J. R. Soc. Interface Published online 11 March 2009...

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*dumulis@pu

Received 9 JaAccepted 5 F

Analysis of dynamic morphogenscale invariance

David M. Umulis*

Agricultural and Biological Engineering, Purdue University, 225 South University Street,West Lafayette, IN 47907, USA

During the development of some tissues, fields of multipotent cells differentiate into distinctcell types in response to the local concentration of a signalling factor called a morphogen.Typically, individual organisms within a population differ in size, but their body plans appearto be scaled versions of a common template. Similarly, closely related species may differ bythree or more orders of magnitude in size, yet common structures between species scale tohave similar proportions. In standard reaction–diffusion equations, the morphogen range hasa length scale that depends on a balance between kinetic and transport processes and not onthe length or size of the field of cells being patterned. However, as shown here for a class ofmorphogen-patterning systems, a number of conditions lead to scale invariance of themorphogen distribution at equilibrium and during the transient approach to equilibrium.Equilibrium scale invariance requires conservation of the total binding site number and totalinput flux. Dynamic scale invariance additionally requires sufficient binding to slow thediffusion of ligand. The equations derived herein can be extended to the study of otherperturbations to gain further insight into the processes regulating the robustness and scalingof morphogen-mediated pattern formation.

Keywords: scale invariance; bicoid; bone morphogenetic protein; systems biology;development; morphogen

1. INTRODUCTION

Morphogens are secreted molecules that are distributedin a spatially non-uniform pattern over a field ofresponding nuclei or cells that read the local concen-tration of the molecules and react accordingly (Wolpert1969). The patterning of tissues by morphogens is adynamic process with kinetics that evolve on multipletime scales, in environments with multiple length scalesand with chemical species that span different concen-tration scales (Lander et al. 2002; Reeves et al. 2006;Umulis et al. 2008). For instance, in some contexts ofbone morphogenetic protein (BMP) patterning, thetime to pattern a tissue is of the order of hours, receptorequilibration is of the order of minutes, the concentra-tions of ligands, receptors and other regulators rangefrom 10K1 nM to hundreds of nM and the lengths rangefrom 5 mm for individual cell sizes to hundreds ofmicrometres for the overall system length (Lander et al.2002; Umulis et al. 2008). Owing to the multiscalenature of morphogen patterning, it is difficult tointuitively understand the short- and long-termbehaviours of the system, and also understand howthe processes that occur on different scales balance tocontrol the dynamic evolution of morphogens.

Scale invariance of morphogen-mediated patterningoccurs in many different contexts and across differentlength and time scales. For instance, scale invariance of

rdue.edu

nuary 2009ebruary 2009 117

morphogen patterning occurs during anterior/posterior(A/P) patterning of Drosophila syncytial blastodermembryos by bicoid (Gregor et al. 2005), during A/Ppatterning of the Drosophila wing disc by decapenta-plegic (Dpp) (Teleman & Cohen 2000), and also duringdorsal/ventral (D/V) patterning of Xenopus (Ben-Zviet al. 2008) and Drosophila embryos by BMPs(D. M. Umulis, M. B. O’Connor & H. G. Othmer,unpublished data). The processes are very differentsince embryonic A/P patterning relies on the transportof the transcription factor Bcd through the syncytialcytoplasm before binding to nuclei at the embryoniccortex, whereas BMPs are transported around andwithin a cellularized tissue that makes up the wing discin Drosophila or the embryo in Xenopus.

Previously, it was suggested that regulation ofbinding site density could lead to scale invariance of amorphogen distribution at equilibrium (Gregor et al.2007; Umulis et al. 2008). However, morphogenpatterning is a dynamic process in a number of differentcontexts (Umulis et al. 2006; Bergmann et al. 2007;Kicheva et al. 2007), and scale invariance of the spatialdistribution of the morphogen is needed not only atsteady state, but also during the transient approach tosteady state. In this paper, mathematical analysis andcomputer simulations are used to investigate the role ofmorphogen-binding sites in the regulation of dynamicscale invariance. Specifically, the analysis is based onthe following biological observations.

doi:10.1098/rsif.2008.0015J. R. Soc. Interface (2009) 6, –117 9 1191

Published online 11 March 2009

This journal is q 2009 The Royal Society9

http://rsif.royalsocietypublishing.org/

Morphogen scale invariance D. M. Umulis1180


—Nuclei located at the embryonic periphery of Droso-phila embryos (approximately a two-dimensional sheetof nuclei wrapped around a three-dimensional embryo)have been implicated to regulate the dynamics of Bcdtransport (Gregor et al. 2005, 2007) along the A/Pembryo axis. Additional evidence suggests (i) thatnuclei rapidly absorb Bcd protein, and (ii) that thedistribution of Bcd is measurably shallower in unferti-lized embryos that lack nuclei located at the periphery(Gregor et al. 2007).

—Nuclei have been shown to dictate the shape of dpERKsignalling (Coppey et al. 2008), which is consistentwith nuclear binding and release of morphogenanalogous to receptor binding and release. Bymeasuring dpERK signalling during the successivenuclear division cycles of Drosophila development,Coppey et al. showed that (i) the range of dpERK isshortened after each successive nuclear division cycle,which doubles the number of nuclei, and (ii) thedpERK distribution is sensitive to nuclear density,which was demonstrated in shackleton mutantembryos that locally disrupt nuclear density.

—The Bcd gradient is scale invariant between relatedspecies in the order Diptera, and the number of nucleiin the blastoderm embryo between the species isconstant even though the embryo size varies threefold.

—The embryos between different species of Diptera differin size but have similar proportions (suggestingconstant geometry/shape; Gregor et al. 2005).

—To demonstrate the scale invariance of wing imaginaldisc patterning by Dpp, Teleman & Cohen (2000)increased wing disc size by ectopic expression ofphosphoinositide 3-kinase family members in theposterior compartment. These mutations affect cellsize but do not significantly change the number of cells(suggesting a constant number of binding sites).Furthermore, the ectopic expression in the posteriorcompartment did not affect the Dpp-secreting cells,which suggests that the total flux of Dpp in the systemwas unchanged (Teleman & Cohen 2000). Lastly, itwas noted by the authors that they did not observe anincrease in receptor density on the cell surface in thePI3K mutants.

While the details between morphogen-mediatedpatterning mechanisms are different, they all rely onthree basic processes: (i) production/secretion froma source, (ii) transport by diffusion, convection ortranscytosis, and (iii) reaction with receptors, otherregulators and/or DNA (Lander et al. 2002; Reeves et al.2006; Umulis et al. 2006, 2008; Coppey et al.2007, 2008; Gregor et al. 2007; Ben-Zvi et al. 2008).Consider the simple reaction–diffusion equation thatdescribes morphogen transport by diffusion withchemical reactions

vu

vtZDDuC

1

tRðu; xÞ on U

Cboundary conditions on vU:

9=; ð1:1Þ

Here, u is the chemical morphogen [M/L3] (quantities in[ ] denote units:M, quantity; L, length; T, time);D is thediffusion coefficient [L2/T ]; DhP2 is the Laplace

J. R. Soc. Interface (2009)

operator [1/L2]; t is a characteristic time scale [T ] forthe chemical reactions; and R is the vector of chemicalreactions [M/L3] that regulate the morphogen. Even in avery simple description of a morphogen-patterningprocess as shown by equation (1.1), it is clear that thesolution will depend on the balance of time scalesassociated with diffusion and chemical reactions, specificboundary conditions and geometry of U.

To identify conditions of morphogen scale invariance,suppose that the solution of equation (1.1) for the dynamicmorphogen distribution can be represented as a productof independent functions of the form uðx; t;p;LÞZPN

nZ0 Lnðp;LÞ$fnðx;p;LÞ$jnðp;L; tÞ, as commonlyoccurs in the solution of linear systems. Many morphogensystems can be represented as the product of thesefunctions, even if one cannot find an exact solution foreach function. Here, u is the concentration; x is theposition; t is the time; p is the vector of parameters; L is ascalar characteristic length for the system; andLn, fn andjn represent the amplitude, shape and dynamics,respectively, for each component of the series solution.Specific examples of Ln, fn and jn are given later herein.Here, one can determine the sensitivity of each of Ln, fnand jn to perturbations of p and L to gain insight into thedominant parameters that affect each characteristic ofmorphogen patterning. With the product solution, it isclear that to ensure dynamic scale invariance, threeconditions must be met: (i) each amplitude (Ln) isindependent of L, (ii) each shape function (fn) stretchesappropriately in the x -direction with respect to changes inthe length L, and (iii) the time-dependent functions (jn)are independent of L.

Herein, the specific choices for concentration, timeand length scales were varied to analyse threestages of dynamic morphogen-mediated patterning:(i) boundary-layer dynamics near the source for veryearly times, (ii) intermediate receptor–ligand dynamicsof binding and release, and (iii) long-time dynamics ofmorphogen patterning at the tissue scale. Each regimeis nonlinear; however, for the boundary-layer and long-time regimes, the system is approximately linear underappropriate assumptions, which allows for moredetailed analysis and the identification of conditionsfor dynamic scale invariance. To explore the validity ofthe approximate analytical solution, the results werecompared with finite-element numerical solutions of thefull system of partial differential equations.

By using the product solution of the linear approxi-mations for a basic model of morphogen production,transport and reaction, a number of experimentallytestable conditions for equilibrium and dynamic scaleinvariance were identified for a class of morphogen-mediated patterning mechanisms. The main result issummarized in box 1 and many additional details andexplanations are provided throughout the text toaccount for numerous additional insights into thedynamics of morphogen-mediated pattern formation.The conservation of the total number of binding sites,total input flux of morphogen molecules and relativelyhigh receptor density can lead to automatic scaleinvariance of the morphogen at steady state and duringthe transient approach to steady state.


Box 1. Summary of main result for dynamic morphogen scale invariance.

A remarkable feature of organism development is the ability of patterning mechanisms to reliably produce consistentproportions between individuals that vary greatly in size. The preservation of proportion or ‘scale invariance’ is manifest innumerous different biological contexts and has been directly demonstrated in one of the earliest patterning events inDrosophila: A/P patterning by the morphogen bicoid (see main text for references). Most morphogens (including bicoid)develop a spatially non-uniform distribution in the course of a few hours; however, there is increasing evidence that manymorphogen distributions never reach steady state and cells continuously respond to the dynamic evolution of themorphogen. Dynamic scale invariance means that the regulation of gene expression occurs at the same relative position andat the same time between tissues/organisms that differ in size. For example, dynamic scale invariance would ensure that theboundary of gene expression for the Bcd target hunchback would occur at 50 per cent embryo length (L) after the sameduration of development regardless of the individual organism sizes.

Preservation of proportion is not a property of reaction–diffusion systems in general, but the analysis herein suggeststhat it may be a property of numerous morphogen-patterning mechanisms that involve the diffusion (with rate D) of amorphogen (C ) that binds to sites (R) with rates k1 and kK1 to form RC

� �, which undergoes endocytosis/decay with rate ke

as shown in the following equations:

ligand:vC

vT ZDv2C

vx2Kk1R$C CkK1RC ; ðB 1Þ

bound receptor:vRC

vT Z k1R$CKðkK1 CkeÞRC ; ðB 2Þ

boundaries: KDvC

vxð0ÞZ j;

vC

vxðLÞZ 0; 0%x%L; ðB 3Þ

and

total receptor: RT ZRCRC : ðB 4Þ

Here T is time; x is position; j is the input flux at the source; and RT is the total binding site density. Substituting scaledposition xZx/L into equation (B 1), and solving for the long-time behaviour of the system, leads to ((2.33) in text)

ligand:vC

vT ZD=L2

1CRT=Km

� �|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

D�

v2C

vx2K ke

RT=Km

1CRT=Km

� �|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

k�

C ;where Km Z ðkK1 CkeÞ=k1;Cboundary conditions; 0%x%1:

ðB 5Þ

For convenience, conditions for steady-state scale invariance are identified first and the results are extended to analysisof the dynamic problem. At steady state, the solution of equation (B 5) is approximately exponential (for the rangeexhibited by most morphogens), which gives

C ZC0expðKx=lÞ; ðB 6Þ

and

lZffiffiffiffiffiffiffiffiffiffiffiffiffiD�=k�

pZ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDKm

keRTL2

s: ðB 7Þ

The steady-state shape of the distribution will be scale invariant if l is independent of the length L. Since D, ke and Km

are constants, this requires RTfLK2, which occurs automatically in patterning contexts that: (i) conserve the total numberof binding sites, receptors or nuclei (constant total numberNT), and (ii) have similar geometries so thatRTZsNT/L

2 wheres is a geometric proportionality constant. The amplitude C0 in equation (B 6) additionally requires that: (iii) the totalmorphogen production at the boundary (molecules/time) is constant ((3.8) in text).

Substituting RTZsNT/L2 into equation (B 5) gives

ligand:vC

vT ZsNT=L

2

Km CsNT=L2|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

f ðLÞ

,DKm

sNT

v2C

vx2KkeC

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

L�independent

: ðB 8Þ

By inspection of equation (B 8), dynamic morphogen scale invariance additionally requires: (iv) that sNT/L2[Km in

which case f(L)/1. This means that the binding site density RT is greater than the Km value (approximate Km valuesrange from 1 to 1000 nM). How does binding lead to dynamic scale invariance? Binding of morphogens to immobile sitesRT

dynamically slows diffusion. Lowering the binding site density by increasing the length L increases the effective diffusion byan amount that exactly offsets the longer time it takes for diffusion in a larger domain. The boost in effective diffusion leadsto a system that is scale invariant dynamically and at steady state.

Morphogen scale invariance D. M. Umulis


1181



input flux

input flux

input flux

zero flux

zero flux

zero flux

surface reactions

surface reactions

surface reactions

(a)

(b)

(c)

Figure 1. Example geometries of morphogen-patterningpathways. (a) Diffusion and nuclear import. (b) Extracellulardiffusion with receptor binding and internalization over a fieldof cells. (c) Diffusion through an extracellular matrix withbinding along the basolateral membrane.



2. RESULTS

While the specific biological mechanisms of morphogen-mediated patterning are very different in differentcontexts (figure 1), the underlying physical processesthat establish a non-uniform morphogen distributionare quite similar. Consider three different morphogenprocesses that occur in Drosophila: (i) diffusion andnuclear import/export of a transcription factor in asyncytium as occurs for Bcd and dpERK signalling(figure 1a; Gregor et al. 2005, 2007; Coppey et al. 2008),(ii) diffusion in a thin fluid layer over a field of cells withreceptor binding and endocytosis as occurs as acomponent of BMP patterning of the dorsal surface(figure 1b; Mizutani et al. 2005; Wang & Ferguson 2005;Umulis et al. 2006), and (iii) diffusion and receptorbinding around columnar epithelial cells as occurs inthe wing imaginal disc (figure 1c; Reeves et al. 2006).In each of the examples, the mechanism of morphogenremoval occurs in a region that can be approximated asa two-dimensional surface: (i) binding to DNA in nucleilocated at the periphery of a syncytial embryo,(ii) receptor binding and endocytosis on a cellularmembrane, and (iii) binding to receptors located alongthe basolateral region of columnar cells. Under appro-priate geometric simplifications and assumptions(details given elsewhere), each morphogen process canbe approximated as a thin sheet that extends in thex -direction. Assume that the surface reactions involve


only binding to a receptor and decay of the receptor–ligand complex, and to simplify the analysis, supposethat the total amount of receptors on the surface isconstant in time. Here, the term receptor is used looselyto more generally mean binding sites that can representregions of specific and non-specific binding of atranscription factor to DNA, the binding to andsubsequent transport through nuclear pore complexes,or binding to receptors located on a cellular membrane.Let Lx, Ly and Lz be the lengths in the x-, y- and z-directions, respectively, and let C be the concentrationof a morphogen in the fluid and RS the concentration ofreceptor on the surface zZ0. Suppose there is a fixedinflux of C on the boundary xZ0, and zero flux on theremaining faces except zZ0, then the governingequations can be written as follows:

vC

vT ZDDC in U; ð2:1Þ

vRS

vT ZfRK k1RSC CkK1RSCK keRS on z Z 0;

ð2:2Þ

vRSC

vT Z k1RSCKðkK1 CkeÞRSC

Z k1RSTCKðk1C CkK1 CkeÞRSC on z Z 0;

ð2:3Þ

KDvC

vzZKk1RSC CkK1RSC on z Z 0; ð2:4Þ

and

KDvC

vx

��0

Z j KDVC Z 0 elsewhere

where

x 2 ½0;Lx �; y2 ½0;Ly�; z 2 ½0;Lz �:

ð2:5Þ

Here, C is the bulk concentration of the morphogen[M/L3]; RS and RSC are the free and bound bindingsites on the surface [M/L2]; T is the time; k1 and kK1

are the forward and reverse binding rates of C to R withunits [(M/L3$T )K1] and [TK1], respectively; and k e isthe decay/removal rate with units [TK1]. Assumingthat the initial condition of free receptors is atequilibrium and the initial levels of C are zero, thenRSTZfR/k e for all time. As a result, the total amountof receptor is constant at every point on the surface zZ0,which leads to the conservation condition RSCRSCZRST (where RST is a constant total level of receptors).

Furthermore, if Lz/Lx ;Ly, then equations(2.2)–(2.4) can be averaged over Lz, which leads to avolumetric receptor density RT;RC and R, whereRTZRSTL

K1z , RCZRSCLK1

z and RZRSLK1z . In this

scenario, the equations for receptor binding on onesurface in a thin gap are completely equivalent to asystem that has a uniform volumetric binding sitedensity, such as binding sites in the extracellular matrixor import/export in nuclei. In view of the boundaryconditions, the solution must be constant in they- direction since the initial conditions are constant in


Table 1. Time and concentration scales. ((bl) denotes boundary layer. Magnitude is a qualitative estimate of the relative rate,concentration or length, and cross-comparisons should only be made between those in the same regime: either boundary layer(bl) or for longer times, i.e. the tRd time scale is slow when compared with other boundary-layer processes, whereas tR isrelatively fast when compared with other processes that regulate the long-time behaviour. Note that the scalings lm,Km, RT, tB,tU and tE are the same for both the boundary-layer and longer time dynamics.)

parameter (symbol) description magnitude units reference

Lx (Lx) system length variable [L] —ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1RT=D

p(lm) minimum diffusion length scale ? [L] —

C0 (C0) scaling concentration of C small [M/L3] Shimmi & O’Connor (2003)(kK1Ck e)/k1 (Km) concentration of half-maximal occu-

pancy? [M/L3] Shimmi & O’Connor (2003)

RT (RT) total binding site concentration variable [M/L3] estimate; Lauffenburger &Linderman (1993)

RTC0/(KmCC0) RC0

� �equilibrium RC concentration variable [M/L3] estimate

L2x=D (tD) diffusion time for Lx slow [T ] Gregor et al. (2005); Lauffenburger

& Linderman (1993)(k1C0)

K1(tC) production of RC by binding fast [T ] estimate

(k1RT)K1 (tB) removal of C by binding fast [T ] estimate

(kK1)K1 (tU) release of C from the binding site fast [T ] Umulis et al. (2006)

(k e)K1

(tE) endocytosis rate of RC slow [T ] Mizutani et al. (2005)

(k1C0CkK1Ck e)K1 (tR) measure of receptor equilibration fast [T ] estimate

C0Lx/j (tj) input flux time scale slow [T ] estimate

boundary-layer groupsl x (l x) boundary-layer length ðl x/LxÞ small [L] —Cd (Cd) scaling concentration of C (bl) small [M/L3] Shimmi & O’Connor (2003)RTCd/(KmCCd) RCd

� �equilibrium RC concentration (bl) ? [M/L3] estimate

l 2x=D (tDd) diffusion time for l x (bl) fast [T ] estimate

(k1Cd)K1

(tCd) production of RC by binding (bl) slow [T ] estimate

(k1CdCkK1Ck e)K1 (tRd) measure of receptor equilibration (bl) slow [T ] estimate

Cdl x/j (tjd) input flux time scale (bl) fast [T ] estimate

Morphogen scale invariance D. M. Umulis 1183


the Ly-direction. This gives the simplified equations

vC

vT ZDv2C

vx 2K k1R$C CkK1RC ; ð2:6Þ

vRC

vT Z k1R$CKðkK1 CkeÞRC ; ð2:7Þ

RT ZRCRC ; ð2:8Þ
and
KDvC

vx

��0

Z jvC

vx

��Lx

Z 0: ð2:9Þ

To understand the solution over different lengthand time scales, equations (2.6)–(2.9) are non-dimensionalized by the general scalings a, b, L and t,which non-dimensionalize the concentration cZC/a,concentration rcZRC=b, position xZx/L and timetiZtK1T (iZ1, 2, 3 for the early-, intermediate- andlong-time behaviour, respectively). Equations (2.6)–(2.9) reduce to

tK1 vc

vtiZ

D

L2

v2c

vx2K k1RTcC k1bcC

kK1b

a

� �rc;

ð2:10Þ

tK1 vrc

vtiZ

k1RTa

bcK½k1acCkK1 Ck e�rc; ð2:11Þ

and

Kvc

vx

��0

ZjL

DaK

vc

vx

��Lx=L

Z 0: ð2:12Þ


2.1. Time-scale analysis

The dynamics of morphogen reaction and transport inequations (2.10)–(2.12) occur on multiple time scales.For instance, one would expect that the time scalefor receptor equilibration is much faster than that fortransport by diffusion and other processes. To gaininsight into the balance of the processes that governmorphogen dynamics on different time scales, one mustmake appropriate choices for t, a, b and L in an effort tofind a solution that captures the fast, intermediate andslow dynamics. The system has many possible lengths,concentration and time scales, and different com-binations can be applied to study the behaviour indifferent regimes. It is not clear a priori what theappropriate choices for t, a and b are for the differentregimes, so that many possible parameter combinationsand a brief description of the parameter group are listedin table 1.

The concentration scale Km is equivalent tothe Michaelis–Menten constant and is defined byKmZ(kK1Cke)/k1. The subscript d denotes that thetime scale is defined for the boundary layer d.

If the concentration of ligand and receptor is initiallyzero everywhere, the early dynamics should begoverned by processes a short distance (l x) from thesource at the boundary. Specifically, the early dynamicswill be governed by the input flux (tjd), short-rangediffusion (tDd) and receptor binding (tB). A shorttime after that, other processes, such as receptor




equilibration (tR), endocytosis (tE) and long-rangediffusion (tD), contribute to the dynamics over the fulllength Lx. In the following sections, a number of differenttime-scale hierarchies are selected to analyse theboundary-layer, intermediate- and long-time regimes,which are applicable to numerous biological pathways.

2.2. Part I: boundary-layer dynamics

It is difficult to select a characteristic length for theboundary layer (l x) a priori, so it is assumed to besufficiently small so that the boundary-layer dynamicsare faster than all the other processes. Choosing LZl xfor the length scale, tjdZCdl x/j, so that t1ZtK1

jd T ; for

the time scale, aZCd and bZRTCd/(KmCCd) for theconcentration scales and substituting them intoequations (2.10)–(2.12) leads to the following equations:

vc

vt1Z

tjd

tDd

v2c

vx2dK

tjd

tBcC

tjd

tB

tRd

tCdcC

tRd

tU

� �rc; ð2:13Þ

vrc

vt1Z

tB

tRd

tjd

tBcK

tjd

tB

tRd

tCdcC

tRd

tUC

tRd

tE

� �rc

� ; ð2:14Þ

and

Kvc

vxd

��0

Ztjd

tDd

� K1

; xd 2 ½0;Lx=l x �: ð2:15Þ

The very early dynamics of ligand (c) and boundreceptor rcð Þ can be easily understood by the analysis ofthe relative time scales in equations (2.13)–(2.15).First, assume that in this scale, Oðtjd; tDdÞ/OðtB; tRd; tCd; tU; tEÞ. All the kinetic terms in equations(2.13) and (2.14) disappear and the very early dynamicsare determined solely by the flux at the boundary (2.15)and diffusion away from the source (2.13). If the flux timescale is much shorter than the diffusion time scaleðtjd/tDdÞ, the ligandaccumulates at the sourcewithoutdiffusing away. If the diffusion time scale in the boundaryis shorter than the input flux time scale ðtDd/tjdÞ, thenthe ligand immediately diffuses away from the source, theflux term is essentially zero and nothing appears tohappen.When the flux and diffusion time scales are of thesame order of magnitude (tDdztjd), the boundary-layerequations give rise to a diffusion equationwith a constantsource at xZ0 and diffusion coefficient tjd/tDd. Thesethree scenarios represent the very short-time behaviourat the source before any appreciable receptor binding.

Shortly after production at the boundary begins,receptor binding (tB) occurs close to the source. Earlyduring patterning, there will be an excess of bindingsites relative to ligand levels (i.e. Cd/RT), whichmeans that tB/tRd. Suppose that OðtDd; tjd; tBÞ/OðtRd; tU; tCd; tEÞ, defining edZtB/tRd and noting thattRd!tCd,tU (by definition), equations (2.13)–(2.15)lead to the following equations, with a source at xdZ0:

vc

vt1Z

tjd

tDd

v2c

vx2dK

tjd

tBc; ð2:16Þ

vrc

vt1Z ed

tjd

tBcK

tjd

tB

tRd

tCdcC

tRd

tUC

tRd

tE

� �rc

� ; ð2:17Þ


and

Kvc

vxd

��0

Ztjd

tDd

� K1

; xd 2 ½0;Lx=l x �: ð2:18Þ

In the t1 time scale, this simplifies to a linearreaction–diffusion equation on a semi-infinite intervalin the limits (lim l x/0, lim ed/0)

vc

vt1Z D

v2c

vx2dK kc ð2:19Þ

and

Kvc

vxd

��0

ZJ ; xd 2 ½0;NÞ: ð2:20Þ

Here, the diffusion coefficient is DZtjd=tDd, the

decay term is kZtjd=tB and the flux is JZtDd=tjd.Solving equation (2.19) with the boundary condition(2.20) gives

cðxd; t1ÞZJffiffiffiffiffiffiffiffiffik=D

q exp K

ffiffiffiffiffiffiffiffiffik=D

qxd

�

! 1K1

21CErf

K2ffiffiffiffiffiffiffiDk

pt1 Cxd

2

ffiffiffiffiffiffiffiffiDt1

q264

375

8><>:

0B@

Cexp 2

ffiffiffiffiffiffiffiffiffik=D

qxd

� Erfc

2ffiffiffiffiffiffiffiDk

pt1 Cxd

2

ffiffiffiffiffiffiffiffiDt1

q264

3759>=>;1CA:

ð2:21Þ

Solutions of the very short-time dynamics given byequation (2.21) are plotted along with the finite-element solution of equations (2.10)–(2.12) in figure 2a.

2.3. Part II: intermediate- andlong-time dynamics

2.3.1. Intermediate dynamics. Following the short-timedynamics in the boundary layer, other processesdetermine the intermediate dynamics. Two importantparameter groups that relate the different time scalesare e1ZtR/tE and e2ZtB/tD. The first (e1) relates theratio of the time scale of receptor equilibration (tR) tothe time scale for the total loss of C by endocytosis/decay of the RC complex (tE), and e2 relates the timescale that C is absorbed by binding sites (tB) to thetime scale for diffusion (tD). As will be shown later, for areasonable morphogen profile, e1=e2zOð1ÞKOð10Þand more evidence for this is given later herein.Substituting aZC0, bZRTC0(KmCC0)

K1 and tZtRso that t2ZtK1

R T , e1 into equations the following andrearranging gives the following equations for theintermediate dynamics:

vc

vt2Z

tR

tD

v2c

vx2K

tR

tBcC

tR

tB½gcCqK e1�rc; ð2:22Þ

vrc

vt2Z cK½gcCq�rc; ð2:23Þ


0.3

0.4

0.2

0.1

0T (s) T (s)

0.01

FEM 0.02

0.03

0.04

0.05

0.06

100 200 300 400 0 100 200 300 400

= 0

= 0.02

= 0.04

(a) (b)

Figure 2. (a) Solutions from the finite-element method (FEM) simulation of equations (2.6)–(2.9) (solid line), boundary-layerequation (2.21) (dashed line) and equation (2.39) (dotted line). (b) Plot of the LN norm for the boundary-layer (solid line) andlong-time (dotted line) solutions versus the FEM solution.



Kvc

vx

��0

ZtD

tj; K

vc

vx

��1

Z 0; x2 ½0; 1�; ð2:24Þ

gZC0

Km CC0

; qZKm

Km CC0

; ð2:25Þ

and

0!g; q!1; qCgZ 1: ð2:26ÞIt appears that on the intermediate time scale,

equation (2.22) is governed principally by the receptorequilibration kinetics; however, it is not clear at thispoint whether the ratio of time scales for receptorequilibration to receptor binding (tR/tB) is large orsmall. If small, the receptor-binding kinetics do not playa major role in the dynamics of c, even on theintermediate time scale. As expected, for equation(2.23), the receptor-binding kinetics dominate theevolution of rc. Equations (2.22)–(2.24) for theintermediate time scale are nonlinear and providelimited additional information about the dynamicevolution of c. However, the intermediate scale leadsnaturally to the slower time scale that corresponds tothe long-time and long-range morphogen patterning.

2.3.2. Long-time dynamics. After the initial boundary-layer formation and initial receptor-binding kineticswhen t2zOðeK1

1 Þ or t2zOðeK12 Þ, the slower processes of

long-range diffusion and ligand endocytosis/decaydictate the dynamics of the morphogen distribution.

Defining t3Ze2t2 and substituting this into equations(2.22) and (2.23) rescales the equations for rc and c forlarge times,

e2tB

tR

vc

vt3Z e2

v2c

vx2KcC ½gcCqK e1�rc ð2:27Þ

and

e2vrc

vt3Z cK½gcCq�rc: ð2:28Þ

Adding equations (2.27) and (2.28) gives the followingequation for the total amount of morphogen (freeCreceptor bound):


tB

tR

vc

vt3C

vrc

vt3Z

v2c

vx2K

e1

e2rc: ð2:29Þ

Taking the limit e2/0 of equations (2.27)–(2.29)with the condition that e1/e2/const. leads to analgebraic equation for rc and the long-time behaviourfor the total amount of ligand. Solving for rc,calculating the derivative of rc with respect to t3 andsubstituting the solution into equation (2.29) gives

rc Zc

gcCq; ð2:30Þ

vc

vt3Z

tR=tB

1CðtR=tBÞðq=ðgcCqÞ2Þ

� �v2c

vx2

Ke1

e2

ðtR=tBÞð1=ðgcCqÞÞ1CðtR=tBÞðq=ðgcCqÞ2Þ

� �c; ð2:31Þ

and

Kvc

vxZ

tD

tj: ð2:32Þ

2.4. Solution of linear approximation forlong-time dynamics

The resulting equation for the long-time behaviour ofunbound c is a nonlinear PDE with no general solution.However, the equation is linear if the level of ligand C0

is small, so that the total amount of bound ligand ðRC Þis far from saturation. Specifically, this requiresC0/Km, so that gc/q, and equation (2.31) becomesa linear non-homogeneous PDE,

vc

vt3Z ~D

v2c

vx2K ~kc; ð2:33Þ

Kdc

dx

��0

Z Jdc

dx

��1

Z 0; ð2:34Þ

~D ZtR=tBK1

; ~k Ze1

e

qK1tR=tBK1

; ð2:35Þ
1Cq tR=tB 2 1Cq tR=tB
and

J Z tD=tj : ð2:36ÞBy introducing v(x,t3)Zc(x,t3)KcE(x), where cE(x) isthe equilibrium solution, the above non-homogeneous


1.0

1.2

1.0

0.8

0.6

0.4

0.2

0

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1.0 0.1 0.2 0.3

0.2 0.4 0.6 0.8 1.0

=0

0.4 0.5

0.4

0.3

0.2

0.1

0 0.5 1.0 1.5

20

15

10

5

0

(a) (b)

(c) (d )

=0.2

=0.4

Figure 3. (a) Example equilibrium solution for the linear approximation (solid line, cE(x)) and finite-element solution of the fullproblem (open circles, cFEM(x)). (b) Comparison of dynamics in the linear approximation (solid line, Clinear) versus the FEMsolution (dotted line, cFEM) of the full system at positions xZ0, 0.2 and 0.4. (c) Accuracy of linear approximation depends on theratio of C0/Km. As C0/Km increases, the approximation and the FEM solution diverge both at (c) equilibrium and(d ) dynamically.T90 is the time it takes to reach 90% of the equilibrium value at xZ0 (solid line), xZ0.2 (dashed line) and xZ0.4(open circles). For the figures herein, the following parameter values are used: DZ20 m2 sK1; k1Z5!10K3 nMK1 sK1; kK1Z1!10K2 sK1; k eZ5!10K4 sK1; NtotZ105 molecules; rZ0.1; LzZ1 mm; and FZ22 molecules minK1.



PDE for c(x,t3) is converted into a simpler homo-geneous PDE in the displacement variable v(x,t3). Theequilibrium solution for equations (2.33)–(2.36) is

cEðxÞZJ

m

emð2KxÞ Cemx

e2mK1

" #ð2:37Þ

and

mZffiffiffiffiffiffiffiffiffi~k=~D

q: ð2:38Þ

Also, the exact transient solution is

cðx;t3ÞZcEðxÞCXNnZ0

An cosðnpxÞexpðKð~kC~DðnpÞ2Þt3Þ;

ð2:39ÞA0 ZK~DJ=~k; ð2:40Þ

andAn ZK2~DJ=ð~kC ~DðnpÞ2Þ; n Z 1; 2;.: ð2:41Þ

The solution for the linear approximation withnZ100 is shown along with a finite-element solutionof equations (2.6)–(2.9) for very early times infigure 2a,b and the equilibrium solution (2.37)


is shown in figure 3a for C0/Km. As shown infigure 2a,b, the long-time behaviour better approxi-mates the finite-element method (FEM) solution afterthe first 100 s (parameters are listed in the caption tofigure 3). Figure 3b shows the transient evolution ofc(x,t3) at xZ0, 0.2, 0.4 for the linear approximationversus the FEM solution for long times. Figure 3a,bdemonstrates that the linear approximation capturesthe equilibrium and dynamic behaviour of the fullnonlinear system when C0/Km. Note that C0 isdetermined by setting xZ0, cE(0)Z1 in equation(2.37). As expected, the linear approximation breaksdown as C0/Km both at equilibrium (figure 3c) anddynamically (figure 3d ).

3. CONDITIONS FOR SCALE INVARIANCE

The solutions for the early- and long-time dynamics ofequations (2.6)–(2.9) provide a framework to investi-gate plausible mechanisms of biological scale invar-iance. To make the analysis more straightforward,conditions for scale invariance of the equilibrium




solution are identified first, followed by a brief analysisof the boundary layer and a more extensive analysis ofthe long-time behaviour of the system. Since the timescale of many morphogen-mediated pathways is of theorder of tens to hundreds of minutes, the long-time andequilibrium behaviours are most relevant for realbiological morphogen pathways.

3.1. Conditions for scale invariance of theequilibrium solution

The equilibrium solution (2.37) can be represented asa product of an amplitude and shape function,cE(x,p,Lx)ZL(p,Lx)$f(x,p,Lx).

Equation (2.37) can be rewritten as

LZJðe2mC1Þmðe2mK1Þ ; fZ

emð2KxÞ Cemx

e2m C1: ð3:1Þ

Now substituting the parameters ~D and ~k intom2Z ~k=~D gives

m2 Zðe1=e2ÞððqK1tR=tBÞ=ð1CqK1tR=tBÞÞ

ðtR=tBÞ=ð1CqK1tR=tBÞ

Ze1

e2qK1z

e1

e2: ð3:2Þ

Earlier, it was speculated that the ratio e1=e2zOð1ÞKOð10Þ for a reasonable morphogen profile. SinceC0/Km in this context, it follows that qK1z1 and thedecay length of the morphogen distribution at equili-brium is determined solely by mz(e1/e2)

1/2. If m[1,the equilibrium solution decays rapidly from the source,and the range of the morphogen would be too short toeffectively pattern a field of cells in a concentration-dependent manner. If m/1, the shape of themorphogen distribution will be essentially flat overthe field of cells and thus neighbouring cells wouldadopt identical fates. For mzOð1Þ, it constrains themaximum of the ratio of time scales to e1=e2zOð10Þ.What does mfOð1Þ mean? Substituting the par-ameters back into m2 gives

m2ze1

e2Z

tD

tE

tR

tBz

keðRT=KmÞD=L2

x

: ð3:3Þ

Equation (3.3) is a ratio of a measure of the diffusiontime against a grouping of binding, equilibration andremoval kinetic time scales. In essence, mfOð1Þ meansthat the kinetic and diffusion time scales are of the sameorder of magnitude and since the distribution ofmorphogen is determined by a balance between thetime scales, it leads to an overall length scale ofmorphogen appropriate for having sufficient range andsufficient slope for patterning.

The shape f will be scale invariant if and only if m isindependent of Lx. It is readily apparent that m isindependent of the system length (Lx) ifRTfLK2

x , sinceD, ke and Km are constant parameters. SubstitutingRTZNT/LxLyLz into equation (3.3) makes the require-ments for scale invariance more transparent in thefollowing equation:

m2zke

DKm

� NTL

2x

LxLyLz

: ð3:4Þ


Given equation (3.4), and since ke, D and Km areconstant, m2 will be independent of Lx and theequilibrium shape function (f) will be scale invariantif NTL

2x=ðLxLyLzÞ is also constant. A number of

scenarios can hypothetically ensure a constant ratioand the two most biologically tenable options arethe following: (i) the total number of binding sites(NT) scales in proportion to LK1

x with fixed Ly, Lz, or(ii) NT remains constant and the product of LxLyLz isproportional to L2

x . While the first possibility cannot beruled out since there may be mechanisms that carefullyregulate NT in proportion to LK1

x , the indirect evidence(detailed in §1) suggests that the second possibility ofconstant NT and appropriate scaling of LxLyLz maycontribute to scale invariance for a number of morpho-gen-patterning mechanisms.

These observations suggest that (i) NT may beconstant between organisms within and betweenspecies at the same developmental stage (Teleman &Cohen 2000; Gregor et al. 2005), (ii) different-sizedorganisms have similar proportions so that LyZrLx ,where r is the proportionality constant (Gregoret al. 2005), and (iii) the total input flux may beconstant (specifically for wing disc patterning by Dpp;Teleman & Cohen 2000). Lastly, it is assumed that theheight of the gap for diffusion (Lz) is constant, which isexpected since the distance between cells, the thicknessof the cortical layer or the thickness of the perivitellinespace are intrinsic to the cell biology that developsthose regions and not the scale of the system. Withthese conditions, m2 can be rewritten as

m2zke

DKm

� NTL

2x

L2xrLz

Zke

DKm

� NT; ð3:5Þ

where

NT ZNT

rLz

hconst: ð3:6Þ

In other words, the shape function f is independentof Lx under conditions of (i) constant total numberof binding sites, (ii) constant gap height Lz, and(iii) uniform dilations in tissue size.

If m is independent of Lx, the amplitude L will beinvariant if J scales appropriately for changes in thesystem length. However, JZtD/tjZjLx/(DC0) andamplitude invariance can only be achieved if jfLK1

x .One plausible mechanism that ensures the proper

scaling of j occurs when the total input number flux F isconstant instead of the flux density j. On a two-dimensional sheet as analysed here, this leads to

FZ

ðLy

0

ðLz

0j dzdy Z jLyLz ; ð3:7Þ

that is,

j ZF

LyLz

: ð3:8Þ

Substituting equation (3.8) into J and noting thatLy/LxZr, a constant, it follows that JZF(rLzDC0)

K1

and so L is independent of Lx. As expected, when theconditions for equilibrium scale invariance are met,the distribution of morphogen shifts in proportionto the length of the system, as shown in figure 4a inx -coordinates.When remapped onto x2[0,1], the profilesare superimposed over each other as shown in figure 4b.




3.2. Analysis of boundary-layer dynamics

For the very early dynamics, the characteristic timescale is related to the flux of molecules into the systemat the boundary (tjdZCdl x/j ). The t1ZT =tjd timescale inherently depends on the choice of the charac-teristic length used to study the boundary-layerdynamics (l x). To determine whether the very earlydynamics are scale invariant, the solution is remappedto actual time (T ) and scaled position x2[0,1].Equation (2.21) then becomes

cðx;T ÞZjLxexp K

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1NT=D

qx

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1NTD

qCd

! 1K1

21CErf

K2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1NTD

qðT =LxÞCLxx

2ffiffiffiffiffiffiffiffiDT

p

24

35

8<:

0@

Cexp 2

ffiffiffiffiffiffiffiffiffiffiffiffik1NT

D

sx

0@

1A

!Erfc2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1NTD

qðT =LxÞCLxx

2ffiffiffiffiffiffiffiffiDT

p

24

359=;1A: ð3:9Þ

In equation (3.9), it is clear that the very earlydynamics in the boundary layer are not scale invariantsince the time (T ) is scaled by the overall length Lx.

3.3. Conditions for long-time dynamicscale invariance

In addition to the steady-state morphogen distribution,the morphogen dynamics are also important in anumber of biological contexts. Equations (2.39)–(2.41)were further analysed to determine (i) the rate ofapproach to the equilibrium solution, and (ii) whether itis possible to achieve scale invariance dynamically, i.e.the morphogen distribution in different-sized domains isproportional for all times during the transient evolutionof the distribution.

Since c(x,t3)ZcE(x)Cv(x,t3) and cE(x) is scaleinvariant by the conditions outlined in §3.1, itfollows that cðx; t3ÞZcEðx;p;LxÞC

PNnZ0 Lnðp;LxÞ!

fnðx;p;LxÞ!jnðp;Lx ; t3Þ will be scale invariant ifeach of the Ln, fn and jn is independent of Lx.

Here, the solution is represented by a sum of termsas shown in the following by rearranging equations(2.39)–(2.41):

vðx; t3ÞZXNnZ0

Lnðp;LxÞfnðx;p;LxÞjnðp;Lx ; t3Þ; ð3:10Þ

L0 ZK~DJ=~k; n Z 0; ð3:11Þ

Ln ZK2~DJ=ð~kC ~DðnpÞ2Þ; n Z 1; 2;.; ð3:12Þfn Z cosðnpxÞ; n Z 0; 1; 2;.; ð3:13Þ

jn Z expðKð~kC ~DðnpÞ2Þt3Þ; n Z 0; 1; 2;.; ð3:14Þand


~D ZtR=tB

1CqK1tR=tB; ~k Z

e1

e2

qK1tR=tB1CqK1tR=tB

: ð3:15Þ

The conditions for scale invariance identified in §3.1make the analysis of dynamic scale invariance straight-forward. First, since RTfLK2

x , there is a directrelationship between actual time (T ) and scaled timet3ZðDKm=N TÞT , whichmeans that if t3 is independentof Lx, then the actual time (T ) is also independent of Lx.

It is immediately apparent that the shape functions(fn) are scale invariant; however, it is not clear whetherthe Ln and jn are independent of Lx. Recalling thatqz1 and that J and m are independent of Lx (§3.1), andsubstituting ~D and ~k into equations (3.11) and (3.12) forLn gives

LnzK2Jðm2 CðnpÞ2ÞK1; ð3:16Þwhich is automatically independent of Lx.

Next, substituting ~k and ~D into equation (3.14) (jn)gives

jn Z exp KtR=tB

1CtR=tBðm2 CðnpÞ2Þt3

� ; n Z 0; 1; 2;.;

ð3:17Þ

and recalling

tR

tBZ

k1ðRTÞk1ðKm CC0Þ

zRT

Km

ZNT=L

2x

Km

;

where NT ZNT=rLz hconst:;

9>=>; ð3:18Þ

the conditions for dynamic scale invariance are moretransparent. Whether or not the morphogen evolutionis scale invariant depends on the ratio of RT to Km.Substituting RT/Km into equation (3.17) gives

jnZexp KRT=Km

1CRT=Km

ðm2CðnpÞ2Þt3�

; nZ0;1;2;.:

ð3:19Þ

If receptor density relative to the Km value is small sothat RT=Km/1, then equation (3.19) becomes

jn|exp KNT=L

2x

Km

ðm2 CðnpÞ2Þt3

!; n Z 0; 1; 2;.;

ð3:20Þ

and the time it takes to reach equilibrium increases inproportion to the length of the system squared. In theother limit ðRT=Km[1Þ;

jn|expðKðm2 CðnpÞ2Þt3Þ; n Z 0; 1; 2;.; ð3:21Þ

and the dynamics of the morphogen distribution arecompletely independent of the length of the system.

Figure 4c (left axis) shows how the value of theleading term associated with each exponential dependson the ratio of RT/Km, which provides an indication ofthe time it takes to reach equilibrium. Figure 4c (rightaxis) shows the time it takes for the FEM solution toreach 90 per cent of the equilibrium values for a givenset of parameters. As RT/Km increases, the exponentialterm becomes more negative, which means the solutionapproaches equilibrium more rapidly until levelling off


1.0(a) (b)

(c) (d )

0.8

0.6

0.4

0.2

010–1 1 101 102 103

0 300

250

200

150

100

50

0

–0.5

–1.0

–1.51 101 102 103

0

–0.5

–1.0

–1.5101 102 103

1.0

0.8

0.6

0.4

0.2

00.2 0.4 0.6 0.8 1.0

Figure 4. (a) Equilibrium solution for the linear approximation versus position for systems with characteristic lengths (Lx) of10 mm (solid line), 100 mm (open circles) and 1000 mm (dotted line). (b) Same as in (a) rescaled to the interval [0, 1]. (c) Plot of theexponential term that depends on the system length versus RT/Km (left axis). Time to reach 90% of the equilibriumconcentration at xZ0 versus RT/Km (right axis) for the finite-element model. (d ) Dynamic scale invariance by conservation of

binding sites is ensured if Lx/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipNT=Km.



at a constant value, which is consistent with thebehaviour of the FEM solution of the full model.However, since RTfLK2

x , for fixed NT, this places limitson the maximum size for which dynamic scale invarianceoccurs automatically. Specifically, dynamic scale invar-

iance requires that Lx/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipNT=Km (figure 4d ).

What does this mean for real biological systems? Theratio tR/tB in equation (3.17) represents the relation-ship between time scales of binding site equilibration tothe time scale of ligand binding and removal from thefreely diffusing pool. While tR/tB emerged as animportant relationship between time scales, it canalso be interpreted as a ratio of the concentration scalesRT/Km. In this context, the ratio of RT to Km relatesthe total binding site density to the concentration ofligand necessary to achieve 50 per cent binding siteoccupancy. A small RT/Km suggests either a very lowconcentration of receptors or very weak binding (eithera slow on-rate or a fast off-rate). With relatively fewbinding sites or weak binding, diffusion is largelyunhindered and changing the binding site density haslittle effect on the dynamics of transport, which leads toan increase in the time it takes to approach steady


state. On the other hand, a large RT/Km suggestseither a high receptor concentration or tight binding.The large number of binding sites transiently slows thediffusion process and decreasing the density of bindingsites leads to a faster effective diffusion. For a range ofparameter values, if the binding site density decreasesðRTfLK2

x Þ as Lx increases, the effective diffusionconstant increases in proportion to L2

x . The boost ineffective diffusion leads to a system that is scaleinvariant both dynamically and at equilibrium.

4. DISCUSSION

In this paper, multiple time scales are used to betterunderstand the dynamics and identify conditions forscale invariance of morphogen-mediated patterning.The principal conditions that lead to scale invariance ofa morphogen distribution are

— conservation of the total number of binding sites forthe morphogen;

— conservation of the total number of morphogenmolecules secreted in a given length of time at theboundary;




— for scale invariance of morphogen gradient interpre-tation, cells (or nuclei) must respond to the totaloccupancy of receptors and not to the surface densityof occupied receptors (appendix A); and

— for dynamic scale invariance, there needs to be anexcess of binding sites on the surface to slow theeffective diffusion.

The properties for dynamic scale invariance arebased on evidence from a number of different pathwaysand the analysis suggests that the simple mechanism ofbinding and release may contribute to scale invariance.While supported by observations, a number of theoreti-cal and experimental studies will further delineatemechanisms of scale invariance in different contexts.The model presented here depends on a number ofassumptions, and future work should be extended toinclude the following: (i) non-uniform binding sitedensity as occurs in wing imaginal disc patterning byDpp (Lecuit & Cohen 1998; Teleman & Cohen 2000;Serpe et al. 2008), (ii) binding site density that changesin time (Umulis et al. 2006; Gregor et al. 2007; Coppeyet al. 2008; Serpe et al. 2008), (iii) feedback fromreceptor signalling (Fujise et al. 2003; Wang &Ferguson 2005; Umulis et al. 2006), (iv) nonlinearextracellular regulation as occurs in dorsoventralpatterning ofDrosophila andXenopus embryos (Mizutaniet al. 2005; Shimmi et al. 2005; O’Connor et al. 2006;Umulis et al. 2006, 2008; Ben-Zvi et al. 2008), (v) moredetailed analysis of nuclear import/export kinetics(Coppey et al. 2007, 2008; Gregor et al. 2007), (vi)transcytosis through columnar epithelial cells (Kruse et al.2004; Bollenbach et al. 2005; Kicheva et al. 2007), and(vii) more realistic geometries of the underlying tissues.The analysis presented herein serves as a starting point forfuture analysis, but it also serves as the common themethat links the diverse morphogen pathways.

The conditions for scale invariance are general enoughthat they can be tested in a number of different contexts.For instance, one couldmeasure the distribution of pMadsignalling and extracellular Dpp–green fluorescentprotein (Kicheva et al. 2007) in the posterior compart-ment of wing discs that ectopically express cyclin-dependent kinases to increase or decrease the numberof cells. Furthermore, additional experiments such aschanging the binding site density (receptors, Cv-2, etc.)during D/V embryonic patterning by BMPs, disruptingthe nuclear density during A/P patterning by Bcd,measuring the time it takes for A/P pattern formation byBcd in different-sized embryos or altering the geometryof a developing tissue would all provide additional insightinto the mechanisms of biological scale invariance.

In summary, morphogens mediated by diffusion,binding and endocytosis/degradation can yield scaleinvariance automatically if the number of binding sitesis conserved for changes in system size. As mentioned inGregor et al. (2007), this may lead to scaling of bicoid.For systems with excess binding sites, scale invariancemay occur dynamically as well as at equilibrium.

I wish to thank Hans Othmer, Michael O’Connor and thereviewers for their helpful comments on this paper. Supportedin part by NIH (GM29123) to H.O.


APPENDIX A

A.1. Ligand decay

In addition to endocytosis or decay of morphogen frombinding sites, one could envision a related but mechan-istically different process where the morphogen has anintrinsic lifetime that may or may not be affected bybinding. For instance, if the morphogen is a transcrip-tion factor that is diffusing over a field of nuclei, thebound and unbound states do not undergo endocytosisas in receptor ligand systems and one would expectsimilar decay times for the bound and unboundmorphogens. In essence, this adds a decay term to theequation for unbound ligand in equation (2.1). Usinganalogous analysis to that developed herein, theextra kinetic step introduces another ratio of timescales e3Zkc/(k1(KmCC0)). This gives

m2 Ze1

e2qK1 C

tR

tB

� K1 e3

e2: ðA 1Þ

If the decay rates in the bound and unbound statesare equal, then e1Ze3. Scale invariance at equilibrium isassured if tR=tB[1, in which case the majority ofmorphogens is in the bound state and the contributionof unbound decay is negligible.

A.2. Scale invariance of receptor–ligandsystems

The above analysis demonstrates how scale invariancecan be achieved if the binding site density scales as L2;however, it is more important that the interpretation ofthe signal scales properly as well. In the case of nucleiand transcription factor systems such as A/P pattern-ing in Drosophila, the local concentration at discretesites is constant for changes in system size so that thescale invariance at the level of occupied binding sites isautomatic. For ligand–receptor systems, the extra-cellular gradient scales by reducing the concentration ofbinding sites. Since receptor concentrations decreasefor increases in size, the surface density of receptor-bound ligand also decreases. Thus, the concentration ofligand-bound receptors is not scale invariant and cellsmust interpret the signal by another mechanism. Onepossibility for morphogen interpretation is to integratethe signal over the entire surface of each cell into adownstream response. In essence, cells count the totalnumber of ligand-occupied receptors to initiate adownstream response as has been shown for activinsignalling (Dyson & Gurdon 1998). Suppose cells arespaced in a square lattice with dimensions l xZaLx andl yZbLy. Mathematically, for a square surface arrangedin a regular array, the total signal is represented by

NCn;mðt3ÞZ rL2x

ððnC1Þa

na

ððmC1Þb

mbRC ðx; t3Þ dhdx

zbKK1m N tot

ððnC1Þa

naC0cðx; t3Þ dx: ðA 2Þ

Here, NCn;mðt3Þ is the total number of occupiedreceptors on the surface of the cell specified by positionn,m, at time t3. The constants a and b correspond to the




fractional length of a cell in scaled coordinates. To showthat the integral of receptors on the surface scales,apply the first mean-value theorem for integrals, whichstates that there exists some c2[a,b] such that thefollowing equality is true:ðb

af ðtÞ dt Z f ðcÞðbKaÞ: ðA 3Þ

This leads to the following expression for the number ofligand-occupied receptors per cell:

NCn;m Z abKK1m N totC0cðx�; t3Þ;

where x� 2 ½na; ðnC1Þa�:

)ðA 4Þ

Then, the mean-value theorem for integration of asingle cell givesððnC1Þa

nacðx; t3ÞdxZ acðx�; t3Þ ðA 5Þ

and ððnC1Þa

nacEðxÞCvðx; t3ÞdxZ acðx�; t 3Þ: ðA 6Þ

Since cE(x) is independent of Lx, it follows that x� is

independent of Lx only if v(x,t3) is independent of Lx bythe conditions shown herein.

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http://dx.doi.org/doi:10.1038/nature07059


http://dx.doi.org/doi:10.1371/journal.pbio.0050046

http://dx.doi.org/doi:10.1103/PhysRevLett.94.018103

http://dx.doi.org/doi:10.1103/PhysRevLett.94.018103

http://dx.doi.org/doi:10.1016/j.ydbio.2007.09.058

http://dx.doi.org/doi:10.1016/j.cub.2008.05.034

http://dx.doi.org/doi:10.1016/j.cub.2008.05.034

http://dx.doi.org/doi:10.1016/S0092-8674(00)81185-X


http://dx.doi.org/doi:10.1242/dev.00379

http://dx.doi.org/doi:10.1073/pnas.0509483102


http://dx.doi.org/doi:10.1016/j.cell.2007.05.026


http://dx.doi.org/doi:10.1126/science.1135774





http://dx.doi.org/doi:10.1016/j.devcel.2005.04.009









http://dx.doi.org/doi:10.1016/S0092-8674(00)00199-9









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