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**|**HHS Author Manuscripts**|**PMC2774827

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- Abstract
- 1 Introduction
- 2 The Mathematical Model
- 3 Matched Asymptotic Expansions for L/S 1
- 4 Effects of Ectopic Receptor Expression
- 5 Signaling Morphogen Concentration for Ectopic Expression at Both Ends
- 6 Asymptotic Behavior for L S
- 7 Conclusion
- References

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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.xPMCID: PMC2774827

NIHMSID: NIHMS134928

See other articles in PMC that cite the published article.

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.

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*.

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.

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.

(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 Ω* _{XY}* in the

As indicated in the Figure 2 (D), *Sog* is synthesized at a constant rate *V _{S}* and only in the ventral region while

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

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 *V _{L}(X)*) while

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 *D _{L}*,

$$\begin{array}{cc}\frac{\partial [L]}{\partial T}=\hfill & {D}_{L}{\nabla}_{\mathit{\text{XY}}}^{2}[L]-{k}_{\mathit{\text{on}}}[L]\{[R]-[\mathit{\text{LR}}]\}+{k}_{\mathit{\text{off}}}[\mathit{\text{LR}}]\hfill \\ \hfill & \hfill -{j}_{\mathit{\text{on}}}[L][S]+({j}_{\mathit{\text{off}}}+\tau {j}_{\mathit{\text{deg}}})[\mathit{\text{LS}}]+{V}_{L}(Y)\end{array}$$

(1)

$$\frac{\partial [\mathit{\text{LR}}]}{\partial T}={k}_{\mathit{\text{on}}}[L]\{[R]-[\mathit{\text{LR}}]\}-({k}_{\mathit{\text{off}}}+{k}_{\mathit{\text{deg}}})[\mathit{\text{LR}}]$$

(2)

$$\frac{\partial [\mathit{\text{LS}}]}{\partial T}={D}_{\mathit{\text{LS}}}{\nabla}_{\mathit{\text{XY}}}^{2}[\mathit{\text{LS}}]+{j}_{\mathit{\text{on}}}[L]\phantom{\rule{thinmathspace}{0ex}}[S]-({j}_{\mathit{\text{off}}}+{j}_{\mathit{\text{deg}}})[\mathit{\text{LS}}]$$

(3)

$$\frac{\partial [S]}{\partial T}={D}_{S}{\nabla}_{\mathit{\text{XY}}}^{2}[S]-{j}_{\mathit{\text{on}}}[L][S]+{j}_{\mathit{\text{off}}}[\mathit{\text{LS}}]+{V}_{S}(Y)$$

(4)

$$\mathbf{\left\{}\begin{array}{c}\hfill {V}_{L}(Y)\hfill \\ \hfill {V}_{S}(Y)\hfill \end{array}\mathbf{\right\}}=\mathbf{\left\{}\begin{array}{c}\hfill {\overline{V}}_{L}H(-Y)\hfill \\ \hfill {\overline{V}}_{S}H(Y)\hfill \end{array}\mathbf{\right\}},{\nabla}_{\mathit{\text{XY}}}^{2}(\phantom{\rule{thinmathspace}{0ex}})=\frac{{\partial}^{2}(\phantom{\rule{thinmathspace}{0ex}})}{\partial {X}^{2}}+\frac{{\partial}^{2}(\phantom{\rule{thinmathspace}{0ex}})}{\partial {Y}^{2}},$$

(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 *k _{on}*,

- a spatially uniform distribution with [
*R*(*X, Y*)] = [];_{0} - a distribution [
*R*(*X, Y*)] = [*R*(_{h}*X*)] uniform in*Y*but with two different segments of uniform receptor expression in*X*as given byfor some positive constant Δ$$[{R}_{h}(X)]=\mathbf{\{}\begin{array}{cc}{\overline{R}}_{0}(1+\mathrm{\Delta})\hfill & (X<{X}_{h})\hfill \\ {\overline{R}}_{0}\hfill & \hfill (X>{X}_{h})\end{array},\mathrm{\Delta}0,$$(6)_{h} - a distribution [
*R*_{2}(*X*)] with three uniform segments of receptor expression in the*X*direction as given byfor some positive constants Δ$$[{R}_{h}(X)]=\mathbf{\{}\begin{array}{cc}{\overline{R}}_{0}(1+{\mathrm{\Delta}}_{\ell})\hfill & \hfill (X<{X}_{\ell})\\ {\overline{R}}_{0}\hfill & \hfill ({X}_{\ell}<X<{X}_{g})\\ {\overline{R}}_{0}(1+{\mathrm{\Delta}}_{g})\hfill & \hfill ({X}_{g}<X)\end{array}$$(7)and Δ_{}._{g}

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 ${\mathrm{\Omega}}_{\mathit{\text{XY}}}=\{(0,{X}_{\text{max}})\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}(-\frac{1}{4}{Y}_{\text{max}},\frac{1}{4}{Y}_{\text{max}})\}$ of the actual domain. Along the boundary Ω* _{XY}* of Ω

$$(X,Y)\in \partial {\mathrm{\Omega}}_{\mathit{\text{XY}}}:\frac{\partial [L]}{\partial n}=\frac{\partial [\mathit{\text{LS}}]}{\partial n}=\frac{\partial [S]}{\partial n}=0$$

(8)

for all *T* > 0 where [*G*]/*n* is the normal derivative of [*G*] and Ω* _{XY}* is the boundary of Ω

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

$$T=0:[L]=[\mathit{\text{LR}}]=[\mathit{\text{LS}}]=[S]=0,$$

(9)

for all (*X, Y*) in Ω* _{XY}*. The system (1) – (9) defines an initial-boundary value problem (IBVP) for the four unknown concentrations [

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

$$t=\frac{D}{{Y}_{\text{max}}^{2}}T,\{x,{x}_{h},{x}_{\ell},{x}_{g},{x}_{\text{max}},y\}=\left\{\frac{X}{{Y}_{\text{max}}},\frac{{X}_{h}}{{Y}_{\text{max}}},\frac{{X}_{\ell}}{{Y}_{\text{max}}},\frac{{X}_{g}}{{Y}_{\text{max}}},\frac{{X}_{\text{max}}}{{Y}_{\text{max}}},\frac{Y}{{Y}_{\text{max}}}\right\},$$

(10)

$$\{{f}_{L},{g}_{L},{h}_{L},{f}_{S},{g}_{S},{h}_{S}\}=\frac{{Y}_{\text{max}}^{2}}{D}\{{k}_{\mathit{\text{off}}},{k}_{\mathit{\text{deg}}},{k}_{\mathit{\text{on}}}{\overline{R}}_{0},{j}_{\mathit{\text{off}}},{j}_{\mathit{\text{deg}}},{j}_{\mathit{\text{on}}}{\overline{R}}_{0}\},$$

(11)

$$\{A,B,C,S,\rho ,{\rho}_{h}\}=\frac{1}{{\overline{R}}_{0}}\{[L],[\mathit{\text{LR}}],[\mathit{\text{LS}}],[S],[R],[{R}_{h}]\},$$

(12)

$$\{{\upsilon}_{L},{\upsilon}_{S},{\overline{\upsilon}}_{L},{\overline{\upsilon}}_{S}\}=\frac{{Y}_{\text{max}}^{2}}{D\phantom{\rule{thinmathspace}{0ex}}{\overline{R}}_{0}}\{{V}_{L},{V}_{S},{\overline{V}}_{L},{\overline{V}}_{S}\},\{{d}_{A},{d}_{C},{d}_{S}\}=\{\frac{{D}_{L}}{D},\frac{{D}_{\mathit{\text{LS}}}}{D},\frac{{D}_{S}}{D}\}$$

(13)

where *D* is the maximum of *D _{L}*,

$$\frac{\partial A}{\partial t}={d}_{A}{\nabla}^{2}A-{h}_{L}A(\rho -B)+{f}_{L}B-{h}_{S}\mathit{\text{AS}}+({f}_{S}+\tau {g}_{S})C+{\upsilon}_{L},$$

(14)

$$\frac{\partial B}{\partial t}={h}_{L}A(\rho -B)-({f}_{L}+{g}_{L})B,$$

(15)

$$\frac{\partial C}{\partial t}={d}_{C}{\nabla}^{2}C+{h}_{S}\mathit{\text{AS}}-({f}_{S}+{g}_{s})C,$$

(16)

$$\frac{\partial S}{\partial t}={d}_{S}{\nabla}^{2}S-{h}_{S}\mathit{\text{AS}}+{f}_{S}C+{\upsilon}_{S},$$

(17)

where now ^{2}( ) = ( ),* _{xx}* + ( ),

$$[R]={\overline{R}}_{0}\rho (x,y),$$

(18)

with

$$(i)\phantom{\rule{thinmathspace}{0ex}}{\rho}_{o}(x,y)=1,$$

(19)

$$(\mathit{\text{ii}})\phantom{\rule{thinmathspace}{0ex}}{\rho}_{h}(x,y)=\mathbf{\{}\begin{array}{cc}{\overline{\rho}}_{h}\equiv 1+\mathrm{\Delta}>1\hfill & \hfill (x<{x}_{h})\\ 1\hfill & \hfill (x>{x}_{h})\end{array},$$

(20)

$$(\mathit{\text{iii}}){\rho}_{2}(x,y)=\mathbf{\{}\begin{array}{cc}{\overline{\rho}}_{1}\equiv 1+{\mathrm{\Delta}}_{\ell}\hfill & \hfill (x{x}_{\ell})\\ 1\hfill & \hfill ({x}_{\ell}x{x}_{g})\\ {\overline{\rho}}_{2}\equiv 1+{\mathrm{\Delta}}_{g}\hfill & \hfill (x{x}_{g})\end{array}.$$

(21)

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

$$\mathbf{\left\{}\begin{array}{c}\hfill {\upsilon}_{L}(y)\hfill \\ \hfill {\upsilon}_{S}(y)\hfill \end{array}\mathbf{\right\}}=\mathbf{\left\{}\begin{array}{c}\hfill {\overline{\upsilon}}_{L}H(-y)\hfill \\ \hfill {\overline{\upsilon}}_{S}H(y)\hfill \end{array}\mathbf{\right\}}$$

(22)

where *H*(*z*) is the Heaviside step function.

The boundary conditions now take the form

$$(x,y)\in \partial \mathrm{\Omega}:\frac{\partial A}{\partial n}=\frac{\partial C}{\partial n}=\frac{\partial S}{\partial n}=0$$

(23)

for *t* > 0 where Ω is the boundary of Ω. The homogeneous initial conditions become

$$t=0:A=B=C=S=0,(x,y)\phantom{\rule{thinmathspace}{0ex}}\epsilon \phantom{\rule{thinmathspace}{0ex}}\mathrm{\Omega}\phantom{\rule{thinmathspace}{0ex}}.$$

(24)

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 ( )/*t* = 0 so that the governing partial differential equations and boundary conditions become

$${\nabla}^{2}A-{h}_{L}A(\rho -B)+{f}_{L}B-{h}_{S}\mathit{\text{AS}}+({f}_{S}+\tau {g}_{S})C+{\overline{\upsilon}}_{L}H(-y)=0,$$

(25)

$${h}_{L}A(\rho -B)-({f}_{L}+{g}_{L})B=0,$$

(26)

$${\nabla}^{2}C+{h}_{S}\mathit{\text{AS}}-({f}_{S}+{g}_{S})C=0,$$

(27)

$${\nabla}^{2}S-{h}_{S}\mathit{\text{AS}}+{f}_{S}C+{\overline{\upsilon}}_{S}H(y)=0,$$

(28)

where we have set *d _{A}* =

$$B(x,y)=\frac{\rho (x,y)A(x,y)}{{\alpha}_{L}+A(x,y)},{\alpha}_{L}=\frac{1}{{h}_{L}}({f}_{L}+{g}_{L}),$$

(29)

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

$${\nabla}^{2}A-\frac{\rho {g}_{L}A}{{\alpha}_{L}+A}-{h}_{S}\mathit{\text{AS}}+({f}_{S}+\tau {g}_{S})C+{\overline{\upsilon}}_{L}H(-y)=0.$$

(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.

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 = * _{L}*/

$$S={\overline{\upsilon}}_{S}S(x,y,\epsilon ),C=\frac{{\overline{\upsilon}}_{S}c(x,y,\epsilon )}{{f}_{S}+{g}_{S}},A=\frac{{\overline{\upsilon}}_{L}}{{\mu}_{L}^{2}}a={\alpha}_{L}{\beta}_{L}a(x,y,\epsilon ),$$

(31)

with

$$\epsilon =\frac{{\overline{\upsilon}}_{L}}{{\overline{\upsilon}}_{S}}=\frac{{\overline{V}}_{L}}{{\overline{V}}_{S}},{\upsilon}_{L}^{*}=\frac{{h}_{S}}{{h}_{L}}\frac{{\overline{\upsilon}}_{L}}{{\sigma}_{L}},{\beta}_{L}=\frac{{\overline{\upsilon}}_{L}}{{g}_{L}},$$

(32)

$${\sigma}_{L}=\frac{{g}_{L}}{{f}_{L}+{g}_{L}},{\mu}_{L}^{2}=\frac{{g}_{L}}{{\alpha}_{L}},{\sigma}_{S}=\frac{{g}_{S}}{{f}_{S}+{g}_{S}},{\alpha}_{S}=\frac{{g}_{S}+{f}_{S}}{{h}_{S}},$$

(33)

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

$$\epsilon \mathbf{\left\{}{\mu}_{L}^{-2}{\nabla}^{2}a-\frac{\rho a}{1+{\beta}_{L}a}+H(-y)\mathbf{\right\}}-({\upsilon}_{L}^{*}\mathit{\text{as}}-c)=0,$$

(34)

$${({f}_{S}+{g}_{S})}^{-1}{\nabla}^{2}c+({\upsilon}_{L}^{*}\mathit{\text{as}}-c)=0,$$

(35)

$${\nabla}^{2}s-({\upsilon}_{L}^{*}\mathit{\text{as}}-c)-{\sigma}_{S}c+H(y)=0,$$

(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

$$B(x,y)=\frac{\rho (x,y){\beta}_{L}a(x,y;\epsilon )}{1+{\beta}_{L}a(x,y;\epsilon )}\equiv b(x,y;\epsilon ).$$

(37)

For ε = * _{L}*/

$$\begin{array}{c}\hfill \{a(x,y;\epsilon ),b(x,y;\epsilon ),c(x,y;\epsilon ),s(x,y;\epsilon )\}\\ ={\displaystyle \sum _{n=0}^{\mathrm{\infty}}\{{a}_{n}(x,y),{b}_{n}(x,y),{c}_{n}(x,y),{s}_{n}(x,y)\}{\epsilon}^{n}.}\hfill \end{array}$$

(38)

The leading terms *a*_{0}(*x, y*), *b*_{0}(*x, y*), *c*_{0}(*x, y*), and *s*_{0}(*x, y*) correspond to the limiting case of ε = 0 (for * _{S}* = ∞). For this limiting case, equations (34)–(36) reduce to

$${\upsilon}_{L}^{*}{a}_{0}{s}_{0}-{c}_{0}=0,{\nabla}^{2}{c}_{0}=0,{\nabla}^{2}{s}_{0}-{\sigma}_{S}{c}_{0}+H(y)=0$$

(39)

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

The second equation in (39) is for *c*_{0}(*x, y*) alone. Together with the homogeneous Neumann condition along the edges of the rectangle, Ω, it requires

$${c}_{0}(x,y)={\overline{c}}_{0}.$$

(40)

To determine the constant _{0}, we integrate the last equation in (39) over the Ω and apply Green’s theorem. The Neumann condition on *s*_{0} along Ω and the result (40) are then used to give

$$\begin{array}{cc}0\hfill & ={\displaystyle \underset{\mathrm{\Omega}}{\iint}\{{\nabla}^{2}{s}_{0}-{\sigma}_{S}{c}_{0}+H(y)\}\mathit{\text{dxdy}}}\hfill \\ \hfill & =-{\sigma}_{S}{\overline{c}}_{0}{x}_{\text{max}}+\frac{1}{2}{x}_{\text{max}},\hfill \end{array}$$

$$c(x,y)~{c}_{0}(x,y)={\overline{c}}_{0}=\frac{1}{2{\sigma}_{S}}.$$

(41)

The value for _{0} in turn simplifies the last equation of (39) to

$${\nabla}^{2}{s}_{0}=\frac{1}{2}\{H(-y)-H(y)\}$$

(42)

with *s*_{0} required to satisfy the homogeneous Neumann condition along Ω. The solution of this BVP is

$${s}_{0}(x,y)={s}_{0}(y)=\mathbf{\{}\begin{array}{cc}{\overline{s}}_{0}+\frac{1}{8}(y+2{y}^{2})\hfill & (y<0)\hfill \\ {\overline{s}}_{0}+\frac{1}{8}(y-2{y}^{2})\hfill & (y>0)\hfill \end{array}$$

(43)

where _{0} is a constant of integration to be determined by the *O*(ε) problem. Note that *s*_{0} is (uniform in *x* and) continuously differentiable but has a simple jump discontinuity in ^{2}*s*_{0}/*y*^{2} across *y* = 0.

Except for the unknown constant _{0}, we have also *a*_{0}(*x, y*) = *a*_{0}(*y*) from the first equation in (44):

$$2{\sigma}_{S}{\upsilon}_{L}^{*}{a}_{0}(y;{\overline{s}}_{0})=2{\sigma}_{S}\frac{{c}_{0}}{{s}_{0}}=\mathbf{\{}\begin{array}{cc}{[{\overline{s}}_{0}+\frac{1}{8}(y+2{y}^{2})]}^{-1}\hfill & (y<0)\\ {[{\overline{s}}_{0}+\frac{1}{8}(y-2{y}^{2})]}^{-1}\hfill & (y>0)\hfill \end{array},$$

(44)

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

$${b}_{0}(x,y)=\frac{\rho (x,y){\beta}_{L}{a}_{0}(y;{\overline{s}}_{0})}{1+{\beta}_{L}{a}_{0}(y;{\overline{s}}_{0})},$$

(45)

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

To determine the unknown constant _{0}, we consider the *O*(ε) problem for *a*_{1}(*x, y*), …, *s*_{1}(*x, y*). The governing equations for these unknowns are

$${({f}_{S}+{g}_{S})}^{-1}{\nabla}^{2}{c}_{1}+[{\nu}_{L}^{*}({a}_{0}{s}_{1}+{a}_{1}{s}_{0})-{c}_{1}]=0,$$

(46)

$${\nabla}^{2}{S}_{1}-{\sigma}_{S}{c}_{1}-[{\nu}_{L}^{*}({a}_{0}{s}_{1}+{a}_{1}{s}_{0})-{c}_{1}]=0,$$

(47)

$${\mu}_{L}^{-2}{\nabla}^{2}{a}_{0}-\frac{\rho {a}_{0}}{1+{\beta}_{L}{a}_{0}}+H(-y)-[{\nu}_{L}^{*}({a}_{0}{s}_{1}+{a}_{1}{s}_{0})-{c}_{1}]=0,$$

(48)

with

$${b}_{1}(x,y)=\frac{\rho (x,y){\beta}_{L}{a}_{1}}{{(1+{\beta}_{L}{a}_{0})}^{2}}.$$

(49)

The unknowns *a*_{1}, *c*_{1} and *s*_{1} are subject to the homogeneous Neumann conditions (8) which also apply to the *O*(ε) terms of the problem.

We begin to determine _{0} by integrating (46) over Ω. Upon application of the two-dimensional divergence theorem and the homogeneous Neumann condition on *s*_{1}, we obtain

$$\int {\displaystyle \underset{\mathrm{\Omega}}{\int}[{\nu}_{L}^{*}({a}_{0}{s}_{1}+{a}_{1}{s}_{0})-{c}_{1}]\mathit{\text{dxdy}}}}=0.$$

(50)

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

$$\int {\displaystyle \underset{\Omega}{\int}\mathbf{\left\{}{\mu}_{L}^{-2}{\nabla}^{2}{a}_{0}-\frac{\rho (x,y){a}_{0}}{1+{\beta}_{L}{a}_{0}}+H(-y)\mathbf{\right\}}\mathit{\text{dxdy}}}}=0$$

or, upon application of the two-dimensional divergence theorem and the homogeneous Neumann condition on *a*_{0},

$$J({\overline{s}}_{0})\equiv {\displaystyle \int {\displaystyle \underset{\mathrm{\Omega}}{\int}\frac{\rho (x,y){a}_{0}(y;{\overline{s}}_{0})}{1+{\beta}_{L}{a}_{0}(y;{\overline{s}}_{0})}\mathit{\text{dxdy}}}}=\frac{{x}_{\text{max}}}{4}.$$

(51)

For any prescribed distribution of *tkv* concentration ρ(*x, y*), this is a condition on *a*_{0}(*y*; _{0}) alone and thus determining _{0} in view of (44).

We may continue the solution process to solve (46) – (48) and the corresponding homogeneous Neumann conditions to determine *c*_{1}, *s*_{1} and *a*_{1}. 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:

*For ε = _{L}/_{S} 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 _{0} 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.

Whenever an outer solution for ε = * _{L}*/

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*] ≤ _{0}, 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*] > _{0} 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 * _{L}* is high for the prescribed

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.

It was pointed out in the previous two sections that for ε = * _{L}*/

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*) = ρ_{o}(*x, y*) 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))

$$[\mathit{\text{LR}}]=\frac{R(X,Y)[L]}{{\overline{R}}_{0}{\alpha}_{L}+[L]}.$$

(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].

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*(_{0}) in (51) and their consequences, focusing on the receptor distributions *R*(*x, y*) = _{0} (so that ρ(*x, y*) = ρ_{o}(*x, y*) 1) and *R*(*x, y*) = *R _{h}*(

*J* (* _{0}*)

For any fixed *y*, we have from the explicit solution for *a*_{0}(*y*) in (44)

$$\frac{\partial {a}_{0}(y;{\overline{s}}_{0})}{\partial {\overline{s}}_{0}}=\mathbf{\{}\begin{array}{cc}-{\overline{c}}_{0}{[{\overline{s}}_{0}+\frac{1}{8}(y+2{y}^{2})]}^{-2}\hfill & (y<0)\hfill \\ -{\overline{c}}_{0}{[{\overline{s}}_{0}+\frac{1}{8}(y-2{y}^{2})]}^{-2}\hfill & (y>0)\hfill \end{array}$$

so that *a*_{0}(*y*; _{0})/_{0} < 0. Since ρ(*x*) is positive, this implies

$$\frac{\mathit{\text{dJ}}({\overline{s}}_{0})}{d{\overline{s}}_{0}}={\displaystyle \int {\displaystyle \underset{\mathrm{\Omega}}{\int}\frac{\rho (x,y)}{{[1+{\beta}_{L}{a}_{0}(y;{\overline{s}}_{0})]}^{2}}\frac{\partial {a}_{0}(y;{\overline{s}}_{0})}{\partial {\overline{s}}_{0}}\mathit{\text{dxdy}}}}<0.$$

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

*J*(_{0}) = *x _{max}*/4

We now examine the root _{0} = ζ of *J*(_{0}) = *x _{max}*/4 for several distributions of the fixed receptor concentrations uniformly in the dorsal-ventral direction:

For this case, the expression for *J*(_{0}) involves only integration of a one-variable function:

$$\begin{array}{cc}{J}_{o}({\overline{s}}_{0})\hfill & \equiv {[J({\overline{s}}_{0})]}_{\rho ={\rho}_{o}=1}={\displaystyle {\int}_{0}^{{x}_{\text{max}}}{\displaystyle {\int}_{-\frac{1}{4}}^{\frac{1}{4}}\left[\frac{{a}_{0}(y;{\overline{s}}_{0})}{1+{\beta}_{L}{a}_{0}(y;{\overline{s}}_{0})}\right]\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{dxdy}}}}\hfill \\ \hfill & \equiv {\displaystyle {\int}_{0}^{{x}_{\text{max}}}\phantom{\rule{thinmathspace}{0ex}}[{I}_{o}({\overline{s}}_{0})]\mathit{\text{dx}}={x}_{\text{max}}\phantom{\rule{thinmathspace}{0ex}}[{I}_{o}({\overline{s}}_{0})]},\hfill \end{array}$$

(53)

where

$${I}_{o}(z)={\displaystyle {\int}_{-\frac{1}{4}}^{\frac{1}{4}}\left[\frac{{a}_{0}(y;z)}{1+{\beta}_{L}{a}_{0}(y;z)}\right]\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{dy}}}.$$

(54)

Thus, with ρ = ρ_{o} = 1, the condition (51) determines _{0} to be ς_{0} with

$$\frac{1}{{x}_{\text{max}}}{[{J}_{o}({\overline{s}}_{0})]}_{{\overline{s}}_{0}={\varsigma}_{0}}={I}_{o}({\varsigma}_{o})=\frac{1}{4}.$$

(55)

The expression (44) for *a*_{0}(*y*; _{0}) can be used to re-write the integral in (54) as

$$\begin{array}{cc}{J}_{o}({\varsigma}_{0})\hfill & =\frac{{x}_{\text{max}}}{2{\sigma}_{s}{\upsilon}_{L}^{*}}\left[{\displaystyle {\int}_{-\frac{1}{4}}^{0}\frac{\mathit{\text{dy}}}{\eta +{\varsigma}_{0}+\frac{1}{8}(y+2{y}^{2})}}+{\displaystyle {\int}_{0}^{\frac{1}{4}}\frac{\mathit{\text{dy}}}{\eta +{\varsigma}_{0}+\frac{1}{8}(y-2{y}^{2})}}\right]\hfill \\ \hfill & \equiv \frac{{x}_{\text{max}}}{2{\sigma}_{s}{\upsilon}_{L}^{*}}[{I}_{m}({\varsigma}_{0})+{I}_{p}({\varsigma}_{0})\equiv {x}_{\text{max}}[{I}_{o}({\varsigma}_{0})],\hfill \end{array}$$

(56)

and

$$\eta =\frac{{\beta}_{L}}{2{\sigma}_{s}{\upsilon}_{L}^{*}}=\frac{1}{2{g}_{L}}\frac{{\sigma}_{L}{h}_{L}}{{\sigma}_{s}{h}_{S}}=\frac{1}{2{g}_{S}}\frac{{\alpha}_{S}}{{\alpha}_{L}}.$$

(57)

The two integrals *I _{m}*(ς

$$\frac{16}{{B}_{m}({\varsigma}_{0})}{\text{tan}}^{-1}\left(\frac{1}{{B}_{m}({\varsigma}_{0})}\right)+\frac{16}{{B}_{p}({\varsigma}_{0})}{\text{tanh}}^{-1}\left(\frac{1}{{B}_{p}({\varsigma}_{0})}\right)=\frac{{\sigma}_{s}{\upsilon}_{L}^{*}}{2}$$

(58)

where

$${B}_{m}(\xi )=8\sqrt{\eta +\xi -\frac{1}{64}},{B}_{p}(\xi )=8\sqrt{\eta +\xi +\frac{1}{64}}.$$

(59)

It is an immediate consequence of Lemma 2 that

*I _{o}* (

With ρ* _{h}*(

$$\begin{array}{cc}{J}_{h}({\overline{s}}_{0})\hfill & \equiv {[J({\overline{s}}_{0})]}_{\rho ={\rho}_{h}(x)}={\displaystyle {\int}_{0}^{{x}_{\text{max}}}{\displaystyle {\int}_{-\frac{1}{4}}^{\frac{1}{4}}\left[\frac{{\rho}_{h}(x){a}_{0}(y;{\overline{s}}_{0})}{1+{\beta}_{L}{a}_{0}(y;{\overline{s}}_{0})}\right]\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{dxdy}}}}\hfill \\ & \equiv {\displaystyle {\int}_{0}^{{x}_{\text{max}}}[{\rho}_{h}(x){I}_{o}({\overline{s}}_{0})]\mathit{\text{dx}}}=({x}_{h}\mathrm{\Delta}+{x}_{\text{max}})[{I}_{o}({\overline{s}}_{0})].\hfill \end{array}$$

(60)

The condition (51) determines _{0} to be ς* _{h}* with

$$\frac{1}{{x}_{\text{max}}}{[{J}_{o}({\overline{s}}_{0})]}_{{\overline{s}}_{0}={\varsigma}_{h}}=(1+{\delta}_{h}\mathrm{\Delta}){I}_{o}({\varsigma}_{h})=\frac{1}{4},$$

(61)

where δ* _{h}* =

$$\frac{16}{{B}_{m}({\varsigma}_{h})}{\text{tan}}^{-1}\left(\frac{1}{{B}_{m}({\varsigma}_{h})}\right)+\frac{16}{{B}_{p}({\varsigma}_{h})}{\text{tanh}}^{-1}\left(\frac{1}{{B}_{p}({\varsigma}_{h})}\right)=\frac{{\sigma}_{s}{\upsilon}_{L}^{*}}{2(1+{\delta}_{h}\mathrm{\Delta})}.$$

(62)

ς* _{h}* [

The claim follows immediately from ${I}_{o}({\varsigma}_{0})=\frac{1}{4}>1/4(1+{\delta}_{h}\u25b3)={I}_{o}({\varsigma}_{h})$ and Lemma 4 for any positive *X _{h}* so that δ

For a piecewise continuous function of *x* alone, ρ(*x, y*) = ρ* _{vd}*(

$$\begin{array}{cc}{[J({\overline{s}}_{0})]}_{\rho ={\rho}_{\upsilon d}(x)}\hfill & \equiv {J}_{\upsilon d}({\overline{s}}_{0})={\displaystyle {\int}_{0}^{{x}_{\text{max}}}{\displaystyle {\int}_{-\frac{1}{4}}^{\frac{1}{4}}\left[\frac{{\rho}_{\upsilon d}(x){a}_{0}(y;{\overline{s}}_{0})}{1+{\beta}_{L}{a}_{0}(y;{\overline{s}}_{0})}\right]\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{dxdy}}}}\hfill \\ \hfill & ={\overline{\rho}}_{\upsilon d}{x}_{\text{max}}{\displaystyle {\int}_{-\frac{1}{4}}^{\frac{1}{4}}\left[\frac{{a}_{0}(y;{\overline{s}}_{0})}{1+{\beta}_{L}{a}_{0}(y;{\overline{s}}_{0})}\right]\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{dy}}}.\hfill \end{array}$$

(64)

With *a*_{0}(*y*; _{0}) given by (44), the integral remaining in (64) can again be evaluated exactly with

$$\begin{array}{cc}{J}_{\upsilon d}({\overline{s}}_{0})& =\frac{{\overline{\rho}}_{\upsilon d}{x}_{\text{max}}}{2{\sigma}_{s}{\upsilon}_{L}^{*}}\left[{\displaystyle {\int}_{-\frac{1}{4}}^{0}\frac{\mathit{\text{dy}}}{{\tilde{s}}_{0}+\frac{1}{8}(y+2{y}^{2})}}+{\displaystyle {\int}_{0}^{\frac{1}{4}}\frac{\mathit{\text{dy}}}{{\tilde{s}}_{0}+\frac{1}{8}(y-2{y}^{2})}}\right]\hfill \\ \hfill & \equiv \frac{{\overline{\rho}}_{\upsilon d}{x}_{\text{max}}}{2{\sigma}_{s}{\upsilon}_{L}^{*}}[{I}_{m}({\overline{s}}_{0})+{I}_{p}({\overline{s}}_{0})]={x}_{\text{max}}{\overline{\rho}}_{\upsilon d}{I}_{o}({\overline{s}}_{0})\hfill \end{array}$$

(65)

where

$${\overline{\rho}}_{\upsilon d}{x}_{\text{max}}={\displaystyle {\int}_{0}^{{x}_{\text{max}}}{\rho}_{\upsilon d}(x)\mathit{\text{dx}}}.{\tilde{s}}_{0}={\overline{s}}_{0}+\frac{{\beta}_{L}}{2{\sigma}_{S}{\upsilon}_{L}^{*}}\equiv {\overline{s}}_{0}+\eta $$

(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:

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

$$[{R}_{A}(X,Y)]=\mathbf{\{}\begin{array}{ccc}{\overline{R}}_{0}\{(1+\mathrm{\Delta})H(-y)+{\mathrm{\Delta}}_{\ell}H(y)\}\hfill & \hfill & (X<{X}_{h})\hfill \\ {\overline{R}}_{0}\{1+{\mathrm{\Delta}}_{g}\}\hfill & \hfill & (X>{X}_{h})\hfill \end{array},$$

(67)

or

$${\rho}_{A}(x,y)]=\frac{1}{{\overline{R}}_{0}}[{R}_{A}(X,Y)]=\mathbf{\{}\begin{array}{ccc}(1+\mathrm{\Delta})H(-y)+{\mathrm{\Delta}}_{\ell}H(y)\hfill & \hfill & (x<{x}_{h})\hfill \\ 1+{\mathrm{\Delta}}_{g}\hfill & \hfill & (x>{x}_{h})\hfill \end{array},$$

(68)

where Δ > 0, Δ_{g} ≥ 0 and 0 ≤ Δ_{} ≤ 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*(_{0}) as defined in (51) continues to be a monotone decreasing function of _{0} and *J*(_{0}) = *x _{max}*/4 has exactly one root in (0, ∞) if

From (37), we have

$$\begin{array}{cc}\frac{[\mathit{\text{LR}}]}{{\overline{R}}_{0}}\hfill & =B(x,y)=b(x,y;\epsilon )=\frac{\rho (x,y){\beta}_{L}a(x,y)}{1+{\beta}_{L}a(x,y)}\hfill \\ \hfill & ~\frac{\rho (x,y){\beta}_{L}{a}_{0}(y;{\overline{s}}_{0})}{1+{\beta}_{L}{a}_{0}(y;{\overline{s}}_{0})}=\frac{\eta \rho (x,y)}{\eta +{s}_{0}(y;{\overline{s}}_{0})}\hfill \end{array}$$

(69)

where *s*_{0}(*y*; _{0}) and *η* are given by (43) and (57), respectively.

- ρ(
*x, y*) = 1: We have for this case_{0}= ς_{0}and therewith$$\begin{array}{cc}\frac{1}{\eta {\overline{R}}_{0}}{[\mathit{\text{LR}}]}_{\rho (x)=1}\hfill & =\frac{1}{\eta}{[b(x,y;\epsilon )]}_{\rho (x)=1}~{\left[\frac{1}{\eta +{s}_{0}(y;{\overline{s}}_{0})}\right]}_{{\overline{s}}_{0}={\varsigma}_{0}}\hfill \\ \hfill & =\mathbf{\{}\begin{array}{cc}{[\eta +{\varsigma}_{0}+\frac{1}{8}(y+2{y}^{2})]}^{-1}\hfill & y<0\hfill \\ {[\eta +{\varsigma}_{0}+\frac{1}{8}(y-2{y}^{2})]}^{-1}\hfill & y>0\hfill \end{array}\hfill & .\end{array}$$(70) - ρ(
*x, y*) = ρ(_{h}*x*): For this case, we have similarly_{0}= ςand therewith_{h}$$\frac{1}{\eta \overline{R}}{[\mathit{\text{LR}}]}_{\rho (x)={\rho}_{h}(x)}=\frac{1}{\eta}{[b(x,y;\epsilon )]}_{\rho (x)={\rho}_{h}(x)}~{\left[\frac{{\rho}_{h}(x)}{\eta +{s}_{0}(y;{\overline{s}}_{0})}\right]}_{{\overline{s}}_{0}={\varsigma}_{h}}.$$(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 *x _{h}* <

$$\begin{array}{cc}\frac{1}{\eta \overline{R}}{[\mathit{\text{LR}}]}_{{\rho}_{h}(x)}\hfill & ~{\left[\frac{1}{\eta +{s}_{0}(y;{\overline{s}}_{0})}\right]}_{{\overline{s}}_{0}={\varsigma}_{h}}\hfill \\ \hfill & =\mathbf{\{}\begin{array}{c}{[\eta +{\varsigma}_{h}+\frac{1}{8}(y+2{y}^{2})]}^{-1}(y0)\hfill \\ {[\eta +{\varsigma}_{h}+\frac{1}{8}(y-2{y}^{2})]}^{-1}(y0)\hfill \end{array}.\hfill \end{array}$$

(72)

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

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.

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* < *x _{h}*

$$\begin{array}{cc}\frac{1}{\eta \overline{R}}{[\mathit{\text{LR}}]}_{{\rho}_{h}(x)}\hfill & ~{\left[\frac{1+\mathrm{\Delta}}{\eta +{s}_{0}(y;{\overline{s}}_{0})}\right]}_{\phantom{\rule{thinmathspace}{0ex}}{\overline{s}}_{0}={\varsigma}_{h}}\hfill \\ \hfill & =\mathbf{\{}\begin{array}{c}(1+\mathrm{\Delta})/[\eta +{\varsigma}_{h}+\frac{1}{8}(y+2{y}^{2})](y0)\hfill \\ (1+\mathrm{\Delta})/[\eta +{\varsigma}_{h}+\frac{1}{8}(y-2{y}^{2})](y0)\hfill \end{array}.\hfill \end{array}$$

(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 _{0} in the denominator. In particular, we have

*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*

$$1+\mathrm{\Delta}>\frac{\eta +{\varsigma}_{h}-\frac{1}{64}}{\eta +{\varsigma}_{0}-\frac{1}{64}}$$

(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.*

For the asymptotic solution to be applicable, we must have ε = * _{L}*/

Distribution of [*LR*] in the anterior-posterior direction at the dorsal midline for the fixed receptor expression *R*_{h}(X) (with ectopic expression in 0 ≤ *X* < *X*_{h} = 0.02 *cm*) for _{L} = 2.5 × 10^{−4}µ*M/s* and **...**

*For these parameter values, we have α _{L} = (g_{L} + f_{L})/h_{L} 4.367 × 10^{−4},* ${\mu}_{L}^{2}={g}_{L}/{\alpha}_{L}\simeq {h}_{L}=4.27\times {10}^{3}$

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* = −*Y*_{max}/4 and decreases monotonically in both directions toward its minimum at *Y* = *Y*_{max}/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* = −*Y*_{max}/4. Shown in Figure 4 is a two-dimensional plot of the steady state signaling gradient [*LR*] for the case *R*(*X, Y*) = *R*_{h}(*X*) with * _{L}* = 0.25 × 10

Two-dimensional plot of the [*LR*] distribution for the fixed receptor expression *R*_{h}(X) (with ectopic expression in 0 ≤ *X* < *X*_{h} = 0.02 *cm*) for _{L} = 2.5 × 10^{−4}µ*M/s* and _{S} = 0.6µ **...**

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 * _{L}* = 0.25 × 10

In Table (2), the signaling morphogen gradient [*LR*] at the dorsal midline are given for * _{L}* = 0.25 × 10

The even rows in Table (2) with δ* _{h}* = 4/11 give the corresponding results for a localized elevated receptor expression R

In the region with elevated receptor expression, 0 ≤ *X* < *X _{h}*, the situation is more complicated. The outer asymptotic steady state solution for the receptor distribution

For * _{S}* = 0.08 µ

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 *X _{h}* of

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 * _{S}* = 0.08 µ

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 < |*Y* − *Y _{max}*/4| <

Distribution of [*LR*] in the anterior-posterior direction at the dorsal midline for the fixed receptor expression *R*_{h}(X) (with ectopic expression in 0 ≤ *X* < *X*_{h} = 0.02 *cm*) for _{L} = 10^{−3}µ*M/s* and **...**

At the lower *Sog* synthesis rate of * _{S}* = 10

While the results for * _{L}* = 10

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 * _{L}* that do not saturate the fixed receptor concentration

On the other hand, for relatively low *Sog* synthesis rates such as * _{S}* = 0.08 µ

For the same rate parameter values but higher *Dpp* synthesis rates at the level of * _{L}* = 10

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:

$$\rho (x,y)={\rho}_{A}(x)=\mathbf{\{}\begin{array}{cc}1+{\mathrm{\Delta}}_{\ell}>1\hfill & (0\le x<{x}_{\ell})\hfill \\ 1\hfill & ({x}_{\ell}<x<{x}_{g})\hfill \\ 1+{\mathrm{\Delta}}_{g}>1\hfill & ({x}_{g}<x\le {x}_{\mathit{\text{max}}})\hfill \end{array}.$$

(75)

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

$$\frac{1}{{x}_{\text{max}}}{J}_{A}({\overline{s}}_{0})\equiv {[\{1+{\delta}_{\ell}{\mathrm{\Delta}}_{\ell}+(1-{\delta}_{g}){\mathrm{\Delta}}_{g}\}{I}_{o}({\overline{s}}_{0})]}_{{\overline{s}}_{0}={\varsigma}_{A}},{\delta}_{k}=\frac{{x}_{k}}{{x}_{\text{max}}}1$$

(76)

where ς* _{A}* is the solution of

$${I}_{o}({\varsigma}_{A})=\frac{\frac{1}{4}}{1+{\mathrm{\Delta}}_{\ell}{\delta}_{\ell}+{\mathrm{\Delta}}_{g}(1-{\delta}_{g})},$$

(77)

or

$$\begin{array}{c}\hfill \frac{16}{{B}_{m}({\zeta}_{A})}{\text{tan}}^{-1}\left(\frac{1}{{B}_{m}({\zeta}_{A})}\right)+\frac{16}{{B}_{p}({\zeta}_{A})}{\text{tanh}}^{-1}\left(\frac{1}{{B}_{p}({\zeta}_{A})}\right)\\ =\frac{{\sigma}_{s}{\upsilon}_{L}^{*}}{2[1+{\mathrm{\Delta}}_{\ell}{\delta}_{\ell}+{\mathrm{\Delta}}_{g}(1-{\delta}_{g})]}\hfill \end{array}$$

(78)

with *B _{m}* and

ς* _{A}* [

This follows from the fact that the right hand side of (77) is less than $\frac{1}{2}$ and *I _{o}* is a monotone decreasing function of

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*_{} < *x* < *x _{g}* where the latter is not over-expressed.

For the uniformly distributed case, we have [*LR*]_{ρ(x)=1}/*η _{0}* is given by (70) as before. For ρ

$$\begin{array}{cc}\frac{1}{\eta {\overline{R}}_{0}}{[\mathit{\text{LR}}]}_{{\rho}_{A}(x)}\hfill & ~{\left[\frac{1}{\eta +{s}_{0}(y;{\overline{s}}_{0})}\right]}_{\phantom{\rule{thinmathspace}{0ex}}{\overline{s}}_{0}={\varsigma}_{A}}\hfill \\ \hfill & =\mathbf{\{}\begin{array}{cc}{[\eta +{\varsigma}_{A}+\frac{1}{8}(y+2{y}^{2})]}^{-1}& (y<0)\\ {[\eta +{\varsigma}_{A}+\frac{1}{8}(y-2{y}^{2})]}^{-1}& (y>0)\end{array}\hfill \\ \hfill & <{\left[\frac{1}{\eta +{s}_{0}(y;{\overline{s}}_{0})}\right]}_{\phantom{\rule{thinmathspace}{0ex}}{\overline{s}}_{0}={\varsigma}_{0}}~\frac{1}{\eta {\overline{R}}_{0}}{[\mathit{\text{LR}}]}_{\rho (x)=1}\hfill \end{array}$$

(79)

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

*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 _{} < X < X_{g} 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*_{}, we have

$$\begin{array}{cc}\frac{1}{\eta {\overline{R}}_{0}}{[\mathit{\text{LR}}]}_{{\rho}_{A}(x)}\hfill & ~{\left[\frac{1+{\mathrm{\Delta}}_{\ell}}{\eta +{s}_{0}(y;{\overline{s}}_{0})}\right]}_{\phantom{\rule{thinmathspace}{0ex}}{\overline{s}}_{0}={\varsigma}_{A}}\hfill \\ \hfill & =\mathbf{\{}\begin{array}{cc}(1+{\mathrm{\Delta}}_{\ell})/[\eta +{\varsigma}_{A}+\frac{1}{8}(y+2{y}^{2})]\hfill & (y<0)\hfill \\ (1+{\mathrm{\Delta}}_{\ell})/[\eta +{\varsigma}_{A}+\frac{1}{8}(y-2{y}^{2})]& (y>0)\hfill \end{array}.\hfill \end{array}$$

(80)

The comparison with the corresponding concentration for a uniform receptor distribution now depends on the magnitude of Δ_{}. In particular, we have

*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*

$$1+{\mathrm{\Delta}}_{\ell}>\frac{\eta +{\varsigma}_{A}-\frac{1}{16}}{\eta +{\varsigma}_{0}-\frac{1}{16}}$$

(81)

*elevates PMad activation in the interval 0 < x < x _{} 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.*

To provide numerical evidence in support of a depressed signaling gradient expression for an ectopic receptor distribution of the type *R _{A}(X)* defined in (7), we consider such a receptor distribution with Δ

Two-dimensional plot of the [*LR*] distribution for the fixed receptor expression *R*_{A}(X) (with the same ectopic expression in 0 ≤ *X* < *X*_{} = 0.02 *cm* and *X*_{g} = 0.035 *cm* < *X* ≤ *X*_{max}) for _{L} = 2.5 × **...**

From the numerical results for the case * _{L}* = 0.25 × 10

Distribution of [*LR*] in the anterior-posterior direction at the dorsal midline for the fixed receptor expression *R*_{A}(X) (with the same ectopic expression in 0 ≤ *X* < *X*_{} = 0.02 *cm* and *X*_{g} = 0.035 *cm* < *X* ≤ *X*_{max}) for **...**

For the higher *Dpp* synthesis rate * _{L}* = 10

For the biologically less realistic case of * _{S}*/

$${\mu}_{L}^{-2}{\nabla}^{2}{a}^{0}-\frac{\rho {a}^{0}}{1+{\beta}_{L}{a}^{0}}+H(-y)=0,$$

(82)

$${({f}_{S}+{g}_{S})}^{-1}{\nabla}^{2}{c}^{0}+({\nu}_{L}^{*}{a}^{0}{s}^{0}-{c}^{0})=0,$$

(83)

$${\nabla}^{2}{s}^{0}-({\nu}_{L}^{*}{a}^{0}{s}^{0}-{c}^{0})-{\sigma}_{S}{c}^{0}+H(y)=0.$$

(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*):

$${b}^{0}(x,y)=\frac{\rho (x){\beta}_{L}{a}^{0}(x,y)}{1+{\beta}_{L}{a}^{0}(x,y)}.$$

(85)

More importantly, (82) is an equation for *a*^{0}(*x, y*) alone and, augmented by the homogeneous Neumann condition along the edges of the rectangular domain, may be solved separately.

*A unique nonnegative solution of the BVP for a ^{0} exists with*

$$0\le {a}^{0}(x,y)\le {a}_{u}$$

(86)

*where the constant a _{u} is given by*

$${a}_{u}=\frac{1}{1-{\beta}_{L}}.$$

Evidently, *a*_{} = 0 is a lower solution for the problem. On the other hand, we have

$$\begin{array}{c}\hfill -{\mu}_{L}^{-2}{\nabla}^{0}{a}_{u}+\frac{\rho (x){a}_{u}}{1+{\beta}_{L}{a}_{u}}-H(-y)\hfill \\ =\frac{[\rho (x)-1]{a}_{u}}{1+{\beta}_{L}{a}_{u}}+1-H(-y)=\rho (x)-1+H(y)>0\hfill \end{array}$$

given ρ ≥ 1. With *a _{u}*/

Having the piecewise *C*^{2} solution for *a*^{0}(* ^{x, y}*), equations (83) and (84) together with the relevant homogeneous Neumann conditions may be solved for

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= 10_{L}^{−3}µ*M/s*and= 0.6µ_{S}*M/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= 10_{S}^{−3}µ*M/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 asincreases.)_{L} - 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*X*) as shown in Table (2) for_{h}= 10_{S}^{−3}µ*M/s*

It is also seen from Table (2) (and the graphs for [*LR*] for the case of * _{L}* = 2.5 × 10

- A very low
*Sog*synthesis ratethat transports to the nearby anterior region only the_{S}*Dpp*and*Dpp-Sog*in a narrow layer adjacent to the location*X*of the discontinuity in_{h}*R*_{h}(X) - Intermediate Sog synthesis rates that manage to shuttle sufficient
*Dpp*and*Dpp-Sog*away from a finite interval (*X*,_{h}*X*) in the posterior region to the anterior region with_{g}*X*substantially smaller than_{g}*X*._{max}

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*R*(_{h}*X*) and (81) for*R*) required for an elevated [_{A}(X)*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.

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.

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