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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Stud Appl Math. Author manuscript; available in PMC 2010 August 1.
Published in final edited form as:
Stud Appl Math. 2009 August 1; 123(2): 175–214.
doi:  10.1111/j.1467-9590.2009.00450.x
PMCID: PMC2774827
NIHMSID: NIHMS134928

Localized Ectopic Expression of Dpp Receptors in a Drosophila Embryo

A.D. Lander,a,b Q. Nie,b,c,d F.Y.M. Wan,b,c,d and Y.-T. Zhangc,d,e

Abstract

Receptor-mediated BMP degradation has been seen to play an important role in allowing for the formation of relatively stable P Mad patterns. To the extent that receptors act as a "sink" for BMPs, one would predict that the localized over-expression of signaling receptors would cause a net flux of freely diffused BMPs toward the ectopic, i.e., abnormally high concentration, receptor site. One possible consequence would be a depression of BMP signaling in adjacent areas since less BMPs are now available for binding with the same normal concentration of receptors at the adjacent areas. However, recent experiments designed to examine this possible effect were inconclusive. In this paper, we investigate the possibility of depression of Dpp signaling outside the area of elevated tkv in a Drosophila embryo by modeling mathematically the basic biological processes at work in terms of a system of nonlinear reaction diffusion equations with spatially varying (and possibly discontinuous) system properties. The steady state signaling morphogen gradient is investigated by the method of matched asymptotic expansions and by numerical simulations.

1 Introduction

For proper functioning of tissues and organs, cells are required to differentiate appropriately for its position. Positional information that instructs cells about their prospective fate is often conveyed by concentration gradients of morphogens, also known as (aka) ligands, bound to cell receptors (bound-morphogens for short). Morphogens/ligands are "signaling" protein molecules that, when bound to appropriate cell receptors, trigger the genetic program to assign/express different cell fates at different concentrations [28, 31]. Morphogen activities are of special importance in understanding the development of a population of uncommitted cells in an embryo to create complex patterns of gene expression in space. This role of morphogens has been the prevailing thought in tissue patterning for over half a century; but only recently have there been sufficient experimental data [6, 27, 28, 31] and adequate analytical studies (see [5, 7, 12, 13] and references therein) for us to begin to understand how various useful morphogen concentration gradients are formed.

Dorsal-ventral (belly-back) patterning in vertebrate and Drosophila embryos is now known to be regulated by bone morphogeneric proteins (BMP). The BMP activity is mainly controlled by several secreted factors including the antagonists chordin and short gastrulation (Sog). In Drosophila fruit flies, seven zygotic genes have been proposed to regulate dorsal-ventral patterning. Among them, decapentaplegic (Dpp) encodes BMP homologues that promotes dorsal cell fates such as amnioserosa and inhibits development of the ventral central nervous system. On the other hand, the chordin homologue Sog promotes the development of central nervous system.

Typically, morphogen concentration gradients are synthesized at certain part of the embryo, followed by their diffusion, binding with receptors (or other non-signaling molecules known collectively as non-receptors) and degradation in appropriate regions [12]. In the above Dpp-Sog system, the production of Dpp is pretty much uniform in the dorsal region and not at all in the ventral region while the opposite is true for Sog. The Dpp activity has been found to have a sharp peak around the midline of the dorsal in the presence of its “inhibitor" Sog (much more so during the transient phase than in steady state). Intriguingly, mutation of Sog results not only in a loss of ventral structure as expected, but the amnioserosa is reduced in addition. This result is paradoxical as the amnioserosa is the dorsal-most tissue and apparently a BMP antagonist is required for maximal BMP signaling [2, 3, 21, 23].

As the system contains many variables, the question of what leads to a sharp (bounded) Dpp concentration peak is difficult to tackle by traditional experimental means. In [20], a quantitative analysis (along with experimental studies) of this phenomenon was undertaken by extending the one-dimensional dynamic Dpp-Sog system formulated in [10] and [18] for the evolution of the morphogen activities in the extracellular space with Dpp and Sog produced in the dorsal and ventral regions, respectively, possibly at different prescribed production rates. The system allows for diffusion, reversible binding and degradation of the two morphogens, Dpp and Sog, as well as reversible binding and degradation of Dpp bound to its signaling cell receptors Thickvein (tkv). The extension consists of allowing the enzyme Tolloid to cleave Dpp-Sog complexes to (degrade Sog and) free up Dpp molecules. Numerical simulations of this relatively simple model for the process of dorsal-ventral patterning in [20] were found to capture the Sog-dependent shuttling of BMPs to the dorsal midline and provide insights into the unusual dynamics of this gradient formation process.

In the model examined in [20], receptor-mediated BMP degradation plays an important role in allowing for the formation of relatively stable PMad patterns. To the extent that signaling receptors act as a "sink" for BMPs, one would predict that the localized over-expression of these receptors would cause a net flux of free BMPs toward the ectopic (abnormally high) receptor concentration site. One possible consequence would be a depression of BMP signaling in adjacent areas since less BMPs are now available for binding with the same concentration of receptors at the adjacent areas as before. Recently, Wang and Ferguson [30] presented experiments in which mRNA for the Dpp receptor tkv was injected in a localized fashion into early embryos. No discernible difference were observed in the PMad patterns that ultimately developed (unless a constitutively active form of the receptor was used).

The experiment of Wang and Ferguson were carried out by RNA injection; it is not possible to know whether the levels of ectopic tkv were substantial compared with endogenous tkv and therefore whether they should have been expected to have any significant influence on BMP degradation. To gain additional information on this issue, GAL4-UAS was used in [20] to express ectopic tkv in the head region of embryos and observed its subsequent effects on PMad staining. As shown in Supplemental Figure S7 of [20] (reproduced from the Supplement of [20] as Figure 1 below), endogenous tkv expression in the embryonic head region is already relatively substantial and can be elevated by expressing wild-type tkv using a bcd-GAL4 driver. When compared with wild-type embryos, those expressing ectopic tkv consistently showed a narrowing and weakening of the PMad staining pattern over a range of 10–12 cell diameters posterior to the border of the bcd domain. Thus, the data are consistent with the supposition on the model earlier that there would be a depression of BMP signaling outside the area of elevated tkv.

Figure 1
Localized expression of Tkv leads to reduced PMad activation in adjacent cells (reproduced from Figure 7 of the SOM for [20]): A) Endogenous tkv expression in a wild-type embryo. Note that expression is elevated in the sections of head relative to the ...

The experimental results of [20] notwithstanding, a closer examination of the biological processes at work suggests some uncertainty regarding the actual effects of a localized over-expression of tkv. Given that there is no shortage of free Dpp throughout the dorsal region of the embryo in steady state, there is no obvious reason for a depression of bound-Dpp concentration outside the area of elevated tkv even if some of the free Dpp has been siphoned off by the ectopic receptors. Furthermore, whether there should be a depression of BMP signaling may depend on the level of Sog synthesis rate given the Sog-dependent shuttling of BMPs to the midline. We will investigate these issues herein by obtaining steady state solutions of a relevant mathematical model for the biological development of interest.

The mathematical model of the aforementioned embryonic development will necessarily be more complex than those previously analyzed by the authors in [12, 13, 18, 20] and references therein. Given the spatial variations of the synthesis rates of Dpp and Sog along the dorsal-ventral axis and the spatial variations of tkv concentration in the anterior-posterior direction, the model must be at least spatially two-dimensional. With Dpp, Sog and the Dpp-Sog complexes diffuse freely in the extracellular space, the model must be multi-diffusional even if we should take the diffusion rates to be (more or less) identical. In Section 2, the roughly cigar shape embryo will be idealized and simplified to make our first analysis tractable. A two-dimensional extension of the extracellular model used in [18] for this simplified domain turns out to be adequate for our purpose. The relevant initial-boundary value problem (IBVP) will be formulated for the idealized problem. With the new mathematical problem similar to that treated in [18] except that it is now spatially two-dimensional, much of the theoretical development in [18] can be extended to assure the existence of a steady state solution. We will therefore focus on obtaining approximate solutions for the steady state problem to gain insight to the steady state behavior in the presence of ectopic receptor expression.

Similar to the simpler one-dimensional case of a uniform receptor expression treated in [18], the restriction of complete immobility of Dpp (as suggested by Eldar et al [5]) is not required for the existence of a steady state behavior for the present problem (see also [20]). For a sufficiently high Sog synthesis rate, we will be able to obtain an outer (asymptotic expansion) steady state solution with respect to the small Dpp-to-Sog synthesis rate ratio for our problem. More remarkably, the effects of ectopic receptor expression for this case can be obtained from the aforementioned outer solution alone without the rather complex inner solutions (and the attendant matching) required in a related problem in [9] or numerical simulations as in [29] to deal with the layer phenomena in the neighborhood of the various receptor concentration and synthesis rate discontinuities. Conditions under which there would be a depression of bound-Dpp concentration posterior to the elevated tkv area can then be analyzed. We will also examine the relatively high Dpp synthesis rate case and show that a regular perturbation solution is sufficient for the determination of the effects of ectopic receptors. The intermediate case of comparable Dpp and Sog synthesis rates admits no useful simplifications and will be investigated by accurate numerical simulations.

2 The Mathematical Model

2.1 Idealized Geometry for the Extracellular Domain

Depending on the stage of development it is in, an embryo may be of different shapes. For the period of development of interest here, the embryo of a Drosophila fruit fly is typically somewhere between the shape of a football and a cigar and may be treated as a prolate spheroid (see Figure 2(A)) for the purpose of analysis. For an extracellular model, we are concerned mainly with biological activities on the surface of the embryo with the various concentration gradients being scalar fields defined on the surface of the prolate spheroid. While we can formulate the equations governing these concentration gradients in terms of the conventional prolate spheroidal coordinates with the z–axis along the length and through the center of the cross section) of the embryo), we will in our preliminary study of this problem simplify it substantially by mapping the relevant part of the surface domain into a rectangle in the Cartesian plane.

Figure 2
(A) Prolate spheroidal surface domain; (B) dorsal and ventral portion of prolate spheroidal surface; (C) half of the mirror-symmetric domain; (D) idealized half domain as a rectangle in the X,Y - plane.

For our investigation, we imagine cutting the prolate spheroidal surface along the one continuous mid-line of both dorsal and ventral part of the embryo (see Figure 2 (B, C)). Given the symmetry of development activities with respect to the dorsal and ventral mid-line, we only need to consider one of the two half prolate spheroidal surfaces resulting from the fictitious cut. We further map the relevant half prolate spheroidal surface (consisting half of the dorsal region and the adjacent half of the ventral extending from the dorsal mid-line to the ventral mid-line) into the rectangular region ΩXY in the X, Y–plane, which spans from −Ymax/4 to Ymax/4 in the polar direction and from 0 to Xmax in the azimuthal direction. Note that Ymax is the circumference of the average cross section of the embryo with the z–axis of the prolate spheroidal coordinate system (which is parallel to the direction of the antero-posterior axis and the x–axis of the Cartesian rectangle ΩXY) as its normal (see Figure 2 (D)).

As indicated in the Figure 2 (D), Sog is synthesized at a constant rate VS and only in the ventral region while Dpp at a constant rate VL only in the dorsal region. Both diffuse away from the respective localized source and bind with each other while Dpp also binds to signaling receptors Thickvein (tkv) which is attached to the cell membrane. Some of these complexes will degrade while others dissociate to free up morphogens and receptors for new binding action. Generally, the receptors also degrade and new one generated at a rate VR. The model in this paper will not have an explicit account for the synthesis, internalization (through endocytosis) and degradation of (free or bound) receptors. As System B in [12] and [17], we limit ourselves to the case of a fixed receptor concentration R(X) corresponding to the case of a receptor synthesis rate matching its degradation rate with internalization implicit in receptor-mediated degradation. (An alternative and equally unrealistic interpretation would be that degradation of a bound Dpp complex destroys only the Dpp molecule and releases the receptor involved for binding with another free Dpp molecule.) The omission of an explicit account of receptor renewal and internalization results in no way affect the usefulness of our analysis; we have already established in [14, 17] that the BVP governing the steady state behavior of more general models which include receptor renewal and internalization can be reduced the corresponding BVP for the simpler fixed receptor system. The effects of the more general case of receptor renewal have also been examined and will be reported in a future publication.

2.2 The Initial-Boundary Value Problem

The allowable set of developmental activities of the two interacting morphogens Dpp and Sog in our model is summarized schematically in Figure 3.

Figure 3
Schematic Diagram of Biological Processes.

For an analytical and computational study of the biological phenomenon of interest, the essential features of these activities are described mathematically by a system of partial differential equations and auxiliary conditions [20]. This approach was first applied to study the development of the Drosophila wing imaginal disc [4, 11, 12]. The three basic biological processes involving Dpp in the wing disc are diffusion for free Dpp molecules, their reversible binding with renewable receptors, and degradation of the Dpp-receptor complexes (aka bound Dpp). The main purpose of [11, 12] was to investigate the role of diffusion in the formation of a Dpp-receptor concentration gradient in the wing disc. That system was extended to include the effect of Sog on the Dpp activity in a dorsal-ventral configuration [10] in an embryo with the cleavage of Dpp-Sog complexes by Tolloid implicitly incorporated into the system through the complete recovery of Dpp after cleavage (while the Sog components degrade). The cleavage and recovery phenomenon has been suggested by previous experimental studies [19, 22]. An even more general system was investigated in [18] where we allowed fractional recovery (in an extracellular model) through the fraction parameter τ, 0 ≤ τ ≤ 1, with τ = 1 corresponding to complete recovery.

The setting for dorsal-ventral patterning in a Drosophila embryo during development with localized over-expression of tkv receptors is different and more complex than those considered in [12, 13, 14, 15, 18, 20]. As shown in the sketch of the dorsal-ventral cross-section of the embryo in Figure 2, Dpp is only produced in the dorsal region (with the temporally uniform rate VL(X)) while Sog is only produced in the ventral region (with the rate VS(X)). With tkv over-expressed along only a part of anterior-posterior direction, the biological development of the embryo is no longer uniform in that direction as it was in the problem investigated in [18]. The appropriate mathematical model for the problem must now be (at least) spatially two-dimensional.

Let [L(X, Y, T)], [S(X, Y, T)], [LS(X, Y, T)] and [LR(X, Y, T)] denote the concentration of Dpp, Sog, Dpp-Sog complexes and Dpp-receptor complexes, respectively. The first three concentrations diffuse with coefficients of diffusion DL, DS, and DLS, respectively, and the concentration of immobile receptor is fixed at every location (X, Y), possibly nonuniformly distributed over the solution domain. Then the system of equations governing the morphogen dynamics as indicated in Figure 3 consists of the following four coupled differential equations, three of them being second order nonlinear partial differential equations (PDE) of the reaction-diffusion type while the other being a first order ordinary diffferential equation (ODE):

[L]T=DLXY2[L]kon[L]{[R][LR]}+koff[LR]jon[L][S]+(joff+τjdeg)[LS]+VL(Y)
(1)

[LR]T=kon[L]{[R][LR]}(koff+kdeg)[LR]
(2)

[LS]T=DLSXY2[LS]+jon[L][S](joff+jdeg)[LS]
(3)

[S]T=DSXY2[S]jon[L][S]+joff[LS]+VS(Y)
(4)

{VL(Y)VS(Y)}={V¯LH(Y)V¯SH(Y)},    XY2()=2()X2+2()Y2,
(5)

where H(z) is the Heaviside unit step function, equal to unity for positive z and zero for z < 0.

In the four differential equations above, the parameters kon, koff and kdeg are the binding rate constant, dissociation rate constant and degradation rate constant, respectively, for Dpp-receptor complexes while jon, joff and jdeg are the corresponding rate constants for Sog-Dpp complexes. The parameter τ in (1) assume a value in the interval [0, 1] giving the fraction of Dpp freed up by the degradation of the bound morphogen complex and returned to the free morphogen pool available for binding with receptors or Sog molecules. There is no return if τ = 0 and complete return if τ = 1. The quantity [R(X, Y)] in (1) and (2) is the initial tkv receptor concentration at the location (X, Y) before the onset of Dpp synthesis and [R(X, Y)] − [LR(X, Y)] being the unoccupied receptor concentration at (X, Y) still available for binding with Dpp. Three special receptor distributions of interest in the subsequent development are:

  1. a spatially uniform distribution with [R(X, Y)] = [R0];
  2. a distribution [R(X, Y)] = [Rh(X)] uniform in Y but with two different segments of uniform receptor expression in X as given by
    [Rh(X)]={R¯0(1+Δ)(X<Xh)R¯0(X>Xh)    ,    Δ>0,
    (6)
    for some positive constant Δh
  3. a distribution [R2(X)] with three uniform segments of receptor expression in the X direction as given by
    [Rh(X)]={R¯0(1+Δ)(X<X)R¯0(X<X<Xg)R¯0(1+Δg)(Xg<X)
    (7)
    for some positive constants Δ[ell] and Δg.

The system of four differential equations (1)(4) above is sixth order in the spatial variables. Given the symmetry with respect to the dorsal and ventral midline, we need only to consider the problem for the rectangular part ΩXY={(0,Xmax)×(14Ymax,14Ymax)} of the actual domain. Along the boundary [partial differential]ΩXY of ΩXY, we have the following homogeneous Neumann conditions:

(X,Y)ΩXY:    [L]n=[LS]n=[S]n=0
(8)

for all T > 0 where [partial differential][G]/[partial differential]n is the normal derivative of [G] and [partial differential]ΩXY is the boundary of ΩXY. The no flux conditions along Y=±14Ymax follow immediately from the symmetry of the developmental activities with respect to the dorsal mid-line and ventral mid-line. The no flux conditions along X = 0 and X = Xmax are more difficult to justify except that there can be no flux in any direction at the poles of the prolate spheroidal shape embryo. A more realistic treatment of the problem using (prolate) spheroidal coordinates will be the subject of a separate investigation.

Until morphogens being generated at T = 0, the biological system was in quiescence so that we have the homogeneous initial conditions

T=0:    [L]=[LR]=[LS]=[S]=0,
(9)

for all (X, Y) in ΩXY. The system (1) – (9) defines an initial-boundary value problem (IBVP) for the four unknown concentrations [L], [LR], [LS] and [S].

2.3 Non-dimensionalization

To reduce the number of parameters in the problem, we introduce the normalized quantities

t=DYmax2T,    {x,xh,x,xg,xmax,y}={XYmax,XhYmax,XYmax,XgYmax,XmaxYmax,YYmax},
(10)

{fL,gL,hL,fS,gS,hS}=Ymax2D{koff,kdeg,konR¯0,joff,jdeg,jonR¯0},
(11)

{A,B,C,S,ρ,ρh}=1R¯0{[L],[LR],[LS],[S],[R],[Rh]},
(12)

{υL,υS,υ¯L,υ¯S}=Ymax2DR¯0{VL,VS,V¯L,V¯S},    {dA,dC,dS}={DLD,DLSD,DSD}
(13)

where D is the maximum of DL, DLS, and DS and R0 is a representative magnitude of [R]. With these normalized quantities, we rewrite the IBVP in the following dimensionless form

At=dA2AhLA(ρB)+fLBhSAS+(fS+τgS)C+υL,
(14)

Bt=hLA(ρB)(fL+gL)B,
(15)

Ct=dC2C+hSAS(fS+gs)C,
(16)

St=dS2ShSAS+fSC+υS,
(17)

where now [nabla]2( ) = ( ),xx + ( ),yy is the Laplacian in the dimensionless variables (x, y) in the rectangle Ω={(0,xmax)×(14,14)}. After normalization, the special receptor distributions will be written as

[R]=R¯0ρ(x,y),
(18)

with

(i)ρo(x,y)=1,
(19)

(ii)ρh(x,y)={ρ¯h1+Δ>1(x<xh)1(x>xh),
(20)

(iii)    ρ2(x,y)={ρ¯11+Δ(x<x)1(x<x<xg)ρ¯21+Δg(x>xg)  .
(21)

For synthesis rates, we will be mainly concerned with the special case

{υL(y)υS(y)}={υ¯LH(y)υ¯SH(y)}
(22)

where H(z) is the Heaviside step function.

The boundary conditions now take the form

(x,y)Ω:    An=Cn=Sn=0
(23)

for t > 0 where [partial differential]Ω is the boundary of Ω. The homogeneous initial conditions become

t=0:A=B=C=S=0,    (x,y)εΩ.
(24)

2.4 Time-Independent Steady State

Similar to what was proved in [18], we expect the various initial concentrations of our embryo to evolve toward a time independent steady state behavior. For this steady state solution, we have [partial differential]( )/[partial differential]t = 0 so that the governing partial differential equations and boundary conditions become

2AhLA(ρB)+fLBhSAS+(fS+τgS)C+υ¯LH(y)=0,
(25)

hLA(ρB)(fL+gL)B=0,
(26)

2C+hSAS(fS+gS)C=0,
(27)

2ShSAS+fSC+υ¯SH(y)=0,
(28)

where we have set dA = dC = dS = 1 to simplify the discussion though the method of analysis employed is also applicable to the more general case. We can solve (26) for B(x, y) in terms of A(x, y) to get

B(x,y)=ρ(x,y)A(x,y)αL+A(x,y),    αL=1hL(fL+gL),
(29)

and use the result in (29) to eliminate B(x, y) from (25) to get

2AρgLAαL+AhSAS+(fS+τgS)C+υ¯LH(y)=0.
(30)

Equations (27), (28) and (30) form a sixth order system of three second order PDE for A(x, y), C(x, y) and S(x, y). Augmented by the boundary conditions (23), this system can be solved by various numerical methods for elliptic boundary value problems. However, to gain insight to the qualitative behavior of the steady state, we will also obtain instead an approximate solution in the context of the method of matched asymptotic expansions.

3 Matched Asymptotic Expansions for VL/VS [double less-than sign] 1

3.1 Re-scaling of Steady State Problem

It is rather typical in the development of Drosophila of interest here that the synthesis rate for Sog is substantially higher than that for Dpp. With ε = [upsilon]L/[upsilon]S = VL/VS [double less-than sign] 1, the BVP for the steady state behavior is amenable to an asymptotic solution by the method of matched asymptotic expansions. For this purpose, we re-scale the dimensionless steady state BVP by observing that both S(x) and C(x) are expected to be O([upsilon]S). On the other hand, we expect A(x) to be O([upsilon]L) at most, in fact quite a bit smaller since available free Dpp should eventually be bound to Sog or receptors given that there is an abundance of Sog molecules. We therefore set

S=υ¯SS(x,y,ε),    C=υ¯Sc(x,y,ε)fS+gS,  A=υ¯LμL2a=αLβLa(x,y,ε),
(31)

with

ε=υ¯Lυ¯S=V¯LV¯S,    υL*=hShLυ¯LσL,    βL=υ¯LgL,
(32)

σL=gLfL+gL,  μL2=gLαL,  σS=gSfS+gS,  αS=gS+fShS,
(33)

and re-write (30), (27) and (28) as

ε{μL22aρa1+βLa+H(y)}(υL*asc)=0,
(34)

(fS+gS)12c+(υL*asc)=0,
(35)

2s(υL*asc)σSc+H(y)=0,
(36)

where we have taken τ = 1 to simplify the presentation though the analysis would apply to other values of τ in (0, 1). The remaining unknown B(x, y) is then given in terms of a(x, y; ε) by (29) written as

B(x,y)=ρ(x,y)βLa(x,y;ε)1+βLa(x,y;ε)b(x,y;ε).
(37)

For ε = [upsilon]L/[upsilon]S [double less-than sign] 1, we seek an outer asymptotic expansion solution in parametric series of ε:

{a(x,y;ε),b(x,y;ε),c(x,y;ε),s(x,y;ε)}=n=0{an(x,y),bn(x,y),cn(x,y),sn(x,y)}εn.
(38)

3.2 Leading Term Outer Solution

The leading terms a0(x, y), b0(x, y), c0(x, y), and s0(x, y) correspond to the limiting case of ε = 0 (for VS = ∞). For this limiting case, equations (34)(36) reduce to

υL*a0s0c0=0,    2c0=0,    2s0σSc0+H(y)=0
(39)

and the boundary conditions (23) applied to the leading term quantities.

The second equation in (39) is for c0(x, y) alone. Together with the homogeneous Neumann condition along the edges of the rectangle, [partial differential]Ω, it requires

c0(x,y)=c¯0.
(40)

To determine the constant c0, we integrate the last equation in (39) over the Ω and apply Green’s theorem. The Neumann condition on s0 along [partial differential]Ω and the result (40) are then used to give

0=Ω{2s0σSc0+H(y)}dxdy=σSc¯0xmax+12xmax,

c(x,y)~c0(x,y)=c¯0=12σS.
(41)

The value for c0 in turn simplifies the last equation of (39) to

2s0=12{H(y)H(y)}
(42)

with s0 required to satisfy the homogeneous Neumann condition along [partial differential]Ω. The solution of this BVP is

s0(x,y)=s0(y)={s¯0+18(y+2y2)(y<0)s¯0+18(y2y2)(y>0)
(43)

where s0 is a constant of integration to be determined by the O(ε) problem. Note that s0 is (uniform in x and) continuously differentiable but has a simple jump discontinuity in [partial differential]2s0/[partial differential]y2 across y = 0.

Except for the unknown constant s0, we have also a0(x, y) = a0(y) from the first equation in (44):

2σSυL*a0(y;s¯0)=2σSc0s0={[s¯0+18(y+2y2)]1(y<0)[s¯0+18(y2y2)]1(y>0),
(44)

which also does not depend on x, and from (37) the leading term solution for B(x, y)

b0(x,y)=ρ(x,y)βLa0(y;s¯0)1+βLa0(y;s¯0),
(45)

which does depend on x (as well as y) through ρ(x, y).

3.3 The O(ε) Problem

To determine the unknown constant s0, we consider the O(ε) problem for a1(x, y), …, s1(x, y). The governing equations for these unknowns are

(fS+gS)12c1+[νL*(a0s1+a1s0)c1]=0,
(46)

2S1σSc1[νL*(a0s1+a1s0)c1]=0,
(47)

μL22a0ρa01+βLa0+H(y)[νL*(a0s1+a1s0)c1]=0,
(48)

with

b1(x,y)=ρ(x,y)βLa1(1+βLa0)2.
(49)

The unknowns a1, c1 and s1 are subject to the homogeneous Neumann conditions (8) which also apply to the O(ε) terms of the problem.

We begin to determine s0 by integrating (46) over Ω. Upon application of the two-dimensional divergence theorem and the homogeneous Neumann condition on s1, we obtain

Ω[νL*(a0s1+a1s0)c1]dxdy=0.
(50)

This relation enables us to simplify the corresponding integral of (48) to

Ω{μL22a0ρ(x,y)a01+βLa0+H(y)}dxdy=0

or, upon application of the two-dimensional divergence theorem and the homogeneous Neumann condition on a0,

J(s¯0)Ωρ(x,y)a0(y;s¯0)1+βLa0(y;s¯0)dxdy=xmax4.
(51)

For any prescribed distribution of tkv concentration ρ(x, y), this is a condition on a0(y; s0) alone and thus determining s0 in view of (44).

We may continue the solution process to solve (46)(48) and the corresponding homogeneous Neumann conditions to determine c1, s1 and a1. While this BVP is now truly two-dimensional given the explicit appearance of ρ(x) in (48), the problem is actually tractable (by the method of eigenfunction expansions for example) because it is linear. However, we will not be concerned with the results for these higher order terms here but only note the following:

Proposition 1

For ε = VL/VS [double less-than sign] 1, a formal leading outer (asymptotic expansion) solution for the re-scaled steady state concentrations of (31) is given by (41), (43), and (44) with the parameter s0 in these expressions determined by (51). The corresponding leading term signaling Dpp-receptor complex is given by (45).

The results deduced from the leading term outer solution of the problem will obviously be modified by higher order terms in the outer asymptotic expansions (38) of the solution.However, the qualitative features of the outer solution are not expected to be changed by such refinements for sufficiently small ε.

3.4 Inner Solution and Receptor Saturation

Whenever an outer solution for ε = VL/VS < 1 is applicable, the leading term solution of Proposition 1 generally captures the qualitative effects of localized ectopic receptor expression with quantitative accuracy improves as ε decreases. However, the formal solution of Proposition 1 may not be applicable even if the condition ε = VL/VS [double less-than sign] 1 is met. Of the two factors limiting its applicability, the possibility of supplementary inner asymptotic expansion solutions adjacent to the solution domain boundaries and discontinuities of system properties turn out not to be an issue for our problem. Given the homogeneous Neumann boundary conditions (8) along [partial differential]Ω, there should not be any sharp gradients (leading to layer solution components comparable to the outer solution in magnitude) adjacent to the edges of Ω. It isalso evident from (30) that there can be at most a finite jump discontinuity in the second derivatives of A(x, y) associated with jump discontinuities of the morphogen synthesis rates VL(X.Y) and VS(X, Y) and possible jump discontinuities of the receptor concentration R(X, Y) = R0ρ(x, y). By (30), (28), and (27), the concentrations A, S and C and their derivatives are expected to be continuous (with [LS] = R0C having even higher continuous derivatives) there. The only observable effect of the various discontinuities is seen from the algebraic relation in (29) for [LR] = R0B (see also (37) and (45)) which is calculated from the solution for A after the implementation of the method of matched asymptotic expansions. These qualitative conclusions from the form of the partial differential equations for the steady state problem are supported by the results of numerical simulations of the original initial-boundary value problem on the evolution of the various concentrations starting from the onset of morphogen synthesis.

The other factor limiting the applicability of the outer asymptotic solution of the Proposition 1 comes from our choice of a model with fixed receptor concentration. For sufficiently large Dpp synthesis rates (but still small compared to the Sog synthesis rate), the Dpp synthesized may form such a high concentration of Dpp-receptor complexes to saturate the fixed receptor concentration. To the extent that our analytical method of solution of for the steady state problem has no built-in mechanism for enforcing the constraint [LR] ≤ R0, the formal asymptotic solution may be an erroneous description of the steady state signaling gradient. As we shall see from an example in a later section, a formal asymptotic solution without enforcing the upper bound on [LR] may result in an (outer) asymptotic solution with [LR] > R0 and/or morphogen concentrations such as [L] and [S] may become negative. As such, the matched asymptotic solution (whose leading term outer solution is summarized in Proposition 1) is not the appropriate steady state solution for our problem when VL is high for the prescribed R0 even if the condition ε = VL/VS < 1 is met.

We note for emphasis that, for the low receptor saturation case, we do not need to consider explicitly the relevant inner solutions of the problems even in the neighborhood of the various synthesis rate and receptor expression discontinuities. For one reason, layer solution components, if any, do not affect the Dpp, Sog and Sog-Dpp complex concentrations (and their first derivatives) in a qualitatively significant way throughout the solution domain. In addition, the Dpp-receptor concentration is computed after the process of matched asymptotic expansion solution for the BVP,. Hence, we will focus our attention on some possible effects of morphogen synthesis rates and ectopic receptor expressions on the signaling morphogen concentration [LR] in the next few sections. These will be deduced from the outer solution when applicable and on numerical simulations of the initial-boundary value problem otherwise.

3.5 Numerical Simulations

It was pointed out in the previous two sections that for ε = VL/VS [double less-than sign] 1, a leading term outer solution suffices for describing accurately the effect of ectopic receptor expression prior to receptor saturation. For a Dpp synthesis rate sufficiently high to result in receptor saturation, effects of ectopic receptor concentration are generally at variance with the matched asymptotic solution even for ε [double less-than sign] 1 and may vary in rather complex ways depending on the values of the remaining system parameters. To uncover other possibilities regarding depressed or elevated [LR] concentration, we will establish in the next section some analytical properties of the outer asymptotic solution which should hold for the exact solution for the low receptor saturation and low Dpp-to-Sog synthesis rate ratio. These results are complemented by numerical simulations of the original IBVP by a finite difference scheme (see [8]). In this finite difference approach, the diffusion terms are approximated by the second order central difference and the adaptive Runge-Kutta-Fehlberg-2–3 method [26] is used for the temporal discretization. Convergence of the calculations and better resolution are observed when the spatial meshes are refined. The overall accuracy of numerical simulations is second order in space and third order in time.

The time evolution simulation code developed for the approach above has been validated by comparing results obtained for the special case of uniform receptor expression (with ρ(x, y) = ρo(x, y) [equivalent] 1) investigated in [18] with those shown in Figure 2 of that paper. The simulation code for the two-dimensional model of this paper when applied to the uniform receptor expression problem for the same set of parameter values as in [18] gives numerical results that are effectively identical to those obtained in Figure 2 of [18] with the corresponding values of [LR] at the dorsal midline agreeing to the three significant figures. As an independent consistency check, the steady state value of [LR] from the simulation code was found to be essentially the same as that calculated from the steady state value of [L] using the steady state relation (see (29))

[LR]=R(X,Y)[L]R¯0αL+[L].
(52)

The validated code for numerical solutions of the initial-boundary value problem for the reaction-diffusion system (14)(17) will be used extensively to study the effects of ectopic receptor expression in the next few sections especially for the problem in [20] which stimulated this research. Typically, simulations were run until T = 20 hrs and the prescribed stringent change tolerance had already been met. The non-monotone approach to steady state and the substantial changes between the initial state and the steady state of the [LR] gradient for our class of Dpp-Sog interaction problems have been documented extensively in [20]. It is therefore prudent to evolve the various morphogen gradients for an unusually long period to ensure steady state. A direct solution for the steady state problem is also possible and are being carried out separately. A time evolution simulation approach is preferred here to facilitate comparison with the one-dimensional results in [20].

4 Effects of Ectopic Receptor Expression

4.1 The Properties of J(s0)

In order to examine the effects of localized over-expression of tkv in the Drosophila embryo as determined by Proposition 1, we establish presently some properties of the function J(s0) in (51) and their consequences, focusing on the receptor distributions R(x, y) = R0 (so that ρ(x, y) = ρo(x, y) [equivalent] 1) and R(x, y) = Rh(x) [equivalent] R0ρh(x) (as given by (6)) in this section.

Lemma 2

J (s0) is a monotone decreasing function of s0.

Proof

For any fixed y, we have from the explicit solution for a0(y) in (44)

a0(y;s¯0)s¯0={c¯0[s¯0+18(y+2y2)]2(y<0)c¯0[s¯0+18(y2y2)]2(y>0)

so that [partial differential]a0(y; s0)/[partial differential]s0 < 0. Since ρ(x) is positive, this implies

dJ(s¯0)ds¯0=Ωρ(x,y)[1+βLa0(y;s¯0)]2a0(y;s¯0)s¯0dxdy<0.

For (43) and (44) to be applicable, we must have s0 > 0 in order for s0(y; s0) and a0(y; s0) to be nonnegative. Hence J(s0) is also positive. By Lemma 2, J(s0) is a monotone decreasing function of s0; hence J(s0) tends to 0 as s0 → ∞. The argument proved the following result:

Proposition 3

J(s0) = xmax/4 has exactly one root in (0, ∞). It has exactly one positive root if J (0) > xmax/4 and no solution if J(0) < xmax/4.

We now examine the root s0 = ζ of J(s0) = xmax/4 for several distributions of the fixed receptor concentrations uniformly in the dorsal-ventral direction:

(a) Uniform Distribution ρ(x, y) = ρo(x, y) [equivalent] 1

For this case, the expression for J(s0) involves only integration of a one-variable function:

Jo(s¯0)[J(s¯0)]ρ=ρo=1=0xmax1414[a0(y;s¯0)1+βLa0(y;s¯0)]dxdy0xmax[Io(s¯0)]dx=xmax[Io(s¯0)],
(53)

where

Io(z)=1414[a0(y;z)1+βLa0(y;z)]dy.
(54)

Thus, with ρ = ρo = 1, the condition (51) determines s0 to be ς0 with

1xmax[Jo(s¯0)]s¯0=ς0=Io(ςo)=14.
(55)

The expression (44) for a0(y; s0) can be used to re-write the integral in (54) as

Jo(ς0)=xmax2σsυL*[140dyη+ς0+18(y+2y2)+014dyη+ς0+18(y2y2)]xmax2σsυL*[Im(ς0)+Ip(ς0)xmax[Io(ς0)],
(56)

and

η=βL2σsυL*=12gLσLhLσshS=12gSαSαL.
(57)

The two integrals Im0) and Ip0) in (56) can be evaluated exactly to give

16Bm(ς0)tan1(1Bm(ς0))+16Bp(ς0)tanh1(1Bp(ς0))=σsυL*2
(58)

where

Bm(ξ)=8η+ξ164,    Bp(ξ)=8η+ξ+164.
(59)

It is an immediate consequence of Lemma 2 that

Lemma 4

Io (z) is also a monotone decreasing function of z.

(b) Uniformly Elevated Anterior Receptor Distribution ρ(x, y) = ρh(x)

With ρh(x) defined in (20), we have instead of (53)

Jh(s¯0)[J(s¯0)]ρ=ρh(x)=0xmax1414[ρh(x)a0(y;s¯0)1+βLa0(y;s¯0)]dxdy0xmax[ρh(x)Io(s¯0)]dx=(xhΔ+xmax)[Io(s¯0)].
(60)

The condition (51) determines s0 to be ςh with

1xmax[Jo(s¯0)]s¯0=ςh=(1+δhΔ)Io(ςh)=14,
(61)

where δh = xh/xmax = Xh/Xmax < 1, or

16Bm(ςh)tan1(1Bm(ςh))+16Bp(ςh)tanh1(1Bp(ςh))=σsυL*2(1+δhΔ).
(62)

(c) Relative Magnitude of ς0 and ςh

With (61) written as

Io(ςh)=14(1+δhΔ)
(63)

where Io(z) as defined in (54), we have then the following relative magnitude of the two roots ς0 and ςh:

Proposition 5

ςh [equivalent] [s0]ρ=ρh > [s0]ρ=1 [equivalent] ς0.

Proof

The claim follows immediately from Io(ς0)=14>1/4(1+δh)=Io(ςh) and Lemma 4 for any positive Xh so that δh = xh/xmax = Xh/Xmax > 0.

(d) Receptor Distribution Uniform in the Dorsal-Ventral Direction

For a piecewise continuous function of x alone, ρ(x, y) = ρvd(x), we have

[J(s¯0)]ρ=ρυd(x)Jυd(s¯0)=0xmax1414[ρυd(x)a0(y;s¯0)1+βLa0(y;s¯0)]dxdy=ρ¯υdxmax1414[a0(y;s¯0)1+βLa0(y;s¯0)]dy.
(64)

With a0(y; s0) given by (44), the integral remaining in (64) can again be evaluated exactly with

Jυd(s¯0)=ρ¯υdxmax2σsυL*[140dys˜0+18(y+2y2)+014dys˜0+18(y2y2)]ρ¯υdxmax2σsυL*[Im(s¯0)+Ip(s¯0)]=xmaxρ¯υdIo(s¯0)
(65)

where

ρ¯υdxmax=0xmaxρυd(x)dx.    s˜0=s¯0+βL2σSυL*s¯0+η
(66)

A particular application of this type of ectopic receptor distributions will be discussed in Section 5.

It is also possible to investigate the effects of ectopic receptor distributions whose ectopicity varies in both the X and Y directions. We illustrate with the following example:

(e) Receptors Concentration Nonuniform in Both X and Y

The effect of a localized over-expression of tkv more akin to the one shown in panels A and B of Figure 1 for the Drosophila embryo is also possible. The distribution in panel A may be approximated by

[RA(X,Y)]={R¯0{(1+Δ)H(y)+ΔH(y)}(X<Xh)R¯0{1+Δg}(X>Xh),
(67)

or

ρA(x,y)]=1R¯0[RA(X,Y)]={(1+Δ)H(y)+ΔH(y)(x<xh)1+Δg(x>xh),
(68)

where Δ > 0, Δg ≥ 0 and 0 ≤ Δ[ell] ≤ 1 + Δ. For this dorsal-ventral nonuniform receptor distribution, Lemma 2 and Proposition 3 continue to hold since their proofs apply to all positive receptor concentration. Hence J(s0) as defined in (51) continues to be a monotone decreasing function of s0 and J(s0) = xmax/4 has exactly one root in (0, ∞) if J(0) > xmax/4.

4.2 Signalling Receptor Concentration

From (37), we have

[LR]R¯0=B(x,y)=b(x,y;ε)=ρ(x,y)βLa(x,y)1+βLa(x,y)~ρ(x,y)βLa0(y;s¯0)1+βLa0(y;s¯0)=ηρ(x,y)η+s0(y;s¯0)
(69)

where s0(y; s0) and η are given by (43) and (57), respectively.

  1. ρ(x, y) = 1: We have for this case s0 = ς0 and therewith
    1ηR¯0[LR]ρ(x)=1=1η[b(x,y;ε)]ρ(x)=1~[1η+s0(y;s¯0)]s¯0=ς0={[η+ς0+18(y+2y2)]1y<0[η+ς0+18(y2y2)]1y>0.
    (70)
  2. ρ(x, y) = ρh(x): For this case, we have similarly s0 = ςh and therewith
    1ηR¯[LR]ρ(x)=ρh(x)=1η[b(x,y;ε)]ρ(x)=ρh(x)~[ρh(x)η+s0(y;s¯0)]s¯0=ςh.
    (71)

As a principal aim of our research effort, we wish to learn whether [LR] is depressed outside the region of elevated receptor expression. For this posterior end range xh < xxmax where the normalized receptor concentration is unit, i.e., ρh(x) = 1, we have

1ηR¯[LR]ρh(x)~[1η+s0(y;s¯0)]s¯0=ςh={[η+ςh+18(y+2y2)]1  (y<0)[η+ςh+18(y2y2)]1  (y>0).
(72)

From this follows the next result which addresses the question that motivated this investigation:

Proposition 6

At low receptor occupation and the outer solution applies, over-expressing Dpp receptors tkv on the anterior half of the embryo reduces PMad activation in cells on the posterior part of the embryo.

Proof

The observation is an immediate consequence of (70), (71) and Proposition 5.

When receptor occupation by Dpp is low so that the asymptotic solution of Section 3 applies, we now see that an elevated receptor concentration in the anterior end of the embryo invariably leads to a depression of signaling bound morphogen concentration posterior to the region of elevated receptor concentration, whether the depression is noticeable depends on the magnitude of the Dpp synthesis rate (with values of all other parameters fixed).

As for the effect of ectopic tkv expression on the signaling [LR] at the anterior end of the embryo (the site of ectopic receptors), we have for 0 < x < xh

1ηR¯[LR]ρh(x)~[1+Δη+s0(y;s¯0)]s¯0=ςh={(1+Δ)/[η+ςh+18(y+2y2)]  (y<0)(1+Δ)/[η+ςh+18(y2y2)]  (y>0).
(73)

The comparison with the corresponding expression for ρ(x) = 1 in the same region now depends on the magnitude of Δ, whether it is sufficiently large to compensate for the reduction by a large s0 in the denominator. In particular, we have

Proposition 7

At low receptor occupation so that the outer asymptotic solution applies, over-expressing Dpp receptors tkv on the anterior end of the embryo by a sufficiently large concentration so that

1+Δ>η+ςh164η+ς0164
(74)

elevates PMad activation in cells on the part of the embryo with the ectopic receptors. The opposite is true if the inequality in (74) is reversed at least for a part contiguous to the dorsal midline.

4.3 Elevated Anterior Receptor Expression

For the asymptotic solution to be applicable, we must have ε = VL/VS [double less-than sign] 1, i.e. VL must be small compared to VS. To illustrate the diverse range of possible steady state configurations in different range of ε (within and outside the range ε [double less-than sign] 1), the 2-D numerical simulation code will be applied in this section to the problem investigated in Figure 5 of [20] which stimulated this research. In addition to VL = 1 nM/s = 10−3µM/s and VS = 0.08 µM/s investigated in that figure, we will also examine cases with VL = 2.5 × 10−4µM/s and with VS = 0.6 µM/s and 10−3µM/s. The remaining values of the different parameters for the problem used in [20] are given in Table (1).

Figure 5
Distribution of [LR] in the anterior-posterior direction at the dorsal midline for the fixed receptor expression Rh(X) (with ectopic expression in 0 ≤ X < Xh = 0.02 cm) for VL = 2.5 × 10−4µM/s and ...
Table (1)
Parameter Values

Remark 8

For these parameter values, we have αL = (gL + fL)/hL [similar, equals] 4.367 × 10−4, μL2=gL/αLhL=4.27×103 and η = 0.00117374…. We will be interested in cases corresponding to the six different combinations of values of VL and VS fixing other parameter values as given in Table (1).

With Dpp synthesized only in the dorsal region and diffused away fromits localized source, the steady state distribution of Dpp expression [L] is expected to reach its maximum at the dorsal midline Y = −Ymax/4 and decreases monotonically in both directions toward its minimum at Y = Ymax/4. Since the steady state [LR] is an increasing function of [L] (see (29) or (69)), it also attains its maximum at the dorsal midline Y = −Ymax/4. Shown in Figure 4 is a two-dimensional plot of the steady state signaling gradient [LR] for the case R(X, Y) = Rh(X) with VL = 0.25 × 10−3 µM/s and VL = 0.6 µM/s. It provides the numerical evidence confirming these qualitative features of the [LR] expression. As such, we will focus our discussion on the signaling gradient along the dorsal midline assisted at times by one-dimensional plots of midline graphs such as the one in Figure 5 for [LR] at Y = −Ymax/4. Given the level of [LR] for a uniform receptor expression reported in Table (2) below (in rows with δh = 0), it is clear that the signaling gradient is now depressed at the posterior end (where there is the same uniform receptor expression) and elevated at the anterior end (where there is an ectopic receptor expression), at least in an interval adjacent to Xh. The elevation and depression become more uniform in X < Xh and X > Xh, respectively, for higher Sog synthesis rates VS.

Figure 4
Two-dimensional plot of the [LR] distribution for the fixed receptor expression Rh(X) (with ectopic expression in 0 ≤ X < Xh = 0.02 cm) for VL = 2.5 × 10−4µM/s and VS = 0.6µ ...
Table (2)
Numerical Solution for [LR] in µM at Dorsal Midline

Starting from quiescence, the non-monotone approach to the steady behavior of the Dpp-Sog interaction has been found to be similar to the one-dimensional problem already discussed extensively in [20]. Hence, it will not be further elaborated herein. Instead, we will present results on the steady state [LR] at different locations of the dorsal midline in the direction of the anero-posterior axis to illustrate the complexity of possible outcomes of the same ectopic receptor expression depending on the magnitude of the two ligand synthesis rates (with all other wing disc rate constants fixed). We will discuss separately the signaling gradient for two particular values of the Dpp synthesis rate, first for VL = 0.25 × 10−3 µM/s and then for VL = 10−3 µM/s for which some observations were given in [20].

4.3.1 Signaling Morphogen Gradient [LR] for VL = 0.25 × 10−3 µM/s

In Table (2), the signaling morphogen gradient [LR] at the dorsal midline are given for VL = 0.25 × 10−3 µM/s. The results for a uniform receptor expression (with R0 = 3 µM) throughout the solution domain are given in the odd rows (with δh = 0) for VS = {0.001, 0.08, 0.6 and ∞} µM/s, respectively. The results for VS = ∞ corresponds to those obtained from the outer asymptotic solution while those for finite VS were obtained by the simulation code for the IBVP. For a uniform receptor distribution, the dorsal midline values of [LR] shown are independent of X as expected and tend to the asymptotic solution as VS increases from 0.08 µM/s to 0.6 µM/s with the latter nearly equal to the asymptotic value. For VS = 1 nM/s = 10−3 µM/s however, we have ε=14 so that the Sog synthesis rate is relatively low leading to a level of Sog concentration in steady state that is too low to shuttle enough Dpp back to the anterior end for additional binding with available receptors to achieve the level of [LR] given by the outer asymtptotic solution.

The even rows in Table (2) with δh = 4/11 give the corresponding results for a localized elevated receptor expression Rh(X) (see (6)). In the interval Xh < XXmax, a comparison of the steady state [LR] concentration for the two different receptor expressions clearly shows a close agreement between the asymptotic and simulation values of [LR] at the dorsal midline near the Xh and less so near the end Xmax. The agreement throughout the interval gets better as VS increases from 0.08 µM/s and more Dpp molecules are shuttled away by a higher concentration of Sog. There is clearly a depression of [LR] concentration in the posterior portion of the solution domain X > Xh (as well as a qualitative agreement even for the case VS = 1nM/s (ε = 1) for which the asymptotic solution is not applicable.)

In the region with elevated receptor expression, 0 ≤ X < Xh, the situation is more complicated. The outer asymptotic steady state solution for the receptor distribution Rh(X) shows a higher steady state [LR] concentration than the corresponding (uniform) concentration for a uniform receptor distribution throughout the sub-interval [0, Xh). On the other hand, the numerical solutions for the IBVP shows a higher [LR] concentration only for VS = 0.6 µM/s; moreover, the elevation in [LR] for this (and other) Sog synthesis rate is nonuniform, closer to the asymptotic solution near Xh and considerably smaller near X = 0. Since the asymptotic steady state solution for [LR] does not depend on VS (or more correctly corresponds to the limiting case of VS = ∞), the lower level of [LR] near X = 0 appears to be due to less Dpp-Sog binding and hence more Dpp-receptor binding resulting in more receptor-mediated degradation of Dpp for the given moderate synthesis rate VL.

For VS = 0.08 µM/s, numerical solutions show an elevated level of [LR] only for the part of the interval [0, Xh) near X = Xh, the location for the abrupt change of receptor expression. For this lower Sog synthesis rate (which is still large compared to the Dpp synthesis rate), [LR] is actually depressed near the end X = 0. The nonuniform distribution in the antero-posterior direction reflects the fact that receptors near Xh has the rights of first refusal to bind with Dpp freed up from the degradation of the Dpp-Sog complexes being transported back to the anterior end. For the given VL and a lower VS such as 0.08 µM/s, the amount of Sog produced does not lead to a sufficiently large concentration of Dpp-Sog to be transported to the anterior end and degrade, freeing up sufficient Dpp to diffuse further away from Xh for the unoccupied receptors near X = 0.

To put it another way, for a fixed Dpp synthesis rate, we get an elevated expression of [LR] at the tip of the anterior compartment away from the location of the abrupt change in receptor expression at Xh of Rh(X) only if the Sog synthesis rate is sufficiently high. When all other biological rate constants of the embryo are held fixed, we need a high level of Sog expression for the formation of a level of [LS] concentration to be transported from the ventral region back to the entire dorsal region, to dissociate and degrade, freeing up enough Dpp to be binding with receptor tkv everywhere in the anterior compartment. At the same time, it appears from numerical solutions of the IBVP that the abrupt change of receptor expression at Xh would invariably results in a boundary layer phenomenon on both sides of the discontinuity of the receptor distribution. The elevation and depression of the steady state signaling [LR] concentration in a narrow region (in the anterior and posterior side, respectively) adjacent to Xh are more pronounced than expected from the informal asymptotic consideration.

4.3.2 Non-signaling Gradients for VL = 0.25 × 10−3 µM/s

When the asymptotic steady state solution is applicable, there is also a close agreement between the asymptotic and simulation results for [LS], [S], and [L] except for the expected spatial nonuniformity in both X and Y directions with the latter similar to those shown in Figure 5 in [18]. The corresponding asymptotic and numerical solutions for [L], [S] and [LS] are given in the Table (3) for VS = 0.08 µM/s to demonstrate the qualitative agreement for this set of parameter values, particularly near X = Xh. The nonuniformity in the X direction is more pronounced for [L] and [S]. The variation with X (computed but not shown here) reduces considerably for VS = 0.6 µM/s or larger.

Table (3)
Asymptotic and Numerical Results at Dorsal Midline

4.3.3 Signaling Morphogen Gradient [LR] for VL = 10−3 µM/s

For a sufficiently high Dpp synthesis rate, there would be more than enough Dpp for binding to saturate the available tkv receptors (which is fixed in our model), at least in an interval 0 < |YYmax/4| < dy centered at the dorsal midline along the posterior end of the embryo in steady state. This is evident from the dorsal midline values of [LR] for the case of a uniform receptor expression shown in Table (4) for VS = 0.6 µM/s and 0.08 µM/s. A graph of [LR] at the dorsal midline for VS = 0.6 µM/s is shown in Figure 6 to illustrate the saturation. Thus the higher Dpp synthesis rate VL = 10−3 µM/s coupled with a sufficiently high Sog synthesis rate leads to a saturation of available receptors in the posterior segment. The asymptotic steady state solution is not applicable in this case as the method of solution does not take into account receptor saturation. In fact, that solution gives a steady state [LR] concentration in excess of the prescribed receptor concentration (see the case VS = ∞ in Table (4)). The inappropriateness of such a solution is also reflected in its (unacceptable) negative steady state concentrations for the free Sog and Dpp molecules (not shown herein).

Figure 6
Distribution of [LR] in the anterior-posterior direction at the dorsal midline for the fixed receptor expression Rh(X) (with ectopic expression in 0 ≤ X < Xh = 0.02 cm) for VL = 10−3µM/s and V ...
Table (4)
[LR] at Dorsal Midline (VL = 10−3µM)

At the lower Sog synthesis rate of VS = 10−3 µM/s, the steady state [LR] at the dorsal midline given in Table (4) is at about 1.98.. µM at the two ends of the embryo, which is nowhere near saturation for the available receptors (with R0 = 3 µM) and seemingly consistent with that predicted by the outer asymptotic solution ([similar, equals] 2.04..). However, the agreement is somewhat fortuitous since the outer asymptotic solution also does not apply in this case given VS = VL so that ε is not small compared to unity and a leading term regular perturbation solution is also mathematically inappropriate. Biologically, the expression of Sog at the ventral region is not sufficiently high so that not enough Dpp is transported to the dorsal region for binding with the available receptors in that region (except in the neighborhood of Xh where it is accomplished at the expense of similar activities near Xh on the posterior side).

While the results for VL = 10−3 µM/s in Table (4) clearly demonstrate the limitation of the outer asymptotic steady state solution obtained in Section 3, they do not in any diminish value of the analytic solution since its applicability can easily be decided by examining whether the concentration of signaling Dpp-receptor complexes meets the restriction [LR] ≤ R0 and whether the remaining concentration gradients are nonnegative. We simply do not use the outer asymptotic solution if either of the two constraints is not met. In this context, it is of some interest to point out that while the asymptotic steady state solution gives an [LR] concentration well in excess of the allowed upper bound for the case of a uniform receptor expression, the corresponding solution for the piecewise constant receptor expression satisfies all the inequality constraints and can therefore be used to study this phase of the embryonic biological development.

4.3.4 Summary

The developments in this section show the complementary nature of the analytical and numerical methods. The former has the advantage of exhibiting more explicitly the dependence of the solution on the various system parameters while the latter applies to a broader region in the parameter space. To simplify our analysis and computation, we have chosen to work with a model with a prescribed tkv receptor concentration fixed for all time. The restriction limits the applicability of the asymptotic solution developed in section 3 which by nature does not take into account the constraint of a fixed receptor concentration during the solution process.

With all other rate parameters fixed, the asymptotic steady state solution provides an adequate description of the signaling morphogen gradient concentration [LR] for moderate Dpp synthesis rates VL that do not saturate the fixed receptor concentration R0 (such as the case VL = 2.5 × 10−4 µM/s reported in Table (2)). For these moderate Dpp synthesis rates, the distribution of [LR] in the anterior-posterior direction obtained by the (more accurate) numerical simulations tends to the asymptotic solution (with a uniform distribution on both sides of the receptor concentration discontinuity) for higher and higher Sog synthesis rates, e.g., for VS = 0.6 µM/s or higher. More importantly (pertaining to the purpose of our investigation), these results established the existence of a depressed [LR] concentration in the posterior end of the embryo when there is an ectopic receptor expression in the anterior end. The asymptotic solution helps delineate more explicitly the mechanism responsible for the depression.

On the other hand, for relatively low Sog synthesis rates such as VS = 0.08 µM/s, the [LR] concentration near the anterior end X = 0 may even be, somewhat surprisingly, lower than the level for the case of a uniform receptor distribution, the elevated level of receptor concentration notwithstanding. An explanation for this somewhat unexpected result was given in subsection 4.3.1.

For the same rate parameter values but higher Dpp synthesis rates at the level of VL = 10−3 µM/s, the available free Dpp eventually saturate the available receptors (of fixed concentration) for the uniform fixed receptor case and at least in the posterior region for the receptor distribution Rh(x) ectopically expressed at the anterior end. For these case, the asymptotic solution gives a signaling gradient [LR] in excess of the available receptors in some region of the embryo (and possibly negative concentration for the Sog and Dpp-Sog gradients) and hence would be inappropriate for the problem. Such a limitation on the asymptotic steady state solution would not be present in a model allowing for receptor renewal.

5 Signaling Morphogen Concentration for Ectopic Expression at Both Ends

5.1 Properties of J(s0)

If we examine panel B of Figure 1 more closely, we would see that there seems to be an over-expression of receptors at both ends of the anterior-posterior axis. To find out what our model would predict for this configuration of receptor over-expression, we consider the following normalized distribution of receptor concentration:

ρ(x,y)=ρA(x)={1+Δ>1(0x<x)1(x<x<xg)1+Δg>1(xg<xxmax).
(75)

For this ρ(x, y), we have instead of (63)

1xmaxJA(s¯0)[{1+δΔ+(1δg)Δg}Io(s¯0)]s¯0=ςA,    δk=xkxmax<1
(76)

where ςA is the solution of

Io(ςA)=141+Δδ+Δg(1δg),
(77)

or

16Bm(ζA)tan1(1Bm(ζA))+16Bp(ζA)tanh1(1Bp(ζA))=σsυL*2[1+Δδ+Δg(1δg)]
(78)

with Bm and Bp given in (59).

Proposition 9

ςA [equivalent] [s0]ρ=ρA > [s0]ρ=ρo [equivalent] ς0.

Proof

This follows from the fact that the right hand side of (77) is less than 12 and Io is a monotone decreasing function of s0 by Lemma 4.

5.2 Signalling Receptor Concentration

We now compare the signaling Dpp for the normal case of a uniformly distributed receptor concentration with one that is over-expressed at the two ends of the anterior-posterior axis, particularly the interval x[ell] < x < xg where the latter is not over-expressed.

For the uniformly distributed case, we have [LR]ρ(x)=1/η R0 is given by (70) as before. For ρ(x) = ρA(x), we have for the range x[ell] < x < xg,

1ηR¯0[LR]ρA(x)~[1η+s0(y;s¯0)]s¯0=ςA={[η+ςA+18(y+2y2)]1(y<0)[η+ςA+18(y2y2)]1(y>0)<[1η+s0(y;s¯0)]s¯0=ς0~1ηR¯0[LR]ρ(x)=1
(79)

where the inequality is a consequence of Proposition 9. The implication of (79) is summarized in the following proposition:

Proposition 10

At low receptor occupation so that the out asymptotic solution applies, over-expressing Dpp receptors tkv at both end of the anterior-posterior axis reduces PMad activation in cells in the region X[ell] < X < Xg outside the location with ectopic receptors.

For the intervals where tkv is over-expressed, the situation is again more complicated. In the range 0 < x < x[ell], we have

1ηR¯0[LR]ρA(x)~[1+Δη+s0(y;s¯0)]s¯0=ςA={(1+Δ)/[η+ςA+18(y+2y2)](y<0)(1+Δ)/[η+ςA+18(y2y2)](y>0).
(80)

The comparison with the corresponding concentration for a uniform receptor distribution now depends on the magnitude of Δ[ell]. In particular, we have

Proposition 11

At low receptor occupation so that the out asymptotic solution applies, over-expressing Dpp receptors tkv at both ends of the anterior-posterior axis by a sufficiently large concentration at the anterior end so that

1+Δ>η+ςA116η+ς0116
(81)

elevates PMad activation in the interval 0 < x < x[ell] while the opposite is true if the inequality in (81) is reversed at least for a region contiguous to the dorsal midline. A corresponding result applies to the posterior end.

5.3 Numerical Results for the Illustrative Example

To provide numerical evidence in support of a depressed signaling gradient expression for an ectopic receptor distribution of the type RA(X) defined in (7), we consider such a receptor distribution with Δ[ell] = Δg = 0.02 cm. Given the the symmetry of RA(X), a similar symmetry expected of the corresponding [LR] expression is confirmed by the two-dimensional plot of its steady state as in Figure 7 for VL = 0.25 × 10−3 µM/s and VS = 0.6µM/s the corresponding one-dimensional plot for its graph along the dorsal midline. Hence, we can limit our discussion to the steady state behavior for the anterior half of the embryo (as in reporting the numerical results shown in Table (5) and Table (6)).

Figure 7
Two-dimensional plot of the [LR] distribution for the fixed receptor expression RA(X) (with the same ectopic expression in 0 ≤ X < X[ell] = 0.02 cm and Xg = 0.035 cm < XXmax) for VL = 2.5 × ...
Table (5)
[LR] at Dorsal Midline
Table (6)
[LR] at Dorsal Midline

From the numerical results for the case VL = 0.25 × 10−3µM/s reported in Table (5), we see a good qualitative agreement between the numerical simulation results for VS = 0.6µM/s and the asymptotic results. As shown in Figure 8, the numerical solution for [LR] along the dorsal midline for VS = 0.6µM/s is nearly uniform within the three subinterval [0, X[ell]), (X[ell], Xg) and (Xg, Xmax]. This is consistent with the prediction by the asymptotic steady state solution. Furthermore, the trend of [LR] is toward the asymptotic solution as VS increases from 10−3µM/s.

Figure 8
Distribution of [LR] in the anterior-posterior direction at the dorsal midline for the fixed receptor expression RA(X) (with the same ectopic expression in 0 ≤ X < X[ell] = 0.02 cm and Xg = 0.035 cm < XXmax) for ...

For the higher Dpp synthesis rate VL = 10−3µM/s, the agreement between the numerical and asymptotic solution is also good but in a different way. With more Dpp available, the same Sog synthesis rate VS = 0.6µM/s proportionally less Dpp toward the anterior end of the embryo, especially near the anterior vertex. Consequently, there is a more pronounced nonuniformity in each of the three subintervals (qualitatively similar to the distribution for Rh(X)), decreasing in magnitude from X[ell] to 0 (and by symmetry from Xg to Xmax). Because of less transport, more Dpp molecules are degraded, especially in the waist region of the embryo X[ell] < X < Xg resulting in a lower [LR] concentration than that for VL = 0.25 × 10−3µM/s. On the other hand, the discrepancy between numerical and asymptotic solution for [LR] is substantially less than those of Table (5) for the lower Dpp synthesis rate case.

6 Asymptotic Behavior for VL [dbl greater-than sign] VS

For the biologically less realistic case of VS/VL = 1/ε [double less-than sign] 1, the structure of the BVP admits a regular perturbation solution in powers of 1/ε. The governing equations for the leading term solution, {a0, …, s0}, corresponds to setting 1/ε = 0 in (34)(36) to get

μL22a0ρa01+βLa0+H(y)=0,
(82)

(fS+gS)12c0+(νL*a0s0c0)=0,
(83)

2s0(νL*a0s0c0)σSc0+H(y)=0.
(84)

Note that the form of equations (35) and (36) is not changed by setting 1/ε = 0; nor is the equation (37) giving b(x, y; ε) in terms a(x, y; ε):

b0(x,y)=ρ(x)βLa0(x,y)1+βLa0(x,y).
(85)

More importantly, (82) is an equation for a0(x, y) alone and, augmented by the homogeneous Neumann condition along the edges of the rectangular domain, may be solved separately.

Proposition 12

A unique nonnegative solution of the BVP for a0 exists with

0a0(x,y)au
(86)

where the constant au is given by

au=11βL.

Proof

Evidently, a[ell] = 0 is a lower solution for the problem. On the other hand, we have

μL20au+ρ(x)au1+βLauH(y)=[ρ(x)1]au1+βLau+1H(y)=ρ(x)1+H(y)>0

given ρ ≥ 1. With [partial differential]au/[partial differential]n = 0, au is an upper solution for the BVP. Hence, a solution of the BVP problem exists and is bounded as in (86) [1, 24, 25]. Uniqueness is proved as in [13].

Having the piecewise C2 solution for a0(x, y), equations (83) and (84) together with the relevant homogeneous Neumann conditions may be solved for c0(x, y) and s0(x, y). Note that the two PDE (83) and (84) are linear in the two unknowns. Actual solutions may be obtained by methods similar to those used in [16, 29]. However, for the purpose of delineating the effects of locally over-expressed tkv on cell signaling, we need only the solution of (82) with which we can calculate [LR] from (85). The numerical solution for the BVP for a0(x, y) is straightforward. Given that μL2 is typically small compared to unity, we note also the singular perturbation structure of (82) with respect to the parameter μL so that asymptotic solution for large μL is also possible. Sample solutions have been obtained for typical sets of parameter values used in [13, 16, 29]. The results on depression of Dpp signaling by localized over-expression of receptors are qualitatively different from that stated in Proposition 6 for ρ(x) = ρh(x) regarding the the abrupt depression of the bound morphogen concentration posterior to the region of the elevated receptor concentration.

7 Conclusion

When the receptor expression is ectopic in the anterior end of an Drosophila embryo, a simple mathematical model based on the essential biological processes for morphogen gradients in Drosophila embryos identified in [12, 20] show that a depression of the signaling Dpp-tkv concentration in the posterior does occur under suitable conditions. At the same time, a lack of Dpp-tkv concentration depression as noted in the work of Wang and Ferguson [30] is now seen to be possible for a number of reasons including:

  • Sufficiently high Dpp and Sog synthesis rates that saturate the fixed receptor expression throughout the posterior portion of the embryo (see the case VL = 10−3 µM/s and VS = 0.6µM/s in Table (4))
  • A sufficiently low Sog synthesis rate that does not shuttle much if any Dpp to the anterior region except in a narrow layer adjacent to the location of receptor expression discontinuity as seen from the case of VS = 10−3µM/s in both Table (2) and Table (4). (Not shown by the Tables is the substantial depression of [LR] in each case is confined to a narrow region with the width of the region becoming narrower as VL increases.)
  • A relatively low elevation of the receptor expression in the anterior region which allows enough Dpp for binding in the posterior region (except for a narrow phenomenon near Xh) as shown in Table (2) for VS = 10−3µM/s

It is also seen from Table (2) (and the graphs for [LR] for the case of VL = 2.5 × 10−4µM/s not shown herein) that the depression in the posterior region (and the elevation in the anterior region) becomes more uniform in X with increasing VS. This suggests that the partial depression over a small subinterval of the posterior end near the location of receptor expression discontinuity for the case of Rh(X) may be the consequences of three different phenomena:

  • A very low Sog synthesis rate VS that transports to the nearby anterior region only the Dpp and Dpp-Sog in a narrow layer adjacent to the location Xh of the discontinuity in Rh(X)
  • Intermediate Sog synthesis rates that manage to shuttle sufficient Dpp and Dpp-Sog away from a finite interval (Xh, Xg) in the posterior region to the anterior region with Xg substantially smaller than Xmax.
  • The receptor expression is elevated at both end (qualitatively similar to the results for RA(X)) leading to an [LR] gradient elevated at both ends and depressed in region of normal receptor expression in between (see Table (5) and Table (6))

The results from our simple mathematical model also offered new insight not previously observed. One important result of this category is the variability of the elevation in [LR] in the region of ectopic receptor expression. From the asymptotic analysis of Sections 3 and 5, we now know that [LR] expression may be elevated, unchanged or even depressed depending on the combinations of the various rate parameter values. In particular, we have from Propositions 7 and 11 that

  • An elevation of the receptor expression in the ectopic region not sufficiently high would result in no change or a depression (rather than an elevation) of signaling morphogen gradient [LR] in the anterior region if it does not meet the condition (such as (74) for Rh(X) and (81) for RA(X)) required for an elevated [LR] expression

It should be noted that some of the results obtained by our asymptotic analysis and numerical simulation are consequence of our simplifying assumption of a fixed receptor expression throughout the embryo which may or may not persist should we allow for receptor synthesis and renewal. It should persist if the receptor-mediated Dpp degradation rate constant is the same as the degradation rate constant of unoccupied receptors. Research results for the more realistic model allowing for receptor renewal will be reported elsewhere. Meanwhile, the results for the fixed receptor expression model reported herein should serve much more than a proof of concept on how we may address the question whether there should be a depression of BMP signaling posterior to the anterior region of ectopic receptor expression and how it should depend on the level of Sog synthesis rate and other rate constants.

Acknowledgment

The research was supported in part by NIH grants P50-GM076516, R01GM067247and R01GM075309. The two NIH R01 grants were awarded through the Joint NSF/NIGMS Initiative to Support Research in the Area of Mathematical Biology. The research of Y.-T. Zhang was partially supported by Oak Ridge Associated Universities (ORAU) Ralph E. Powe Junior Faculty Enhancement Award and by NSF research grant DMS-0810413.

References

1. Amann H. On the existence of positive solutions of nonlinear boundary value problems. Indiana Univ. Math. J. 1971;Vol. 21:125–146.
2. Ashe HL, Levine M. Local inhibition and long-range enhancement of Dpp signal transduction by Sog. Nature. 1999;Vol. 398:427–431. [PubMed]
3. Bier E. A unity of opposites. Nature. 1999;Vol. 398:375–376. [PubMed]
4. Crick FC. Diffusion in embryogenesis. Nature. 1970;Vol. 225:40–42.
5. Eldar A, Dorfman R, Weiss D, Ashe H, Silo BZ, Barkai N. Robustness of MmP morphogen gradient in Drosophila embryonic patterning. Nature. 2002;Vol. 419:304–308. [PubMed]
6. Entchev EV, Schwabedissen A, Gonzalez-Gaitan M. Gradient formation of the TGSF-beta homolog Dpp. Cell. 2000;Vol. 103:981–991. [PubMed]
7. Gurdon JB, Bourillot PY. Morphogen gradient interpretation. Nature. 2001;Vol. 413:797–803. [PubMed]
8. Gustafsson B, Kreiss H-O, et al. New York: Wiley Inc.; 1995. Time Dependent Problems and Difference methods.
9. Iron D, Syed A, Theisen H, Lukacsovich T, Naghibi M, Marsh JL, Wan FYM, Nie Q. The role of feedback in the formation of morphogen territories. Math. Biosci. & Eng. (MBE) 2008;5(No2):277–298. [PMC free article] [PubMed]
10. Kao J, Nie Q, Teng A, Wan FYM, Lander AD, Marsh JL. Can morphogen activity be enhanced by its inhibitors?. In: Bathe KJ, editor. Proc. 2nd MIT Conf. on Comp. Mech.; Elsevier Science Ltd; 2003. pp. 1729–1734.
11. Kerszberg M, Wolpert L. Mechanisms for positional signalling by morphogen transport: a theoretical study. J. Theor. Biol. 1998;Vol. 191:103–114. [PubMed]
12. Lander AD, Nie Q, Wan FYM. Do Morphogen Gradients Arise by Diffusion? Developmental Cell. 2002;Vol. 2:785–796. [PubMed]
13. Lander AD, Nie Q, Wan FYM. Spatially distributed morphogen production and morphogen gradient formation. Math. Biosci. Eng. (MBE) 2005;Vol. 2:239–262. [PubMed]
14. Lander AD, Nie Q, Wan FYM. Internalization and end flux in morphogen gradient formation. J. Comp. Appl. Math. 2006:232–251.
15. Lander AD, Nie Q, Wan FYM. Membrane associated non-recptors and morphogen gradients. Bulletin of Math. Bio. 2006 to appear.
16. Lander AD, Nie Q, Wan FYM, Xu J. Diffusion and Morphogen Gradient Formation - Part I: Extracellular Formulation. Center for Complex Biological Systems (CCBS) preprint, UCI. 2002
17. Lou Y, Nie Q, Wan FYM. Nonlinear eigenvalue problems in the stability analysis of morphogen gradients. Studies in Appl. Math. 2004;Vol. 113:183–215.
18. Lou Y, Nie Q, Wan FYM. Effects of Sog on Dpp-receptor binding. SIAM J. Appl. Math. 2005;Vol. 65:1748–1771. [PMC free article] [PubMed]
19. Marques G, Musacchio M, Shimell MJ, Wunnenberg-Stapleton K, Cho KW, O’Connor MB. Production of a DPP activity gradient in the early Drosophila embryo through the opposing actions of the SOG and TLD proteins. Cell. 1997;Vol. 91:417–426. [PubMed]
20. Mizutani CM, Nie Q, Wan FYM, Zhang Y-T, Vilmos P, Bier E, Marsh JL, Lander AD. Formation of the BMP activity gradient in the Drosophila embryo. Dev. Cell. 2005;Vol. 8:915–924. with Supplement. [PMC free article] [PubMed]
21. Oelgeschlager M, Larrain J, Geissert D, Roberts EM. The evolutionarily conserved bmp-binding protei twised gastrulation promotes bmp signalling. Nature. 2000;Vol. 405:757–762. [PMC free article] [PubMed]
22. Piccolo S, Agiusa E, Lu B, Goodman S, Dale L, De Roberts E. Cleavage of chordin by xolloid metalloprotease suggests a role for proteolytic processing in the regulation of spemann organizer activitiy. Cell. 1997;Vol. 91:407–416. [PMC free article] [PubMed]
23. Ross JJ, Shimmi O, Vilmos P, Petryk A, Kim H, Gaudenz K, Hermanson S, Ekker SC, O’Connor MB, Marsh JL. Twisted gastrulation is a conserved extracellular BMP antagonist. Nature. 2001;Vol. 410:479–483. [PubMed]
24. Sattinger DH. Monotone Methods in Nonlinear Elliptic and Parabolic Boundary Value Problems. Indiana University Math. J. 1972;Vol. 21:981–1000.
25. Smoller J. Shock Waves and Reaction-Diffusion Equations. New York: Springer Verlag Inc.; 1982.
26. Stoer J, Bulirsch R. New York: Springer; 1993. Introduction to Numerical Analysis.
27. Teleman AA, Cohen SM. Dpp gradient formation in the Drosophila wing imaginal disc. Cell. 2000;Vol. 103:971–980. [PubMed]
28. Teleman AA, Strigini M, Cohen SM. Shaping morphogen gradients. Cell. 2001;Vol. 105:559–562. [PubMed]
29. Vargas B. Ph.D. Dissertation. Irvine: University of California; 2006. Dec, Leaky boundaries and morphogen gradient formation.
30. Wang Y-C, Ferguson EL. Spatial bistability of Dpp-receptor interactions during Drosophila dorsal-ventral patterning. Nature. 2005;Vol. 434:229–234. [PubMed]
31. Wolpert L, Beddington R, Brockes J, Jessel T, Lawrence P, Meyerowitz E. Principles of Development. 2nd Ed. Oxford Unviersity Press; 2002.