris provided by the fold change of the input that it isneeded to change the output from 10% to 90% of themaximal value. When n = 1, this requires an 81-foldincrease of the input, almost two orders of magnitude.On the other hand, for n = 2 the same change can begenerated by only a 9-fold change in c. As the valueof n increases, turning a system from 10 to 90% ‘on’requires progressively smaller fold-changes of input.

Gene regulatory regions with homotypic bindingsites for cooperatively binding transcription factorscan display an increase in the sharpness of thetranscriptional response to morphogen gradients. Aclear example of this strategy was provided by studiesof Bcd-dependent regulation of hunchback (hb).23

In a number of cases, the emerging patterns changevery abruptly, sometimes over just a couple cells, alength scale which is considerably less than the lengthscale on which the concentration of a morphogenchanges significantly. Explaining such abrupt changessolely based on a direct response to a morphogengradient requires very high values of n, which is



inconsistent with experimentally measured values thatrarely exceed n = 5. On the other hand, a mechanismbased on a positive feedback loop, which relies ononly modest cooperativity, can generate patterns ofessentially arbitrary sharpness.67

Positive Feedback Circuit: BistabilityTo illustrate the patterning function of circuits withpositive feedback loops, we analyze a hypotheticalgene regulatory region with three binding sites fortwo activators: one site for the morphogen A and twosites for factor B, which is the protein product of thegene controlled by this region. We assume that thereis no cooperativity between factors A and B. On theother hand, binding of B is highly cooperative and isdescribed by a Hill function with n = 2. If we assumethat this regulatory region is active when either oneof the factors is bound, the probability of finding theregion in a configuration favorable for transcriptionis given by

P{A or B bound} = 1 − (1 − P{A})(1 − P{B})= P{B} + P{A}(1 − P{B}).

We can use this expression to construct adynamical model for the concentration of factor B(for clarity, we suppress the dependence of CA and CBon space and time)

dCB

dt= kB,p

deg

[Cmax

B

(C2

B

K2B + C2

B

+ CA

KA + CA(1 − C2

B

K2B + C2

B

))− CB

].

This model has two interesting properties.67

First, the steady state profile of B can change veryabruptly as a function of coordinate. Second, CB canexhibit persistent responses to transient inputs, e.g., tomorphogen concentrations with pulse-like dynamics.In other words, the system has memory. Both of theseproperties stem from the fact that it has multiplesteady states at zero morphogen concentration. WhenCA = 0, the differential equation for CB becomes:

dCB

dt= kB,p

deg

[Cmax

B

(C2

B

K2B + C2

B

)− CB

]. (1)

The number and stability of steady statesin this equation can be determined by analyzingthe intersection of sigmoidal synthesis and lineardegradation terms (Figure 8). When Cmax

B > 2KB, the

5

(a) (b)

(c) (d)

(e) (f)

B

CBmax < 2KB CB

max < 2KB

A

A

B

4

3f(CB) f(CB)

CB CB

2

1

0

0 1 2CB

CA

3 4 5 0 1 2CB

3 4 5

5

4

3

2

1

0

4

3

2

1

00 0.2 0.4 0.6 0.8 0

5

4

3

2

1

00 0.25 0.5

x0.75 1

FIGURE 8 | Spatial signal processing by the positive feedback loop.(a) Possible configurations of a gene regulatory region controlled bytwo activators, one of which has a cluster of homotypic binding sites.(b) Structure of the positive feedback circuit. (c) Geometrical analysis ofsteady states and their stability at zero level of morphogenconcentration. A case with a single stable steady state. Modelparameters are as follows (see text for details): Cmax

B = 3, KA = 10,KB = 2. (d) Geometrical analysis of steady states and their stability atzero level of morphogen concentration. A case with steady statemultiplicity. Parameters are as in (c), but Cmax

B = 5. (e) Parametricdependence of steady states on morphogen concentration (CA ). The‘off’ steady state (blue) disappears beyond critical concentration of themorphogen. Stable and unstable steady states are shown by solid anddashed curves, respectively. (f) Discontinuous spatial profile of thesteady state generated by a positive feedback responding to amorphogen gradient.

curves intersect three times, which means that thereare three distinct concentrations of factor B at whichits synthesis and degradation balance each other. Byanalyzing the signs of the time derivative of CB at eachof these concentrations, we see that the intermediatesteady state is unstable (Figure 8(b)) and the othersare stable.

Consider now what happens to each ofthese steady states as a function of morphogenconcentration, CA. The ‘off’ state exists only belowa critical value of morphogen concentration, denotedby CA,crit. At this concentration, the ‘off’ state collideswith the intermediate unstable steady state and boththem disappear. In contrast, the ‘on’ state remains



stable at all morphogen concentrations, approachingCmax

B when CAKA.This steady state picture is useful in analyzing

the spatial response of an autoregulating circuit to amorphogen gradient. For simplicity, let us assume thatthe gradient is at steady state, given by a monoton-ically decaying function CA,s(x). If the system startsin a state with CB = 0 for x, then the steady statevalue of CB at a given value of x depends on whetherthe morphogen concentration at this location is belowor above the critical concentration, CA,crit. All pointsthat correspond to morphogen concentrations belowCA,crit will reach the steady state that corresponds tothe ‘off’ state of the positive feedback, in its bistableregime. At the same time, wherever CA,s(x) > CA,crit,the system will move to the steady state that is ‘related’to the ‘on’ state at CA = 0. Because of a discontinuouschange in the steady solution as a function of mor-phogen concentration, the steady state profile of CB

will exhibit a discontinuity as a function of position.As a consequence the spatial distribution of CB willbe infinitely sharp.

The same analysis can be used to understand theability of the positive feedback system to remembertransient inputs. Consider a system that has reacheda steady state distribution of CB. What happens whenthe morphogen concentration is now reduced to zerothroughout the system? Based on analysis of the steadystates with CA = 0, we see that the concentration ofCB at all x located to the right of the value that cor-responds to CA,crit will relax back to the ‘off’ state(CB = 0). On the other hand, CB for all x to the left ofthis value, will change only slightly, adjusting to the‘on’ state (Figure 8(c)).

Thus, a positive feedback circuit can convert atransient signal into a stable pattern of transcriptionalactivity. This property is essential for developmen-tal patterning systems, which are commonly activatedby transient inputs. For instance, the Bcd gradientis degraded toward the onset of cellularization andsome of the gene expression patterns established byBcd are maintained by mechanisms that depend onpositive feedback or dual repressor circuit, which alsosupports bistability.24,68 Importantly, similar mecha-nisms have been recently shown to play essentially thesame roles in vertebrate systems.69,70

CONCLUSION

We reviewed a number of simple models that can beused as building blocks in the mathematical analysis ofmorphogen gradients and their effects on gene expres-sion. Biological systems necessarily use a combinationof multiple strategies for the formation and inter-pretation of morphogen gradients.21,33 For example,understanding even the earliest steps of pattern forma-tion by the Dl gradient requires simultaneous analysisof coherent and incoherent feedforward loops andpositive autoregulation.34 As another example, gradi-ent formation in the terminal system relies on a cascadeof reaction–diffusion systems, the output of which isinterpreted by gene regulatory regions that respondto a combination of graded and uniform signals.56,71

The differential contributions of multiple layers ofregulation in such complex systems can be exploredusing the models reviewed in the sections above. Spe-cific examples of this approach can be found in recentpublications.24,25,28,37,38,63,71,72

We focused on patterns in one spatial dimension.However, it is clear that a combination of multiplegraded signals, distributed along different axes of thetissue and acting through gene regulatory regions sim-ilar to ones discussed above can generate a varietyof two-dimensional patterns.73,74 Patterns with mul-tiple peaks, such as the striped expression patterns ofthe pair rule genes, can be described using models inwhich a single gene is controlled by multiple regulatoryregions, which respond to different combinations oftranscription factors. As a first approximation, theseregions can be modelled independently, similar to theapproaches outlined in this review.75

To summarize, the emergence of patterns ofincreasing complexity can be understood based onsimple physicochemical models. Connecting thesemodels to experimentally observed patterns requiresa quantitative analysis of inductive signals and tran-scriptional responses as a function of time, space, andgenetic background. Supported by quantitative experi-ments, mechanism-based, quantitative models provideunique insights into the robustness and evolution ofdevelopmental pattern formation processes. The earlyDrosophila embryo, with its simple anatomy, and awealth of available and rapidly evolving genetic andimaging tools provides a fertile ground for testing thisinterdisciplinary approach.

ACKNOWLEDGMENTS

S.Y.S. thanks Alexander Berezhkovskii, Alistair Boettiger, Oliver Grimm, Johannes Jaeger, Jitendra Kanodia,Yoosik Kim, Ulrike Lohr, Dmitri Papatsenko, Michael Levine, Cyrill Muratov, John Reinitz, Christine Rushlow,



Trudi Schupbach, Ruth Steward, and Eric Wieschaus for helpful discussions. S.Y.S. also thanks Jitendra Kanodia,Bomyi Lim, and Yoosik Kim for invaluable help with preparing the figures.

REFERENCES1. Wolpert L. Positional information and spatial pattern

of cellular differentiation. J Theor Biol 1969, 25:1–47.

2. Wolpert L. One hundred years of positional informa-tion. Trend Genet 1996, 12:359–364.

3. Ashe HL, Briscoe J. The interpretation of morphogengradients. Development 2006, 133:385–394.

4. Wartlick O, Kicheva A, Gonzalez-Gaitan M. Mor-phogen gradient formation. Cold Spring Harb PerspectBiol 2009, 1:a001255.

5. Turing AM. The chemical basis of morphogenesis. PhilTrans Roy Soc Lond 1952, B237:37–72.

6. Struhl G, Struhl K, Macdonald PM. The gradient mor-phogen Bicoid is a concentration-dependent transcrip-tional activator. Cell 1989, 57:1259–1273.

7. Driever W, Nusslein-Volhard C. A gradient of Bicoidprotein in Drosophila embryos. Cell 1988, 54:83–93.

8. Driever W, Nusslein-Volhard C. The Bicoid pro-tein determines position in the Drosophila embryoin a concentration-dependent manner. Cell 1988,54:95–104.

9. Steward R. Relocalization of the Dorsal protein fromthe cytoplasm to the nucleus correlates with its function.Cell 1989, 59:1179–1188.

10. Roth S, Stein D, Nusslein-Volhard C. A gradient ofnuclear localization of the Dorsal protein determinesdorsoventral pattern in the Drosophila embryo. Cell1989, 59:1189–1202.

11. Rushlow CA, Han KY, Manley JL, Levine M. Thegraded distribution of the Dorsal morphogen is initi-ated by selective nuclear import transport in Drosophila.Cell 1989, 59:1165–1177.

12. Porcher A, Dostatni N. The Bicoid morphogen system.Curr Biol 2010, 20:R249–R254.

13. Hong JW, Hendrix DA, Papatsenko D, Levine MS.How the Dorsal gradient works: Insights frompostgenome technologies. Proc Natl Acad Sci 2008,105:20072–20076.

14. Furriols M, Casanova J. In and out of Torso RTK sig-nalling. EMBO J 2003, 22:1947–1952.

15. Briscoe J. Making a grade: Sonic Hedgehog signallingand the control of neural cell fate. EMBO J 2009,28:457–465.

16. Cornell RA, Ohlen TV. Vnd/nkx, ind/gsh, and msh/msx: conserved regulators of dorsoventral neural pat-terning? Curr Opin Neurobiol 2000, 10:63–71.

17. Gregor T, Wieschaus E, McGregor AP, Bialek W, TankDW. Stability and nuclear dynamics of the Bicoid mor-phogen gradient. Cell 2007, 130:141–153.

18. Abu-Arish A, Porcher A, Czerwonka A, Dostatni N,Fradin C. High mobility of bicoid captured by fluo-rescence correlation spectroscopy: implication for therapid establishment of its gradient. Biophys J 2010,99:L33–35.

19. Chung K, Kim Y, Kanodia JS, Gong E, Shvartsman SY,Lu H. A microfluidic array for large-scale ordering andorientation of embryos. Nat Method 2011, 8:171–177.

20. Kanodia JS, Rikhy R, Kim Y, Lund VK, DeLotto R,Lippincott-Schwartz J, Shvartsman SY. Dynamics of theDorsal morphogen gradient. Proc Natl Acad Sci 2009,106:21707–21712.

21. Jaeger J, Blagov M, Kosman D, Kozlov KN, Manu,Myasnikova E, Surkova S, Vanario-Alonso CE, Sam-sonova M, Sharp DH, et al. Dynamical analysis of regu-latory interactions in the gap gene system of Drosophilamelanogaster. Genetics 2004, 167:1721–1737.

22. Jaeger J, Surkova S, Blagov M, Janssens H, Kosman D,Kozlov KN, Manu, Myasnikova E, Vanario-AlonsoCE, Samsonova M, et al. Dynamic control of positionalinformation in the early Drosophila embryo. Nature2004, 430:368–371.

23. Lebrecht D, Foehr M, Smith E, Lopes FJ, Vanario-Alonso CE, Reinitz J, Burz DS, Hanes SD. Bicoidcooperative DNA binding is critical for embryonic pat-terning in Drosophila. Proc Natl Acad Sci U S A 2005,102:13176–13181.

24. Manu, Surkova S, Spirov AV, Gursky VV, Janssens H,Kim AR, Radulescu O, Vanario-Alonso CE, Sharp DH,Samsonova M, et al. Canalization of gene expression inthe Drosophila blastoderm by gap gene cross regulation.Plos Biol 2009, 7:591–603.

25. Umulis DM, Shimmi O, O’Connor MB, Othmer HG.Organism-scale modeling of early Drosophila pattern-ing via bone morphogenetic proteins. Dev Cell 2010,18:260–274.

26. Bollenbach T, Pantazis P, Kicheva A, Bokel C, Gonzalez-Gaitan M, Julicher F. Precision of the Dpp gradient.Development 2008, 135:1137–1146.

27. Kicheva A, Pantazis P, Bollenbach T, Kalaidzidis Y, Bit-tig T, Julicher F, Gonzalez-Gaitan M. Kinetics ofmorphogen gradient formation. Science 2007, 315:521–525.

28. Parker DS, White MA, Ramos AI, Cohen BA,Barolo S. The cis-regulatory logic of Hedgehog gradientresponses: key roles for gli binding affinity, competition,and cooperativity. Sci Signal 2011, 4:ra38.

29. Perry MW, Boettiger AN, Bothma JP, Levine M.Shadow enhancers foster robustness of Drosophila gas-trulation. Curr Biol 2010, 20:1562–1567.



30. Janssens H, Hou S, Jaeger J, Kim AR, Myasnikova E,Sharp D, Reinitz J. Quantitative and predictive model oftranscriptional control of the Drosophila melanogastereven skipped gene. Nat Genet 2006, 38:1159–1165.

31. Perkins TJ, Jaeger J, Reinitz J, Glass L. Reverse engineer-ing the gap gene network of Drosophila melanogaster.PLoS Comput Biol 2006, 2:e51.

32. Vakulenko S, Manu, Reinitz J, Radulescu O. Size reg-ulation in the segmentation of Drosophila: interactinginterfaces between localized domains of gene expressionensure robust spatial patterning. Phys Rev Lett 2009,103:168102.

33. Papatsenko D. Stripe formation in the early fly embryo:principles, models, and networks. Bioessays 2009,31:1172–1180.

34. Zinzen RP, Senger K, Levine M, Papatsenko D. Com-putational models for neurogenic gene expression in theDrosophila embryo. Curr Biol 2006, 16:1358–1365.

35. Ashyraliyev M, Siggens K, Janssens H, Blom J,Akam M, Jaeger J. Gene circuit analysis of the ter-minal gap gene huckebein. PLoS Comput Biol 2009,5:e1000548.

36. Umulis DM, Serpe M, O’Connor MB, Othmer HG.Robust, bistable patterning of the dorsal surface of theDrosophila embryo. Proc Natl Acad Sci U S A 2006,103:11613–11618.

37. Eldar A, Dorfman R, Weiss D, Ashe H, Shilo BZ,Barkai N. Robustness of the BMP morphogen gradi-ent in Drosophila embryonic patterning. Nature 2002,419:304–308.

38. Mizutani CM, Nie Q, Wan FY, Zhang YT, Vilmos P,Sousa-Neves R, Bier E, Marsh JL, Lander AD. Forma-tion of the BMP activity gradient in the Drosophilaembryo. Dev Cell 2005, 8:915–924.

39. Eldar A, Shilo BZ, Barkai N. Elucidating mechanismsunderlying robustness of morphogen gradients. CurrOpin Genet Dev 2004, 14:435–439.

40. Boettiger AN, Levine M. Synchronous and stochasticpatterns of gene activation in the Drosophila embryo.Science 2009, 325:471–473.

41. Crick FH. Diffusion in embryogenesis. Nature 1970,225:420–422.

42. Lander AD, Nie W, Wan FY. Do morphogen gradientsarise by diffusion? Dev Cell 2002, 2:785–796.

43. Berezhkovskii AM, Sample C, Shvartsman SY. Howlong does it take to establish a morphogen gradient?Biophys J 2010, 99:L59–61.

44. Gordon P, Sample C, Berezhkovskii AM, Muratov CB,Shvartsman SY. Local kinetics of morphogen gradients.Proc Natl Acad Sci 2011, 108:6157–6162.

45. Berezhkovskii AM, Sample C, Shvartsman SY. Forma-tion of morphogen gradients: local accumulation time.Phys Rev E 2011, 83:051906.

46. Little SC, Tkacik G, Kneeland TB, Wieschaus EF, Gre-gor T. The formation of the Bicoid morphogen gradient

requires protein movement from anteriorly localizedmRNA. PLoS Biol 2011, 9:e1000596.

47. Grimm O, Coppey M, Wieschaus E. Modelling theBicoid gradient. Development 2010, 137:2253–2264.

48. Drocco JA, Grimm O, Tank DW, Wieschaus EF.Measurement and perturbation of morphogen life-time: Effects on gradient shape. Biophys J 2011,101:1807–1815.

49. Li WX. Functions and mechanisms of receptor tyro-sine kinase Torso signaling: lessons from Drosophilaembryonic terminal development. Dev Dyn 2005,232:656–672.

50. Stein D, Stevens LM. The torso ligand, unmasked? SciSTKE 2001, 4:PE2 (Review).

51. Lloyd TE, Atkinson R, Wu MN, Zhou Y, Pennetta G,Bellen HJ. Hrs regulates endosome membrane invagi-nation and tyrosine kinase receptor signaling inDrosophila. Cell 2002, 108:261–269.

52. Sprenger F, Nusslein-Volhard C. Torso receptor activ-ity is regulated by a diffusible ligand produced at theextracellular terminal regions of the Drosophila egg.Cell 1992, 71:987–1001.

53. Casanova J, Struhl G. The torso receptor localizesas well as transduces the spatial signal specifyingterminal body pattern in Drosophila. Nature 1993,362:152–155.

54. Chen Y, Struhl G. Dual roles for patched in sequesteringand transducing hedgehog. Cell 1996, 87:553–563.

55. Berezhkovskii AM, Coppey M, Shvartsman SY. Signal-ing gradients in cascades of two-state reaction-diffusionsystems. Proc Natl Acad Sci 2009, 106:1087–1092.

56. Coppey M, Boettiger AN, Berezhkovskii AM, Shvarts-man SY. Nuclear trapping shapes the terminal gra-dient in the Drosophila embryo. Curr Biol 2008,18:915–919.

57. Kim Y, Coppey M, Grossman R, Ajuria L, Jimenez G,Paroush Z, Shvartsman SY. MAPK substrate compe-tition integrates patterning signals in the Drosophilaembryo. Curr Biol 2010, 20:446–451.

58. Watson JD, Baker TA, Bell SP, Gann A, Levine M,Losick R. Molecular Biology of the Gene. Cold SpringHarbor: Benjamin Cummings; 2007.

59. Liang HL, Nien CY, Liu HY, Metzstein MM, Kirov N,Rushlow C. The zinc-finger protein Zelda is a key acti-vator of the early zygotic genome in Drosophila. Nature2008, 456:400–U467.

60. Liberman LM, Stathopoulos A. Design flexibility in cis-regulatory control of gene expression: synthetic andcomparative evidence. Dev Biol 2009, 327:578–589.

61. Bintu L, Buchler NE, Garcia HG, Gerland U, Hwa T,Kondev J, Kuhlman T, Phillips R. Transcriptional reg-ulation by the numbers: applications. Curr Opin GenetDev 2005, 15:125–135.

62. Dresch JM, Liu XZ, Arnosti DN, Ay A. Thermody-namic modeling of transcription: sensitivity analysis



differentiates biological mechanism from mathematicalmodel-induced effects. BMC Syst Biol 2010, 4:35–45.

63. Fakhouri WD, Ay A, Sayal R, Dresch J, Dayringer E,Arnosti DN. Deciphering a transcriptional regulatorycode: modeling short-range repression in the Drosophilaembryo. Mol Syst Biol 2009,6:article 341.

64. Ochoa-Espinosa A, Yu D, Tsirigos A, Struffi P, Small S.Anterior-posterior positional information in the absenceof a strong Bicoid gradient. Proc Natl Acad Sci 2009,106:3823–3828.

65. Lohr U, Chung H-R, Beller M, Jackle H. Antagonisticaction of Bicoid and the repressor Capicua determinesthe spatial limits of Drosophila head gene expressiondomains. Proc Natl Acad Sci 2009, 106:21695–21700.

66. Alon U. Network motifs: theory and experimentalapproaches. Nat Rev Genet 2007, 8:450–461.

67. Lewis J, Slack JM, Wolpert L. Thresholds in develop-ment. J Theor Biol 1977, 65:579–590.

68. Gardner TS, Cantor CR, Collins JJ. Construction of agenetic toggle switch in Escherichia coli. Nature 2000,403:339–342.

69. Acloque H, Ocana OH, Matheu A, Rizzoti K, WiseC, Lovell-Badge R, Nieto MA. Reciprocal repressionbetween Sox3 and snail transcription factors defines

embryonic territories at gastrulation. Dev Cell 2011,21:546–558.

70. Saka Y, Smith JC. A mechanism for the sharp transi-tion of morphogen gradient interpretation in Xenopus.BMC Dev Biol 2007, 7:47.

71. Kim Y, Andreu JM, Lim B, Chung K, Terayama M,Berg CA, Jimenez G, Lu H, Shvartsman SY. Gene regu-lation by MAPK substrate competition. Dev Cell 2011,20:880–887.

72. Papatsenko D, Levine M. The Drosophila gap gene net-work is composed of two parallel toggle switches. PloSOne 2011, 6:e21145.

73. Papatsenko D, Goltsev Y, Levine M. Organization ofdevelopmental enhancers in the Drosophila embryo.Nucleic Acids Res 2009, 37:5665–5677.

74. Zeitlinger J, Zinzen RP, Stark A, Kellis M, Zhang HL,Young RA, Levine M. Whole-genome ChIP-chip anal-ysis of dorsal, twist, and snail suggests integration ofdiverse patterning processes in the Drosophila embryo.Gene Dev 2007, 21:385–390.

75. Clyde DE, Corado MS, Wu X, Pare A, Papatsenko D,Small S. A self-organizing system of repressor gradientsestablishes segmental complexity in Drosophila. Nature2003, 426:849–853.


advanced review mathematical models of morphogen gradients ...shvartsmanlab.com/pdf/wires...

Documents