Geometry and topology of parameter space: investigating measures of robustness in regulatory networks

Abstract

The concept of robustness of regulatory networks has been closely related to the nature of the interactions among genes, and the capability of pattern maintenance or reproducibility. Defining this robustness property is a challenging task, but mathematical models have often associated it to the volume of the space of admissible parameters. Not only the volume of the space but also its topology and geometry contain information on essential aspects of the network, including feasible pathways, switching between two parallel pathways or distinct/disconnected active regions of parameters. A method is presented here to characterize the space of admissible parameters, by writing it as a semi-algebraic set, and then theoretically analyzing its topology and geometry, as well as volume. This method provides a more objective and complete measure of the robustness of a developmental module. As a detailed case study, the segment polarity gene network is analyzed.

1 Introduction

For biological networks, the concept of robustness often expresses the idea that the system's regulatory functions should operate correctly under a variety of situations. The network should respond appropriately to various stimulii and recognize meaningful ones (either harmful or favorable), but it should also ignore small (not meaningful) variations in the environment as well as inescapable fluctuations in the abundances of biomolecules involved in the network [1–3].

While it is difficult to define this robustness property in a precise form, it has been associated to the space of admissible kinetic parameters, its volume [3], and the effect of paramater perturbations on the qualitative behavior of the system [1,2,4]. Some methods for parameter sensitivity have been developed [5,6], based essentially on derivatives of variables or fluxes with respect to the system's parameters. The volume of the parameter space can be used as an indication of “how many” parameter combinations are possible, and these are related to the ability of the network to work under a variety of situations. For instance, parameters may range through different orders of magnitude, representing very different environments. A small parameter space volume is a clear indication of low robustness, as the model will require precise tuning to reproduce any features. Hence robustness is associated to larger volumes. However, size may not always be a reliable measure for robustness; other quantities, such as shape, also play a very significant role, as illustrated in Fig. 1. In the context of systems with uncertain parameters, for instance, it is quite useful to have an idea of the distribution, or shape, of the sets of good or bad parameters. In [4] statistical analysis of a chemotaxis network indicates that there are two regions of the uncertain parameter space with a high concentration of bad parameters (thus suggesting a feasible parameter space of the form Fig. 1d). Analysis of the shape or geometry of the admissible parameter set gives an indication not only of its size, but also how far perturbations around each parameter disrupt the network. A robust biological network will admit small fluctuations in its parameters without changing its qualitative behavior. So, a robust network will be associated to a system whose parameter set has few “narrow pieces” and “sharp corners”. In such sets, reasonable parameter fluctuations may occur without leaving the set, hence maintaining the network's qualitative behavior (compare Fig. 1a, b). The topology, and in particular the simple-connectedness or not of the feasible parameter set is also important (compare Fig. 1c, d). However, even if the set is connected, it may exhibit low robustness if it is composed of several pieces with lower dimension connecting faces, as in Fig. 1e. In fact, as will be seen later, this is one of the situations that happen in our example.

Fig. 1.

The role of geometry and topology in robustness. Regions a and b have the same volume, but b is less robust: the same perturbation leads out of the space. Regions c and d also have the same volume, but d is not a simply connected set, hence less robust. Region e, although connected, is composed of two pieces that touch only along a small face

To illustrate the importance of parameter space geometry, and the insight it brings to understanding the network, the model of the segment polarity network developed by von Dassow and collaborators [3] will be analyzed as a “case study” of our approach. The segment polarity network is part of a cascade of gene families responsible for generating the segmentation of the fruit fly embryo [7]. Genes in earlier stages are transiently expressed, but the segment polarity genes maintain a stable pattern for about three hours. It has been suggested that the segment polarity genes constitute a robust developmental module, capable of autonomously reproducing the same behavior or generating the same gene expression pattern, in response to transient inputs [3,8,9]. This robustness would be due to the nature of interactions among genes, rather than the kinetic parameters of the reactions. The model [3] describes the interactions among the principal segment polarity genes, is continuous, and involves cell-to-cell communications and around 50 parameters which are essentially unknown. The authors of [3] explored the model by randomly choosing 240,000 parameter sets out of which about 1,192 (or 0.5%) sets were consistent with the generation (at steady state) of the wild type pattern. To explore the robustness of the network as a property of its interactions, Albert and Othmer [9,10] developed a Boolean model of the segment polarity network, a discrete logical model where each species has only two states (0 or 1; “OFF” or “ON”), but no kinetic parameters need to be defined. This Boolean model is amenable to various methods for systematic robustness analysis [11–13]. Ingolia [8] focused on the properties of the (slightly changed) model [3] in individual cells, such as bistability, and extrapolated necessary conditions on parameters to the full intercellular model.

We propose a different approach, that retains the information contained on the kinetic parameters, but partially approximates the model by a logical form with various possible ON levels and species-dependent activation parameters. The admissible set of parameters of the model [3] is analyzed by constructing a cylindrical algebraic decomposition. Among other conclusions, our analysis completely explains the two “missing links” in von Dassow et al. original model, namely: why the segment polarity pattern can not be recovered without the negative regulation of engrailed by Cubitus repressor protein, and why the autocatalytic wingless activation pathway vastly increases the network robustness.

The present approach shows that, in contrast to volume only estimates, the topology and geometry of parameter sets provide reliable quantitative measures of robustness of a system. Some of our work may be seen as a “global” counterpart to the local analysis done in [14] in which eigenvector analysis was used to study the “stiff/sloppy” character of good parameter sets. In separate work [15], we study evolutionary implications of the geometric structure, with a focus on a measure of robustness that is related to having low rate of exit from the region under random walk [16].

2 The segment polarity network model

The principal segment polarity genes [7] are engrailed (en), wingless, patched, cubitus interruptus and hedgehog. The wild type expression pattern for these five segment polarity genes is experimentally well characterized, and is stably maintained for a period of about three hours (approximately during stages 8–11 of embryonic development) [17]. The pattern is periodically repeated (every four cells, in the early stages 6–8 of embryonic development), and defines the positions of the parasegments in the embryo of the fly. In wild type, both engrailed and hedgehog are expressed in every third cell [18], while wingless is expressed in every cell anterior to an en-expressing cell [18]. Further experimental observations show that cubitus is expressed in all but the cell expressing en [19], and patched is strongly expressed in every cell surrounding en-expressing cells [18], but not expressed in en-expressing cells. The boundaries of the parasegments are formed between the two cells expressing wingless and engrailed.

The model proposed by von Dassow et al. [3] (Appendix B) describes the concentrations of these five mRNAs and corresponding proteins in a four cell parasegment of the fly embryo, subject to periodic boundary conditions (see also Fig. 6). From now on, each cell is assumed to have a square shape, with four faces (see Appendix E). Nine of these mRNAs and proteins are considered to have a homogeneous concentration throughout each cell: engrailed mRNA and protein (en and EN), wingless mRNA and (internal) protein (wg and IWG), patched mRNA (ptc), cubitus mRNA, active and repressor proteins (ci, CI, and CN), and hedgehog mRNA (hh). Each of these variables has a distinct concentration in each cell (X_i , i = 1, . . ., 4). In addition, there are three other proteins whose concentration varies in each of the four cell faces: external wingless protein (EWG), patched protein (PTC) and hedgehog protein (HH). For each of these variables, the concentration in cell i at face j is denoted X_{i, j}, i = 1, . . ., 4, j = 1, . . ., 4. Thus, overall there are: n 9 × 4 + 3 × 4 × 4 = 84 variables. Throughout the paper, the following notation will be used (prime denotes transpose):

$$X={(X_1,X_2,X_3,X_4)}^{\prime },\text{for}X\in \{en,\mathrm{EN},wg,\mathrm{IWG},ptc,ci,\mathrm{CI},\mathrm{CN},hh\}.$$

and

$$X={(X_{1,1},X_{1,2},X_{1,3},X_{1,4},X_{2,1},\dots ,X_{4,4})}^{\prime },\text{for}X\in \{\mathrm{EWG},\mathrm{PTC},\mathrm{HH}\}.$$

Fig. 6.

Four cells in a parasegment, with periodic boundary conditions in both dimensions. Each cell has four membranes. The relative values of Wingless in each cell (EWGi) are shown

The full vector of concentrations is:

$$x=(e{n}^{\prime },{\mathrm{EN}}^{\prime },w{g}^{\prime },{\mathrm{IWG}}^{\prime },{\mathrm{EWG}}^{\prime },pt{c}^{\prime },{\mathrm{PTC}}^{\prime },c{i}^{\prime },{\mathrm{CI}}^{\prime },{\mathrm{CN}}^{\prime },h{h}^{\prime },{\mathrm{HH}}^{\prime }).$$

To formulate the mathematical model, define a vector of species concentrations $(x\in {\mathbb{R}}_{\ge 0}^n)$ and a parameter vector $(p\in {\mathbb{R}}_{\ge 0}^r)$, together with a set of outputs ($y\in {\mathbb{R}}_{\ge 0}^m$, the measured gene expression levels). Introduce functions $f:{\mathbb{R}}_{\ge 0}^n\times {\mathbb{R}}_{\ge 0}^r\to {\mathbb{R}}^n$ and $h:{\mathbb{R}}_{\ge 0}^n\to {\mathbb{R}}_{\ge 0}^m$, where ${\mathbb{R}}_{\ge 0}=\{x\in \mathbb{R}:x_i\ge 0,\text{for all}i\}$, and consider the system with outputs

$$\frac{dx}{dt}=f(x,p)$$ (1)

$$y=h\left(x\right)$$ (2)

where the function h(x) could be, for instance, a vector listing the concentration of wingless, engrailed, and other of the segment polarity mRNAs which have been experimentally measured. Or, in other words, y is “the phenotype corresponding to the genotype x”. The function f is, for instance, as shown in Appendix B for von Dassow et al.'s model. The wild type expression pattern for the segment polarity genes can be viewed as (one of) the steady state solution of system (1).

2.1 The wild type pattern set

An output function is typically composed of variables (or combinations of variables) that are known or available from measurements. Following [3,9] (and references therein), as well as the discussion above, the wild type expression pattern, in each group of four cells, is characterized mainly by the expression of engrailed, hedgehog, and wingless. Here we will further add expression of cubitus and patched to incorporate further experimental evidence [18,19]. These five mRNAs are among the most well documented, so we will consider the output function $h:{\mathbb{R}}_{\ge 0}^n\to {\mathbb{R}}_{\ge 0}^{20}$ to be the state of these five variables. In addition, experimental data is typically of the form “expressed”/“not expressed”, which may be best translated as “0” if concentration is below a certain threshold ε, or “1” if concentration is above the threshold. Then we have:

$$y=h\left(x\right)=\left(\begin{array}{c}\hfill h_{en}\left(x\right)\hfill \\ \hfill h_{hh}\left(x\right)\hfill \\ \hfill h_{wg}\left(x\right)\hfill \\ \hfill h_{ci}\left(x\right)\hfill \\ \hfill h_{ptc}\left(x\right)\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {\text{sign}}_{\epsilon }\left(en\right)\hfill \\ \hfill {\text{sign}}_{\epsilon }\left(hh\right)\hfill \\ \hfill {\text{sign}}_{\epsilon }\left(wg\right)\hfill \\ \hfill {\text{sign}}_{\epsilon }\left(ci\right)\hfill \\ \hfill {\text{sign}}_{\epsilon }\left(ptc\right)\hfill \end{array}\right),$$ (3)

where sign_ε(r) = 0 if r < ε and sign_ε(r) = 1 if r ≥ ε. From experimental measurements, the wild type phenotype is characterized as follows. It is well known [17] that both en and hh are expressed in every third cell, so the desired steady state outputs for these variables are

$$h_{en}\left(x\right)={(0,0,1,0)}^{\prime },h_{hh}\left(x\right)={(0,0,1,0)}^{\prime },\text{for}x\in {\mathbb{R}}^n.$$ (4)

Wingless mRNA is only expressed in every second cell, to the left of en, so that its desired steady state output is

$$h_{wg}\left(x\right)={(0,1,0,0)}^{\prime },\text{for}x\in {\mathbb{R}}^n.$$ (5)

Cubitus and patched mRNAs are typically expressed in all but those cells expressing en [18,19], so the desired steady state output for ci and ptc is of the form:

$$h_{ci}\left(x\right)={(1,1,0,1)}^{\prime },h_{ptc}\left(x\right)={(1,1,0,1)}^{\prime },\text{for}x\in {\mathbb{R}}^n.$$ (6)

The boundaries of the parasegments are expected to form between every second and third cells. The set that contains all states which yield an output satisfying (4)–(6) will be the set representing the wild type pattern for the segment polarity model (Appendix B). From the definition of h (and taking into account the assumptions below for simplicity of analysis), the set of wild type states is of the form:

$$\mathcal{W}=\{x\in {\mathbb{R}}_{\ge 0}^n:en={(0,0,N_3,0)}^{\prime },hh={(H_1,H_2,H_3,H_4)}^{\prime },wg={(W_1,W_2,W_3,W_4)}^{\prime },ci={(U_1,U_2,0,U_4)}^{\prime },ptc={(T_1,T_2,T_3,T_4)}^{\prime },\text{with}{W}_4<W_{1,3};W_{1,3,4},H_{1,2,4}<\epsilon ;W_2,N_3,H_3\ge \epsilon ;T_4=T_2;T_{1,2},U_{1,2,4}\ge \epsilon ;T_3<\epsilon ;\}.$$ (7)

The value ε is a threshold for mRNA (or protein) concentration above which the gene (or protein) is considered expressed. In [3] it was assumed that a gene/protein is expressed when it reaches 10% of its maximal concentration. As discussed below, in the model discussed in this paper the maximal concentration will be 1, hence we will set ε = 0.1.

Remark For simplicity, we have imposed some additional conditions when writing down the set $\mathcal{W}$, compared to merely asking that (4)–(6) hold. These conditions are:

$$\begin{array}{cc}\hfill & e{n}_i=0,i=1,2,4\text{rather than}e{n}_i<\epsilon \hfill \\ \hfill & c{i}_3=0\text{rather than}c{i}_3<\epsilon \hfill \\ \hfill & w{g}_4\le \text{min}\{w{g}_1,w{g}_3\}\hfill \\ \hfill & pt{c}_4=pt{c}_2.\hfill \end{array}$$

These are mild assumptions which however allow many analytic calculations to be carried out explicitly (further discussion can be found in Appendix H), and provide intuition into the dynamics of the segment polarity model. Furthermore, as will become clear in our analysis, these assumptions are verified for many sets of feasible parameters—more precisely, for all sets of parameters except in subsets of lower dimension.

3 Steady states define the feasible parameter space

Previous studies [3,8] have tested the parameter space by randomly choosing sets of parameters and simulating the continuous model. If the corresponding trajectory reaches a steady state, and if this steady state is compatible with the experimentally observed wild type gene pattern, then the given set of parameters is said to be a “solution” to the modeling problem.

A more efficient and complete study of the parameter space can be devised, by first solving the algebraic equations of the model at steady state, and writing the steady state solutions as a function of the parameters. On the other hand, the steady state solutions are known—the set of elements representing the wild type pattern is denoted by $\mathcal{W}-\text{so}$, one can then look for parameters that yield this pattern. Since many sets of parameters may be expected to yield the wild type pattern, this procedure provides a family of conditions defining regions of “good” or feasible parameters “p” for wild type steady states $x\in \mathcal{W}$.

The problem of characterizing the sets of feasible parameters is then reduced to finding all possible parameter vectors p which correspond to a system having a steady state in $\mathcal{W}$.

This will be the set of “good” parameters:

$$G=\{p\in {\mathbb{R}}_{\ge 0}^r:\exists x\in \mathcal{W}\mathrm{s}.\mathrm{t}.f(x,p)=0\}.$$ (8)

3.1 Large Hill coefficients: approximating the continuous model

To find the set G, a straightforward approach would be to solve the steady-state equations for the original system, thus obtaining expressions for x in terms of p:

$$f(x,p)=0\iff x=F\left(p\right),$$

and compare these expressions to the desired form (in $\mathcal{W}$):

$$F\left(p\right)\in \mathcal{W}\iff p\in G.$$

A possible drawback of this method is that explicit solutions x = F(p) for the original system and then explicit formulas for G may not be easy to compute. On the other hand, many of the equations in the model [3] involve terms of the form (see also Appendix B):

$$\varphi (X,\kappa ,\nu )=\frac{X^{\nu }}{{\kappa }^{\nu }+X^{\nu }}\text{or}\psi (X,\kappa ,\nu )=\frac{{\kappa }^{\nu }}{{\kappa }^{\nu }+X^{\nu }},$$

meaning that, if species X is above a certain threshold κ, the function ϕ is active (ON), but the function ψ is inactive (OFF). The exponent ν, also known as the Hill coefficient, characterizes the steepness of an OFF/ON transition.

As found in [20], the model is more robust when the coefficients ν are large. Indeed, for ν in the interval [5.0, 10.0], together with some constraints on other parameters (as detailed in Table 1 in [20]), the proportion of “good parameters” sets increased to 4 out of 5. For large enough exponents, this saturation function becomes very steep, and ϕ becomes practically insensitive to the actual value of ν. Let ν be very large and approximate ϕ by a step function: it is not unreasonable to expect this approximation to capture a large part of the feasible parameter space. This is also the basis of the typical on/off logical interpretation of gene expression. Any such term ϕ(X, κ, ν), for large ν, may thus be replaced by a multivalued step function of the form:

$${\theta }^{+}(X,\kappa )=\{\begin{array}{cc}0,\hfill & X<\kappa \hfill \\ [0,1],\hfill & X=\kappa \hfill \\ 1,\hfill & X>\kappa .\hfill \end{array}\phantom{\}}$$ (9)

Table 1.

Relative volume of G

Region	Volume
G Auto	322 × 10–5
G CI	8.5 × 10–5
G CI,WG	1.5 × 10–5

Fraction of G corresponding to each of the three wingless activation pathways: auto-regulation only (G_Auto), cubitus regulation only (G_CI), and both cubitus and wingless regulation (G_CI,WG). <mi></mi>. Total number of parameter sets generated: 1 × 10⁷. Number of feasible parameter sets: 33276

Define also the symmetric step function ${\theta }^{-}(X,\kappa )=1-{\theta }^{+}(X,\kappa )$. The approximation of ϕ(X, κ, ν) by a step function inevitably has a discontinuity at X = κ. Defining θ⁺ to be multivalued at κ is one way to represent all the states that the function ϕ may take as X is in smaller and smaller neighborhoods of κ, as ν becomes larger and larger. Thus:

$$\underset{\nu \to \infty }{\text{lim}}\varphi (X,\kappa ,\nu )={\theta }^{+}(X,\kappa ),\underset{\nu \to \infty }{\text{lim}}\psi (X,\kappa ,\nu )=1-{\theta }^{+}(X,\kappa )={\theta }^{-}(X,\kappa ),$$ (10)

where, for X = κ, we interpret (10) as saying that the corresponding limit belongs to the interval [0, 1]. A composite function of ϕ and ψ also frequently appears in the continuous equations (Appendix B):

$$\varphi (X_a\psi (X_b,{\kappa }_b,{\nu }_b),{\kappa }_a,{\nu }_a).$$

This function can be simplified in terms of step functions to:

$${\theta }^{+}(X_a{\theta }^{-}(X_b,{\kappa }_b),{\kappa }_a)=\{\begin{array}{cc}{\theta }^{+}(X_a,{\kappa }_a){\theta }^{-}(X_b,{\kappa }_b),\hfill & X_b\ne {\kappa }_b\hfill \\ {\theta }^{+}(X_a[0,1],{\kappa }_a),\hfill & X_b={\kappa }_b,\hfill \end{array}\phantom{\}}$$ (11)

since

$$X_b>{\kappa }_b\Rightarrow {\theta }^{-}(X_b,{\kappa }_b)=0\Rightarrow {\theta }^{+}(X_a{\theta }^{-}(X_b,{\kappa }_b),{\kappa }_a)={\theta }^{+}(0,{\kappa }_a)=0,X_b={\kappa }_b\Rightarrow {\theta }^{-}(X_b,{\kappa }_b)=[0,1]\Rightarrow {\theta }^{+}(X_a{\theta }^{-}(X_b,{\kappa }_b),{\kappa }_a)={\theta }^{+}(X_a[0,1],{\kappa }_a),X_b<{\kappa }_b\Rightarrow {\theta }^{-}(X_b,{\kappa }_b)=1\Rightarrow {\theta }^{+}(X_a{\theta }^{-}(X_b,{\kappa }_b),{\kappa }_a)={\theta }^{+}(X_a,{\kappa }_a).$$

As an example, consider the equation governing engrailed from the original model which can be found in [3,20] (or in Appendix B). In this model the concentration of engrailed in cell i (en_i), is positively regulated by external Wingless protein (EWG_i) and negatively regulated by Cubitus repressor protein (CN_i) concentrations (further notation is found in Appendix A):

$$\frac{de{n}_i}{dt}=\frac{1}{H_{en}}(\varphi ({\mathrm{EWG}}_{\underset{‒}{i}}\psi ({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}en},{\nu }_{\mathrm{CN}en}),{\kappa }_{\mathrm{WG}en},{\nu }_{\mathrm{WG}en})-e{n}_i).$$

For large exponents ν, this simplifies to the equation:

$$\frac{de{n}_i}{dt}=\frac{1}{H_{en}}({\theta }^{+}({\mathrm{EWG}}_{\underset{‒}{i}}{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}en}),{\kappa }_{\mathrm{WG}en})-e{n}_i).$$

To analyticaly study the space of feasible parameters for the segment polarity network model [3], we will thus consider that all exponents ν are large, and apply method (10) to simplify the original system of equations. In addition, as discussed, the system is assumed to be at steady state, in which case the gene expression pattern must satisfy:

$$e{n}_i={\theta }^{+}({\mathrm{EWG}}_{\underset{‒}{i}}{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}en}),{\kappa }_{\mathrm{WG}en}).$$

Applying (10) and then solving the system at steady state yields the set of algebraic equations (52)–(63), which characterize the gene expression pattern of the segment polarity network according to our approximation of the von Dassow et al. model. In particular, note that the cubitus mRNA equation becomes

$$c{i}_i=U_i{\theta }^{-}({\mathrm{EN}}_i,{\kappa }_{\mathrm{EN}ci}),i=1,\dots ,4,$$

where the choice of the parameters $U_i\in [0,1]$ is explained in Appendix C. Furthermore, in characterizing the set of feasible parameters, it will become clear that allowing distinct U_i enlarges the space of possible parameters. Asymmetry in cubitus expression (i.e., distinct values U_i for each i = 1, . . ., 4) could be due, for instance, to some of the pair rule genes. Sloppy paired, or a combination of Runt and Factor X, regulate the transition from pair rule to segment polarity genes expression, and induce asymmetric anterior/posterior parasegment expression [21].

Finally, note that the maximal expression levels of wg are written in terms of the parameters α_CIwg and α_WGwg. From Eq. (54), there are several possible combinations of the step functions, each leading to a different value for wg₂. These possibilities are given by:

$$w=\frac{R_{\mathrm{CI}}{\alpha }_{\mathrm{CI}wg}+R_{\mathrm{WG}}{\alpha }_{\mathrm{WG}wg}}{1+R_{\mathrm{CI}}{\alpha }_{\mathrm{CI}wg}+R_{\mathrm{WG}}{\alpha }_{\mathrm{WG}wg}},$$ (12)

where $R_{\mathrm{CI}},R_{\mathrm{WG}}\in [0,1]$ reflect the possible values of the step functions. Two different pathways for wingless activation can be identified. Indeed, wingless can be activated by Cubitus only (in which case R_CI > 0, R_WG = 0), or by Wingless only (R_CI = 0, R_WG > 0), or by both Cubitus and Wingless (R_CI, R_WG > 0).

4 Missing link: engrailed regulation by Cubitus repressor

A first result from our model formulation is the explanation of a “missing link” in a first version of the model proposed by von Dassow et al. [3]. In this first version, engrailed was regulated only by EWG, and no feasible parameter sets were found. Indeed, below (Theorem 1) we prove that, for any set of parameters, the mechanism for wingless regulation generates a strong symmetry in the steady state expression of external Wingless. This mechanism effectively prevents any asymmetry arising in en due to EWG only.

Theorem 1 Assume wg^WT = (w₁, w₂, w₃, w₄)′, with 0 ≤ w₄ ≤ min{w₁, w₃} ≤ max{w₁, w₃} < w₂. Then, at steady state:

$$EW{G}_{\underset{‒}{4}}^{WT}\le \text{min}\{EW{G}_{\underset{‒}{1}}^{WT},EW{G}_{\underset{‒}{3}}^{WT}\}\le \text{max}\{EW{G}_{\underset{‒}{1}}^{WT},EW{G}_{\underset{‒}{3}}^{WT}\}<EW{G}_{\underset{‒}{2}}^{WT}.$$ (13)

The proof is based on the following Lemma which is shown in Appendix E.

Lemma 4.1 Let wg = (wg₁, wg₂, wg₃, wg₄). There exist constants β_max > β_med > β_min > 0 and γ_max > γ_med > γ_min > 0 such that:

$$\begin{array}{cc}\hfill EW{G}_{\underset{‒}{1}}& =w{g}_1{\beta }_{\text{max}}+(w{g}_2+w{g}_4){\beta }_{\text{med}}+w{g}_3{\beta }_{\text{min}},\hfill \\ \hfill EW{G}_{\underset{‒}{2}}& =w{g}_2{\beta }_{\text{max}}+(w{g}_1+w{g}_3){\beta }_{\text{med}}+w{g}_4{\beta }_{\text{min}},\hfill \\ \hfill EW{G}_{\underset{‒}{3}}& =w{g}_3{\beta }_{\text{max}}+(w{g}_2+w{g}_4){\beta }_{\text{med}}+w{g}_1{\beta }_{\text{min}},\hfill \\ \hfill EW{G}_{\underset{‒}{4}}& =w{g}_4{\beta }_{\text{max}}+(w{g}_1+w{g}_3){\beta }_{\text{med}}+w{g}_2{\beta }_{\text{min}}.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill IW{G}_1& =w{g}_1{\gamma }_{\text{max}}+(w{g}_2+w{g}_4){\gamma }_{\text{med}}+w{g}_3{\gamma }_{\text{min}},\hfill \\ \hfill IW{G}_2& =w{g}_2{\gamma }_{\text{max}}+(w{g}_1+w{g}_3){\gamma }_{\text{med}}+w{g}_4{\gamma }_{\text{min}},\hfill \\ \hfill IW{G}_3& =w{g}_3{\gamma }_{\text{max}}+(w{g}_2+w{g}_4){\gamma }_{\text{med}}+w{g}_1{\gamma }_{\text{min}},\hfill \\ \hfill IW{G}_4& =w{g}_4{\gamma }_{\text{max}}+(w{g}_1+w{g}_3){\gamma }_{\text{med}}+w{g}_2{\gamma }_{\text{min}}.\hfill \end{array}$$

Now, consider the steady state equation that would result if no dependence of engrailed on CN is assumed in Eq. (52):

$$e{n}_i^{\mathrm{WT}}={\theta }^{+}({\mathrm{EWG}}_{\underset{‒}{i}}^{\mathrm{WT}},{\kappa }_{\mathrm{WG}en})$$

Compare to states in $\mathcal{W}$:

$$e{n}_3=N_3\text{and}e{n}_i=0,\text{for}i=1,2,4.$$

Then, from the definition of θ⁺, for consistency in our model it is necessary that:

$$\begin{array}{cc}\hfill & {\mathrm{EWG}}_{\underset{‒}{i}}^{\mathrm{WT}}\le {\kappa }_{\mathrm{WG}en},\text{for}i=1,2,4\hfill \\ \hfill & {\mathrm{EWG}}_{\underset{‒}{3}}^{\mathrm{WT}}\ge {\kappa }_{\mathrm{WG}en}.\hfill \end{array}$$

However, by (13), the inequalities for i = 2 (and possibly i = 1) and i = 3 are incompatible. This means that, due to the symmetry in Wingless=distribution, such a simple regulation of en can never lead to the segment polarity pattern. Thus engrailed requires regulation by some other factor, in this case repression by the Cubitus protein (CN), as in (52). In order to obtain repression of en in the second cell, recalling (11) one can now ask:

$$\begin{array}{cc}\hfill {\mathrm{EWG}}_{\underset{‒}{1}}^{\mathrm{WT}}\le {\kappa }_{\mathrm{WG}en}\text{or}& {\mathrm{CN}}_1^{\mathrm{WT}}\ge {\kappa }_{\mathrm{CN}en},\hfill \\ \hfill & {\mathrm{CN}}_2^{\mathrm{WT}}\ge {\kappa }_{\mathrm{CN}en},\hfill \\ \hfill {\mathrm{EWG}}_{\underset{‒}{3}}^{\mathrm{WT}}\ge {\kappa }_{\mathrm{WG}en}\text{and}& {\mathrm{CN}}_3^{\mathrm{WT}}\le {\kappa }_{\mathrm{CN}en},\hfill \\ \hfill {\mathrm{EWG}}_{\underset{‒}{4}}^{\mathrm{WT}}\le {\kappa }_{\mathrm{WG}en}\text{or}& {\mathrm{CN}}_4^{\mathrm{WT}}\ge {\kappa }_{\mathrm{CN}en},\hfill \end{array}$$

that is, CN is responsible for repression in the second, and possibly in the first, cells. This means that, at steady state, CN must be expressed in both the first and second cells. This in turn requires the presence of Patched protein in both the first and second cells. On the other hand, from Appendix F, we know that a steady state $x\in \mathcal{W}$, implies $pt{c}_1^{\mathrm{WT}}={\mathrm{PTC}}_1^{\mathrm{WT}}$ and $pt{c}_3^{\mathrm{WT}}={\mathrm{PTC}}_3^{\mathrm{WT}}=0$. While patched expression is typically weaker in the first than in second and fourth cells (see [3]), this shows that it is nevertheless necessary, that is, the segment polarity gene pattern obtains only when T₁ is above the expression threshold. The discussion on CN leads to the following conclusion:

Lemma 4.2 Consider system (1) and assume that, at steady state, the system is in $\mathcal{W}$, that is ci^WT = (U₁, U₂, 0, U₄)′ and ptc^WT = (T₁, T₂, T₃, T₂)′, with U_1,2,4 ≥ ε and T₃ < ε. Then $PT{C}_{3,T}^{WT}=0$ and $C{I}_3^{WT}=C{N}_3^{WT}=c{i}_3^{WT}=0$ and $PT{C}_{2,T}^{WT}=PT{C}_{4,T}^{WT}>0$. Also $PT{C}_{1,2,4}^{WT}\ge {\kappa }_{PTCCI}$ and

$$C{N}_i^{WT}=U_i\frac{Q_i{H}_{CI}{C}_{CI}}{1+Q_i{H}_{CI}{C}_{CI}},i=1,2,4,C{I}_i^{WT}=U_i-C{N}_i=U_i\frac{1}{1+Q_i{H}_{CI}{C}_{CI}},$$ (14)

with

$$Q_i\in \{\begin{array}{cc}\left\{1\right\},\hfill & ifPT{C}_{i,T}>{\kappa }_{PTCCI}\hfill \\ [0,1],\hfill & ifPT{C}_{i,T}={\kappa }_{PTCCI}.\hfill \end{array}\phantom{\}}$$

5 A cylindrical algebraic decomposition of the parameter space

The algebraic equations f(x, p) = 0 together with $x\in \mathcal{W}$ impose constraints on the set of good parameters (G), though not providing as yet explicit conditions on p. An explicit characterization of the parameters p may be obtained by calculating a cylindrical algebraic decomposition (CAD) of G: this is a special type of representation of G through a hierarchy of inequalities on the parameters. Suppose that a first family of parameters, say {p₁, . . ., p_r1} (with 1 ≤ r₁ < r) may take values in a product of intervals L₁ × · · · × L_r1 (where each interval may be open, closed or mixed). Then a CAD is defined as follows:

$$\begin{array}{cc}\hfill S_1& =L_1\times \cdots \times L_{r_1}\subset {\mathbb{R}}^{r_1}\hfill \\ \hfill S_j& =\{(\widehat{p},p_j)\in {\mathbb{R}}^j:\widehat{p}={(p_1,\dots ,p_{j-1})}^{\prime }\in S_{j-1},f_j\left(\widehat{p}\right)\prec p_j\prec g_j\left(\widehat{p}\right)\}\subset {\mathbb{R}}^j\hfill \end{array}$$ (15)

for j = r₁ + 1, . . ., r, where $f_j,g_j:S_{j-1}\to {\mathbb{R}}_{\ge 0}$ are continuous functions such that f_j(p̂) ≤ g_j (p̂) for all $\widehat{p}\in {\mathbb{R}}^{j-1}$, the symbol ≺ denotes either < or ≤, and S_r = G.

Computing the cylindrical algebraic decomposition of a set is a complex problem, but various standard algorithms are available [22,23]. Several software packages have been developed, for instance QEPAD [24], (based in [25]) and in Mathematica [26]. See also [27] for an overview of available software, current applications, and many other related references. Common applications of CADs include computation of the controllable or reachabable sets in hybrid systems, see for instance [28,29], where the latter includes an application to a genetic network in the fly wing. Constructing a CAD involves the use of symbolic computation and, while various improvements have been achieved, it still is a time consuming problem. For instance, the estimated maximum time for the algorithm [22] is dominated by “2^2kN”, where N is the length of the input formula and 0 < k ≤ 8. Fortunately, in view of these computational complexity difficulties, in the present example it is relatively easy to directly compute a CAD without using general methods, and we will do so.

The computation of the feasible parameter set G for the segment polarity network is detailed in Appendix H. The CAD of G can be used to answer several questions regarding geometry and topology of the feasible parameter space. First, the volume of G can be estimated, relative to the unitary hypercube [0, 1]^r (r = 31), as described in Sect. 6. Second, the topology of G can be analyzed, to find out its connectedness (e.g., simple-connectedness; or composed of various disconnected components). To summarize, we show that G can be written as a union of several regions:

$$G=G_{\mathrm{I}}\cup \cdots \cup G_{\mathrm{V}a}\cup G_{\mathrm{V}b}\cup \cdots \cup G_{\mathrm{VIII}a}\cup G_{\mathrm{VIII}b}\cup G_{\text{Auto}}.$$

These 13 regions are all connected and, in particular, G_Va to G_VIIIb are connecting faces and have a lower dimension (see Fig. 1e and Theorem 2, below).

Each of these regions has a CAD with nine levels, as listed next. The levels S₁, . . ., S₆ are the same for all the regions G_k, $k\in \{\mathrm{I},\dots ,\mathrm{VIII}b,\text{Auto}\}$. The form of the last three levels depends on each region: $S_7^k$, $S_8^k$, $S_9^k$. At the base of the CADs, S₁ is a product of intervals for r₁ = 23 parameters (while r = 31) defined as follows:

$$S_1:\{\begin{array}{c}{p}_{\mathrm{CI}}=(H_{\mathrm{CI}},C_{\mathrm{CI}},U_1,U_2,U_3,U_4),\in [5,100]\times (0,1]\times {[\epsilon ,1]}^2\times (0,1]\times [\epsilon ,1],\hfill \\ p_{\mathrm{PTC}-\mathrm{HH}}=(H_{\mathrm{PTC}},H_{\mathrm{HH}},{\left[\mathrm{PTC}\right]}_0,{\left[\mathrm{HH}\right]}_0,r_{LM\mathrm{PTC}},r_{LM\mathrm{HH}},{\kappa }_{\mathrm{PTCHH}})\in {[5,100]}^2\times {(0,1]}^5,\hfill \\ {\alpha }_{\mathrm{CI}wg},{\alpha }_{\mathrm{IWG}wg}\in [\frac{\epsilon }{1-\epsilon },10],\hfill \\ {\kappa }_{\mathrm{EN}ci},{\kappa }_{\mathrm{EN}hh},{\kappa }_{\mathrm{CN}hh}\in (0,1].\hfill \end{array}\phantom{\}}$$ (16)

Levels S₂ to S₄ are characterized as follows:

$$S_2=\{(\widehat{p},{\kappa }_{\mathrm{PTCCI}}):\widehat{p}\in S_1,0<{\kappa }_{\mathrm{PTCCI}}\le \text{min}\{F_{{\mathrm{PTC}}_{1,\mathrm{T}}},F_{{\mathrm{PTC}}_{2,\mathrm{T}}}\}\},$$ (17)

$$S_3=\left\{(\widehat{p},{\kappa }_{\mathrm{CN}ptc}):\widehat{p}\in S_2,\text{max}\left\{\frac{U_i{Q}_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}\right\}\le {\kappa }_{\mathrm{CN}ptc}\le 1\right\},$$ (18)

$$S_4=\left\{(\widehat{p},{\kappa }_{\mathrm{CI}ptc}):\widehat{p}\in S_3,0<{\kappa }_{\mathrm{CI}ptc}\le \text{min}\left\{\frac{U_i}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}\right\}\right\},$$ (19)

where F_PTC1,T, F_PTC2,T are the solutions of the algebraic equations (58) for the set of parameters p̂. Let $w_2\in \{\frac{{\alpha }_{\mathrm{CI}}}{1+{\alpha }_{\mathrm{CI}}},\frac{{\alpha }_{\mathrm{WG}}}{1+{\alpha }_{\mathrm{WG}}},\frac{{\alpha }_{\mathrm{WG}}+{\alpha }_{\mathrm{CI}}}{1+{\alpha }_{\mathrm{WG}}+{\alpha }_{\mathrm{CI}}}\}$. Then levels S₅ and S₆ are characterized as:

$$S_5=\{(\widehat{p},{\kappa }_{\mathrm{WG}en}):\widehat{p}\in S_4,0<{\kappa }_{\mathrm{WG}en}\le w_2{\beta }_{\text{med}}\},$$ (20)

$$S_6=\left\{(\widehat{p},{\kappa }_{\mathrm{CN}en}):\widehat{p}\in S_5,0<{\kappa }_{\mathrm{CN}en}\le \underset{i=1,2}{\text{min}}\left\{\frac{U_i{Q}_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}\right\},\text{if}{\kappa }_{\mathrm{WG}en}\in [w_2{\beta }_{\text{min}},w_2{\beta }_{\text{med}}];\text{or}0<{\kappa }_{\mathrm{CN}en}\le \underset{i=1,2,4}{\text{min}}\left\{\frac{U_i{Q}_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}\right\},\text{if}{\kappa }_{\mathrm{WG}en}\in [0,w_2{\beta }_{\text{min}}]\right\}$$ (21)

Finally, the last three levels have different expressions depending on the region of the parameter space:

$$S_7^k=\{(\widehat{p},{\kappa }_{\mathrm{WG}wg}):\widehat{p}\in S_6,f_{\mathrm{WG}wg}^k\left(\widehat{p}\right)\prec {\kappa }_{\mathrm{WG}wg}\prec g_{\mathrm{WG}wg}^k\left(\widehat{p}\right)\},$$ (22)

$$S_8^k=\{(\widehat{p},{\kappa }_{\mathrm{CN}wg}):\widehat{p}\in S_7^k,f_{\mathrm{CN}wg}^k\left(\widehat{p}\right)\prec {\kappa }_{\mathrm{CN}wg}\prec g_{\mathrm{CN}wg}^k\left(\widehat{p}\right)\},$$ (23)

$$S_9^k=\{(\widehat{p},{\kappa }_{\mathrm{CI}wg}):\widehat{p}\in S_8^k,f_{\mathrm{CI}wg}^k\left(\widehat{p}\right)\}\prec {\kappa }_{\mathrm{CI}wg}\prec g_{\mathrm{CI}wg}^k\left(\widehat{p}\right)\},$$ (24)

where $k\in \{\mathrm{I},\dots ,\mathrm{VIII}b,\text{Auto}\}$ and the functions $f_j^k$ and $g_j^k(j\in {\{}_{\mathrm{WG}wg}{,}_{\mathrm{CN}wg}{,}_{\mathrm{CI}wg}\})$ are listed in Tables 4 and 5. These CADs have a biological interpretation (see discussion in Sect. 5.1): from Eqs. (52)–(63) subject to (4) and (5), several parameters are free to take any values, within physiological restrictions only—these form S₁. The parameters defined at the levels S₂ to S₆ have constraints which depend only on the family of parameters defined in S₁. The last levels, $S_7^k$ to $S_9^k$ define different regions of G, with the property that each region is associated to the activation of a particular biological pathway: G_Auto corresponds to activation of wingless on the second cell by the autocatalytic pathway only. In regions G_I to G_VIIIb the Cubitus pathway also promotes activation of wingless.

Table 4.

Parameter κ_WGwg in each region

Region	Intervals
Gk, k = i, . . . , viiib	[w2;C γmax, 1] (cubitus pathway only)
	[w2;C, W γmed, w2;C,W γmax] (both cubitus and wingless pathways)
G Auto	[w2; W γmed, w2; W γmax]

In regions G_I to G_VIIIb the cubitus pathway contributes to wingless mRNA regulation. In region G_Auto only the wingless pathway contributes to wingless mRNA regulation. Define: $w_{2;C}=\frac{{\alpha }_{\mathrm{CI}}}{1+{\alpha }_{\mathrm{CI}}},w_{2;C,W}=\frac{{\alpha }_{\mathrm{WG}}+{\alpha }_{\mathrm{CI}}}{1+{\alpha }_{\mathrm{WG}}+{\alpha }_{\mathrm{CI}}}$, and $w_{2;W}=\frac{{\alpha }_{\mathrm{WG}}}{1+{\alpha }_{\mathrm{WG}}}$

Table 5.

Parameters κ_CNwg and κ_CIwg in each region

Region	Intervals
	κ CNwg	κ CIwg
GI (maxi=1,4 Ũi < Ũ2)	[Ũ2, 1]	[maxi=1,4 (Ui – Ũi), U2 – Ũ2]
GII (Ũ4 < Ũ2 < Ũ1)	[Ũ2, Ũ1]	[U4 – Ũ4, U2 – Ũ2]
GIII (Ũ2 < mini=1,4 Ũi)	[Ũ2, mini=1,4 Ũi]	(0, U2 – Ũ2]
GIV (Ũ1 < Ũ2 < Ũ4)	[Ũ2, Ũ4]	[U1 – Ũ1, U2 – Ũ2]
GVa (Ũ4 = Ũ2 < Ũ1)	{Ũ2}	(0, U2 – Ũ2]
GVb (Ũ4 = Ũ2 < Ũ1)	[Ũ2, Ũ1]	{U2 – Ũ2}
GVIa (Ũ4 < Ũ2 = Ũ1)	{Ũ2}	[U4 – Ũ4, U2 – Ũ2]
GVIb (Ũ4 < Ũ2 = Ũ1)	[Ũ2, 1]	{U2 – Ũ2}
GVIIa (Ũ1 = Ũ2 < Ũ4)	{Ũ2}	(0, U2 – Ũ2]
GVIIb (Ũ1 = Ũ2 < Ũ4)	[Ũ2, Ũ4]	{U2 – Ũ2}
GVIIIa (Ũ1 < Ũ2 = Ũ4)	{Ũ2}	[U1 – Ũ1, U2 – Ũ2]
GVIIIb (Ũ1 < Ũ2 = Ũ4)	[Ũ2, 1]	{U2 – Ũ2}
G Auto	(0, Ũi] or	[Ũi – Ũi, 1] (each i = 1, 2, 4)

In regions G_I to G_VIII the cubitus pathway contributes to wingless mRNA regulation. In region G_Auto only the wingless pathway contributes to wingless mRNA regulation. Define ${\tilde{U}}_i=\frac{U_i{Q}_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}$, and note that $U_i-{\tilde{U}}_i=\frac{U_i}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}$

In the next lemma it is shown that each region G_k is topologically equivalent to a unitary closed hypercube, hence topologically trivial. However, some regions (G_Va to G_VIIIb) have a lower dimension. This is clear by observing Table 5: in each of these eight regions, either of the parameters κ_CNwg or κ_CIwg is a single point (as opposed to a non-trivial interval).

To simplify notation, for i = 1, . . ., 4 define

$${\tilde{U}}_i=U_i\frac{Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}$$

and observe that CN_i = Ũ_i and CI_i = U_i — Ũ_i.

Let ≺ denote either of the symbols < or ≤ and define

$$L_j=\{x\in \mathbb{R}:a_j\prec x\prec b_j\}.$$

Let $\mathcal{I}$ denote the unitary interval (open, closed or mixed):

$$\mathcal{I}=\{x\in \mathbb{R}:0\prec x\prec 1\}.$$

Theorem 2 Consider the sets $S_9^k,k\in \{I,\dots ,VIIIb,Auto\}$, obtained from (24). Then (with r = 31):

1. $S_9^k$, k = Auto, I, II, III, IV are homeomorphic to ${\mathcal{I}}^r$;

2. $S_9^k$, k = Va, Vb, VIa, VIb, VIIa, VIIb, VIIIa, VIIIb are homeomorphic to ${\mathcal{I}}^{r-1}\times \left\{1\right\}$.

Proof Consider first case (i). To argue by induction, note that each L_j is clearly homeomorphic to the interval $\mathcal{I}$ (with the same open, closed or mixed boundaries as L_j), for j = 1, . . ., 23. Since a_j < b_j for all j = 1, . . ., 23, simply consider the bijective function ${\varphi }_j:L_j\to \mathcal{I}$ given by ϕ_j(p) = (b_j − p)/(b_j − a_j). Then the set S₁ = L₁ × · · · × L₂₃ is homeomorphic to the product ${\mathcal{I}}^{23}$, by considering the bijective function ${\phi }_1:S_1\to {\mathcal{I}}^{23}$ given by φ₁(p₁, . . ., p₂₃) = (ϕ₁, . . ., ϕ₂₃(p₂₃))′.

For i ≥ 2, assume that S_i−1 is homeomorphic to ${\mathcal{I}}^{i-1}$. Note that f_i(p) < g_i(p) for all i = 2, . . ., 9, for $S_9^k$ with k = Auto, I, II, III, IV. Next, define the following continuous function:

$${\phi }_i:S_{i-1}\times \mathcal{I}\to S_{i-1}\times \mathbb{R},{\phi }_i(p,t)=(p,f_i\left(p\right)+t(g_i\left(p\right)-f_i\left(p\right))).$$

For each fixed p, f_i(p) < g_i(p) + t (g_i(p) − f_i(p)) < g_i(p) for all $t\in \mathcal{I}$. Therefore, φ_i maps into S_i. On the other hand, since g_i(p) − f_i(p) > 0 for all $p\in S_{i-1}$, φ_i has an inverse function defined on S_i and continuous, given by:

$${\phi }_i^{-1}:S_i\to S_{i-1}\times \mathcal{I},{\phi }_i^{-1}(p,y)=\left(p,\frac{y-f_i\left(y\right)}{g_i\left(p\right)-f_i\left(p\right)}\right).$$

So S_i is homeomorphic to $S_{i-1}\times \mathcal{I}$, and therefore, by inductive hypothesis, to ${\mathcal{I}}^i$.

Next, consider case (ii). From Table 5 it is clear that, in regions G_Va to G_VIIIb, exactly one of the parameters κ_CNwg or κ_CIwg has a single point as an interval. So, the previous argument is valid up to S_r−1 and then the last parameter is a point. This finishes the proof. □

5.1 Two wingless mRNA activation pathways

Following the model of von Dassow et al., there are two possible parallel pathways for wingless activation: either by the Cubitus interruptus protein (CI), or through auto-activation; both pathways could be simultaneously activating wingless production. Since the activation constants α_CIwg and α_WGwg, are free parameters, in each of the three cases $w{g}_2^{\mathrm{WT}}$ will have a different ON level as calculated from (12). Computation of EWG and IWG depends on $w{g}_2^{\mathrm{WT}}$, so each of these three cases must be separately analyzed for feasibility. For both pathways, exact analytic computation of PTC_{i, j} and HH_{i, j} (i, j = 1, . . ., 4) is also carried out (see Appendix F). When CI and CN contribute to regulate wingless expression, it is easy to see from (5), (14) and (54) that:

$$U_i\frac{1}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}\le {\kappa }_{\mathrm{CI}wg}\text{or}{U}_i\frac{Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}\ge {\kappa }_{\mathrm{CN}wg},$$ (25)

for i = 1, 3, 4, and

$$U_2\frac{1}{1+Q_2{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}\ge {\kappa }_{\mathrm{CI}wg}\text{and}{U}_2\frac{Q_2{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}{1+Q_2{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}\le {\kappa }_{\mathrm{CN}wg}.$$ (26)

From observation of (25) and (26) it is clear that the situations Ũ₂ = Ũ₁ or Ũ₂ = Ũ₄ define some critical regions. To see this, for simplicity consider Q_i = 1 (which is true for many parameter sets, i.e., κ_PTCCI < min{F_PTC1,T, F_PTC2,T}). Then Ũ₂ = Ũ₁ is compatible with the wild type pattern only when κ_CIwg = U₁ − Ũ₁ = U₂ − Ũ₂ or κ_CNwg = Ũ₁ = Ũ₂. Moreover, recalling the definition of step function, it follows that in this case θ⁺ (CI₂θ⁻ (CN₂, κ_CNwg), κ_CIwg) = [0, 1] and wg may in fact be either activated or not in cell 2. The set G can thus be partitioned into four “strictly feasible” components, divided by the critical hyperplanes Ũ₂ = Ũ₁ or Ũ₂ = Ũ₄ (Fig. 2). Note that, for parameters belonging to these hyperplanes: $w{g}_2=\frac{R_{CI}{\alpha }_{CI}+R_{WG}{\alpha }_{WG}}{1+R_{CI}{\alpha }_{CI}+R_{WG}{\alpha }_{WG}}$ where R_CI is any number in the interval [0, 1]. Thus there are multiple possible steady states, and some of these may not provide the right phenotype. These hyperplanes correspond, in fact, to regions G_V to G_VIII.

Fig. 2.

Projection of set G into the (U₁, U₂, U₄) space (case Q_i = 1, i = 1, 2, 4). The critical hyperplanes U₂ = U₄ (yellow/light grey) and U₂ = U₁ (blue/dark grey) define four strictly feasible components in Fig. 3, compares the regions where CI/CN is active (G_II) or only WG is active (G_Auto), both polyhedrons.

When only the wingless auto-activation pathway contributes to wingless activation, the necessary conditions are (when Q_i = 1):

$$\left(U_2\frac{1}{1+Q_2{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}<{\kappa }_{\mathrm{CI}wg}\text{or}{U}_2\frac{Q_2{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}{1+Q_2{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}>{\kappa }_{\mathrm{CN}wg}\right)\text{and}{\mathrm{IWG}}_2\ge {\kappa }_{\mathrm{WG}wg}.$$ (27)

Comparing (26) and (27) it is not surprising that, for each triple (Ũ₁, Ũ₂, Ũ₄), κ_CIwg and κ_CNwg belong to complementary intervals, since they define whether or not the CI/CN pathway is active. The projection on the (κ_CIwg, κ_CNwg, κ_WGwg)-dimensions, shown

6 Geometry, volume and the second missing link

From the CAD (16)–(24) it is very easy to compute the (relative) volume of G. Following a Monte Carlo approach, the free parameters (16) and also (17)–(19) are chosen first, from a uniform distribution in their respective intervals. Then the quintuple of parameters κ_WGen, κ_CNen, κ_CIwg, κ_CNwg, and κ_WGwg are randomly chosen (from a uniform distribution) in the interval [0, 1]. It is then checked whether the whole parameter set falls inside or outside G. If the parameter set falls inside G, we also check which wingless activation pathways are compatible, as well as the region (U₁, U₂, U₄). This Monte Carlo approach provides an estimate of the volume of G, when projected into the (κ_CNen, κ_WGen, κ_CIwg, κ_CNwg, κ_WGwg) dimensions. In addition, we also obtain an estimate of the fraction of the feasible parameter sets that correspond to either of the wingless activation pathways (Table 1).

The volume of this five-dimensional cube occupied by feasible parameter sets is only about 0.3%. Interestingly, we also found that the vast majority of the feasible parameter space corresponds to auto-regulation of wingless. As illustrated by the polyhedrons in Fig. 3, region G_Auto is much larger than the others. Here, regions G_I to G_VIIIb are re-grouped according to the biological pathway, following the indications in Tables 3 and 4. Region G_CI corresponds to wingless mRNA regulation by cubitus only, and region G_{CI, WG} corresponds to regulation by both cubitus and wingless proteins. We have: G_CI υ G_CI,WG = G_I υ · · · υ G_VIIIb, where each G_i intersects G_CI and G_{CI, WG}.

Fig. 3.

Projection of G on the space (κ_CIwg, κ_CNwg, κ_WGwg), of the fibre over the point represented by “*” in Fig. 2. This point corresponds to choosing values for (U₁, U₂, U₄) in region G_II. Depending on the choice of parameters S₁, the interval for κ_WGwg in G_Auto may or may not intersect the other two. The values w₂;. are defined in Table 4

Table 3.

Parameters common to all regions: κ_PTCCI, κ_CIptc, κ_CNptc, κ_WGen, and κ_CNen

Parameters	Intervals
κ PTCCI	$(0,\text{min}\{F_{{\mathrm{PTC}}_{1,\mathrm{T}}},F_{{\mathrm{PTC}}_{2,\mathrm{T}}}\}]$
κ CIptc	$\left(0,\text{min}\left\{\frac{U_i}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}\right\}\right]$
κ CNptc	$\left(\text{max}\left\{\frac{U_i{Q}_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}\right\},1\right]$
κ WGen	(0, w2βmed]
κ CNen	$\left(0,{\text{min}}_{i=1,2}\left\{\frac{U_i{Q}_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}\right\}\right]\text{if}{\kappa }_{\mathrm{WG}en}\in [w_2{\beta }_{\text{min}},w_2{\beta }_{\text{med}}]$
	$\left(0,{\text{min}}_{i=1,2,4}\left\{\frac{U_i{Q}_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}\right\}\right]\text{if}{\kappa }_{\mathrm{WG}en}\in (0,w_2{\beta }_{\text{min}}]$

Let $w_2\in \left\{\frac{{\alpha }_{\mathrm{CI}}}{1+{\alpha }_{\mathrm{CI}}},\frac{{\alpha }_{\mathrm{WG}}}{1+{\alpha }_{\mathrm{WG}}},\frac{{\alpha }_{\mathrm{WG}}+{\alpha }_{\mathrm{CI}}}{1+{\alpha }_{\mathrm{WG}}+{\alpha }_{\mathrm{CI}}}\right\}$

The large difference observed between G_CI, G_{CI, WG} and G_Auto explains the second “missing link” in the first version of von Dassow et al. model, namely the wingless autocatalytic activation. Note that the presence of this link greatly increases the total volume of the feasible parameter space: in fact the region G_Auto is 96% of the total feasible volume.

6.1 Geometry: parameter distributions

To further analyze the geometry of the feasible parameter space, one may ask how the parameters are distributed in their intervals. For instance, is each parameter p_i more likely to attain high or low values more frequently, and can a specific “tendency” for each parameter p_i be identified. An answer to this question is obtained by computing the marginal distribution of each parameter from a family of randomly generated parameters in the full parameter space G. Taking all the parameter sets generated proviously to compute the relative volume of G, and computing a histogram for each parameter, the result shown on Fig. 4 is obtained. As expected, many parameters have a uniform distribution, as their values do not influence the final outcome of the network in any particular way (for instance, most half-lives and diffusion-related parameters). Other parameters exhibit a marked tendency for higher (e.g., κ_CNptc), medium (e.g., κ_WGwg) or lower (e.g., κ_CIptc) values. All the parameters that exhibit a marked tendency are listed in Table 2, and classified according to their function in the network: for instance, κ_CNptc represents the repression of ptc by CN, and therefore, high values of κ_CNptc correspond to a weak repression.

Fig. 4.

Parameter histograms out of 33276 parameter sets (refer to model equations for explanation of parameters). The notation and scales follow those of Fig. 6 in [20]. In the y-axis, the histograms are all normalized to their maximal value. The half-lifes (denoted Hx) range between 5 and 100 mins in a linear scale. The coefficients aCIwg and aWGwg range between ε/(1 ε) = 1/9 and 10.0 also in a linear scale. All other parameters range between 10⁻³ and 1, and shown in log₁₀ scale

Table 2.

Comparison between the constraints identified by von Dassow and Odell [20], and the exact constraints given by the regions defined above

Parameter	Description	Tendency [20, Table 1]	Tendency (within G)	Tendency (GCI + GCI,WG)
κ WGen	WG activation of en	Moderate	Moderate
κ CNen	CN repression of en	Strong	Strong
κ WGwg	WG autoactivation	Moderate	Moderate
κ CIwg	CI activation of wg	Weak	—	Strong
κ CNwg	CN repression of wg	Strong	—	Weak
κ CIptc	CI activation of ptc	Strong	Strong
κ CNptc	CN repression of ptc	Weak	Weak
κ ENci	EN repression of ci	Moderate	—
κ PTCCI	PTC stimulation of CI cleavage	Strong	Strong
κ ENhh	EN activation of hh	Weak	—
κ CNhh	CN repression of hh	Strong	—
C CI	Maximal cleavage rate of CI	Rapid	Rapid
H IWG	Half-life of intracellular WG	Short	Short
r endo WG	Rate of WG endocytosis	Slow	—
r exo WG	Rate of WG exocytosis	Moderately slow	Moderately fast
r MxferWG	Rate of WG cell-to-cell exchange	Slow	Slow
α WGwg	Maximal WG autocatalytic rate	—	Moderately rapid

Total number of parameters generated in G: 33276

A very similar analysis was performed by von Dassow and Odell [20], who also plotted the distribution of their family of feasible parameters to determine possible constraints for each parameter. Overall, our results agree very well with those of von Dassow and Odell: most tendencies found by these authors (see Fig. 6 and Table 1 of [20]) are confirmed by our parameter analysis. There are only six exceptions, where our analysis showed no tendency (compare columns 3 and 4 of Table 2), suggesting that these five parameters can, in fact, take values in a larger set, implying that the parameter space is larger than estimated in [20]. From these exceptions, κ_ENci, κ_ENhh, κ_CNhh, and r_{endo WG} all belong to the group of parameters which can be freely chosen. The other parameters are κ_CNwg, κ_CIwg, which depend on the U_i regions, and again our analysis shows that this pair has no preferred tendency.

A more detailed examination of the conditions on κ_CIwg and κ_CNwg turns out to be very illuminating. First, note that κ_CIwg and κ_CNwg actually define the pathway through which wingless is activated. That is, in each of the regions G_Auto, G_CI or G_{CI, WG}, κ_CIwg and κ_CNwg belong to distinct intervals as a function of U_i. Thus, it may be expected that the distribution of these parameters varies in each region. By plotting the histograms for κ_CIwg and κ_CNwg for each region alone (Fig. 5), we note that these show a marked tendency only outside region G_Auto, for low κ_CIwg and high κ_CNwg. The tendency of κ_CIwg and κ_CNwg outside G_Auto is, however, the opposite of that observed by von Dassow and Odell, a fact that can be explained once again by the “second missing link”. Since the volume of G_Auto is about 96% of G, it dominates the overall tendency. Indeed, since all feasible parameter sets in [3,20] were found only after adding the autocatalytic wingless activation link, it can be inferred that those parameters belong to region G_Auto.

Fig. 5.

Comparison of histograms for the pair κ_CIwg, κ_CNwg, in the whole feasible parameter space (G) and in the regions corresponding to the auto-catalytic pathway (G_Auto), or the cubitus pathway (G_CI + G_{CI, WG}). The x-axis is shown in log₁₀ scale

7 Discussion and conclusions

Analysis of the feasible parameter set, by estimating its volume, identifying connected components, and studying its geometric properties, are valuable tools for establishing and quantifying robustness in regulatory networks. The concept of robustness, in the sense that the system's regulatory functions should operate correctly under a variety of situations, is closely related to the parameter space and the effect of parameter perturbations. In this context, our analysis of the model of the segment polarity network proposed in [3] shows that its feasible parameter space is composed of a single connected component, indicating a high robustness. However, we have found one topological and one geometric property which may contribute to lower robustness. First, there are two distinct regions in the parameter space (which correspond to two adjacent “cubes” , see Fig. 3) associated with two different biological pathways: either auto-catalytic activation of wingless or activation by cubitus proteins only. Second, we have identified two lower dimensional planes of critical parameters (Ũ₂ = Ũ₁ and Ũ₂ = Ũ₄), where the model may fail to generate the wild type pattern. Since they form a lower dimensional space, these critical parameters are not likely to be an operating mode of the network. Nevertheless, the critical planes separate part of the feasible parameter space into four regions. An implication of this topological characterization is a diminished capacity of the network to respond well to environmental perturbations. Random fluctuations may drive the system through one of the critical planes, and possibly lead to a break down of the network or a different phenotype.

The reason why the planes Ũ₂ = Ũ₁ and Ũ₂ = Ũ₄ are critical can be traced in large part to an incompatibility of cubitus repression functions in the second cell: CN₂ should be present to repress engrailed expression, but should be absent to enhance CI₂ activation of wingless. To increase the network's robustness to environmental fluctuations, the segment polarity model should account for engrailed regulation by other factor than cubitus. One possibility is to include regulation by pair-rule gene products, such as sloppy paired, as explored both in [8,9]. An external factor, again possibly from the pair-rule genes, will also play a major role in establishing asymmetry in the cubitus levels (U_i). These contribute to a larger admissible parameter space, and together with an improved engrailed regulation, will greatly enhance robustness of the segment polarity network in maintaining its pattern. An extension of the current analysis including the regulation by sloppy paired can be found in [15].

Comparing the volume estimates for the regions G_Auto and G_CI or G_{CI, WG} shows that the first accounts for about 96% of the total feasible volume. Thus it seems much more likely that wild type expression in this model of the segment polarity network is achieved through the wingless auto-activation pathway. In the absence of the auto-activation link, von Dassow et al. failed to observe any feasible parameter set in their numerical experiments. However, as soon as the auto-activation pathway was added (the second “missing link” in the model [3]), immediately a significant percentage of feasible parameter sets were observed. This is not surprising, as elucidated by our analysis: while wingless auto-activation is not strictly necessary to establishing the segment polarity genes pattern, it does greatly increase the probability that the pattern is achieved, by increasing the volume of the feasible parameter space. At the same time, the parameter histogram for the activity threshold κ_WGwg is very sharp, when compared with the other parameters (and our results are in clear agreement with the original study by von Dassow et al. [20]). This indicates that fine tuning of κ_WGwg is essential to maintenance of the asymmetric wg pattern. Note that (cf. Theorem 1) wingless protein concentration is only sligthly higher in the wg-expressing cell (2nd cell) than in its two immediate neighbours (1st and 3rd cells). Thus, fine tuning of κ_WGwg is necessary to promote auto-activation in the 2nd cell but prevent auto-activation in the 1st and 3rd cells.

The analysis developed in this paper can be applied to other systems and regulatory networks, to systematically characterize and explore the admissible space of parameters, its topology and geometry. The method presented here assumes there is a (fixed) set of target states or “pattern” (ω) to be reproduced (or avoided) by the system, typically a desired steady state of the system. This pattern should also satisfy a family of algebraic equations (x = f(x; p)), on the variables and parameters of system. These equations can be those characterizing the system at steady state for instance, but can also include other constraints as long as they are written in this form. To obtain a family of equations that is easier to deal with, the functions f(x; p) may be simplified using reasonable approximations. For instance, sigmoidal type functions may be approximated by piecewise constant step functions. These equations are then symbolically solved with respect to the parameters. As a result, one obtains a family of inequalities characterizing the set of parameters compatible with the desired pattern and constraints. It is not guaranteed that an nonempty set of compatible parameters exists, as this depends on the constraints. Computation of a cylindrical algebraic decomposition is the main difficulty of this method. In general one may expect that it will work best with smaller/medium systems (on the order of 10–20 variables).

Our results emphasize that robustness of a regulatory module should not be measured simply as a function of the volume of its admissible parameter space. The geometry (for instance, convexity or existence of sharp points) and topology (connectedness) of the parameter space play fundamental roles in measuring robustness. These provide reliable information on how the network's interactions contribute to its robustness or fragility, and serve as measures to classify robust regulatory modules.

Appendix A: Notation

The original model can be found in [3,20]. In order to make our work more clear, we include the notation as well as the original equations below. Without loss of generality (the geometry remains unchanged), each cell is assumed to have four faces (Fig. 6), rather than six as in the original model [3]. The model reproduces a parasegment of four cells and uses repetition of this group of four cells to reproduce the embryo's anterior/posterior axis (A/P axis in Fig. 6), and the circular ventral/dorsal axis (V/D axis in Fig. 6). Because intercellular diffusion is only considered along the A/P axis (left/right), and because cells repeat in the orthogonal V/D direction (up/down), it is indeed equivalent to consider symmetric four-sided or six-sided hexagonal cells.

A saturation function, and its horizontal reflexion, are introduced:

$$\begin{array}{cc}\hfill \varphi (X,\kappa ,\nu )& =\frac{X^{\nu }}{{\kappa }^{\nu }+X^{\nu }},\hfill \\ \hfill \psi (X,\kappa ,\nu )& =1-\varphi (X,\kappa ,\nu ).\hfill \end{array}$$

The subscripted variables are as follows:

$$\begin{array}{cc}\hfill X_i& =\text{concentration of species}X\text{on cell}i\left(\text{when homogeneous throughout the cell}\right),\hfill \\ \hfill X_{i,j}& =\text{concentration of species}X\text{on cell}i,\text{at face}j,\hfill \\ \hfill {\kappa }_{XY}& =\text{threshold for activation of species}Y,\text{induced by species}X,\hfill \\ \hfill n(i,j)& =\text{index of neighbor to cell}i,\text{at face}j,\hfill \\ \hfill X_{n(i,j),j+2}& =\text{concentration of species}X\text{on cell face apposite to}i,j,\hfill \\ \hfill X_{i,\mathrm{T}}& =\sum _{j=1}^4{X}_{i,j}=\text{total concentration of species}X\text{on cell}i,\hfill \\ \hfill X_{\underset{‒}{i}}& =\sum _{j=1}^4{X}_{n(i,j),j+2}=\text{total concentration of species}X\text{presented to cell}i\text{by its neighbors}.\hfill \end{array}$$

Appendix B: Original equations

From [3,20], the model equations are:

$$\frac{de{n}_i}{dt}=\frac{1}{H_{en}}(\varphi ({\mathrm{EWG}}_{\underset{‒}{i}}\psi ({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}en},{\nu }_{\mathrm{CN}en}){\kappa }_{\mathrm{WG}en},{\nu }_{\mathrm{WG}en})-e{n}_i)$$ (28)

$$\frac{d{\mathrm{EN}}_i}{dt}=\frac{1}{H_{\mathrm{EN}}}(e{n}_i-{\mathrm{EN}}_i)$$ (29)

$$\frac{dw{g}_i}{dt}=\frac{1}{H_{wg}}\left(\frac{{\alpha }_{\mathrm{CI}wg}\varphi ({\mathrm{CI}}_i\psi ({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}wg},{\nu }_{\mathrm{CN}wg}),{\kappa }_{\mathrm{CI}wg},{\nu }_{\mathrm{CI}wg})+{\alpha }_{\mathrm{WG}wg}\varphi ({\mathrm{IWG}}_i,{\kappa }_{\mathrm{WG}wg},{\nu }_{\mathrm{WG}wg})}{1+{\alpha }_{\mathrm{CI}wg}\varphi ({\mathrm{CI}}_i\psi ({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}wg},{\nu }_{\mathrm{CN}wg}),{\kappa }_{\mathrm{CI}wg},{\nu }_{\mathrm{CI}wg})+{\alpha }_{\mathrm{WG}wg}\varphi ({\mathrm{IWG}}_i,{\kappa }_{\mathrm{WG}wg},{\nu }_{\mathrm{WG}wg})}-w{g}_i\right)$$ (30)

$$\frac{d{\mathrm{IWG}}_i}{dt}=\frac{1}{H_{\mathrm{WG}}}(w{g}_i-{\mathrm{IWG}}_i+r_{\text{endo}}{H}_{\mathrm{IWG}}{\mathrm{EWG}}_{i,\mathrm{T}}-H_{\mathrm{WG}}{r}_{\text{exo}}{\mathrm{IWG}}_i)$$ (31)

$$\frac{d{\mathrm{EWG}}_{i,j}}{dt}=\frac{1}{4}{r}_{\text{exo}}{\mathrm{IWG}}_i-r_{\text{endo}}{\mathrm{EWG}}_{i,j}+r_M({\mathrm{EWG}}_{n(i,j),j+2}-{\mathrm{EWG}}_{i,j})+r_{LM}({\mathrm{EWG}}_{i,j-1}+{\mathrm{EWG}}_{i,j+1}-2{\mathrm{EWG}}_{i,j})-\frac{{\mathrm{EWG}}_{i,j}}{H_{\mathrm{WG}}}$$ (32)

$$\frac{dpt{c}_i}{dt}=\frac{1}{H_{ptc}}(\varphi ({\mathrm{CI}}_i\psi ({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}ptc},{\nu }_{\mathrm{CN}ptc}),{\kappa }_{\mathrm{CI}ptc},{\nu }_{\mathrm{CI}ptc})-pt{c}_i)$$ (33)

$$\frac{d{\mathrm{PTC}}_{i,j}}{dt}=\frac{1}{H_{\mathrm{PTC}}}\left(\frac{1}{4}pt{c}_i-{\mathrm{PTC}}_{i,j}-{\kappa }_{\mathrm{PTCHH}}{H}_{\mathrm{PTC}}{\left[\mathrm{HH}\right]}_0{\mathrm{HH}}_{n(i,j),j+2}{\mathrm{PTC}}_{i,j}\right)+r_{LM\mathrm{PTC}}({\mathrm{PTC}}_{i,j-1}+{\mathrm{PTC}}_{i,j+1}-2{\mathrm{PTC}}_{i,j})$$ (34)

$$\frac{dc{i}_i}{dt}=\frac{1}{H_{ci}}(\varphi (B_i\psi ({\mathrm{EN}}_i,{\kappa }_{\mathrm{EN}ci},{\nu }_{\mathrm{EN}ci}),{\kappa }_{\mathrm{B}ci},{\nu }_{\mathrm{B}ci})-c{i}_i)$$ (35)

$$\frac{d{\mathrm{CI}}_i}{dt}=\frac{1}{H_{\mathrm{CI}}}(c{i}_i-{\mathrm{CI}}_i-H_{\mathrm{CI}}{C}_{\mathrm{CI}}{\mathrm{CI}}_i\varphi ({\mathrm{PTC}}_{i,\mathrm{T}},{\kappa }_{\mathrm{PTCCI}},{\nu }_{\mathrm{PTCCI}}))$$ (36)

$$\frac{d{\mathrm{CN}}_i}{dt}=\frac{1}{H_{\mathrm{CI}}}(H_{\mathrm{CI}}{C}_{\mathrm{CI}}{\mathrm{CI}}_i\varphi ({\mathrm{PTC}}_{i,\mathrm{T}},{\kappa }_{\mathrm{PTCCI}},{\nu }_{\mathrm{PTCCI}})-{\mathrm{CN}}_i)$$ (37)

$$\frac{dh{h}_i}{dt}=\frac{1}{H_{hh}}(\varphi ({\mathrm{EN}}_i\psi ({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}hh},{\nu }_{\mathrm{CN}hh}),{\kappa }_{\mathrm{EN}hh},{\nu }_{\mathrm{EN}hh})-h{h}_i)$$ (38)

$$\frac{d{\mathrm{HH}}_{i,j}}{dt}=\frac{1}{H_{\mathrm{HH}}}\left(\frac{1}{4}h{h}_i-{\mathrm{HH}}_{i,j}-{\kappa }_{\mathrm{PTCHH}}{H}_{\mathrm{HH}}{\left[\mathrm{PTC}\right]}_0{\mathrm{PTC}}_{n(i,j),j+2}{\mathrm{HH}}_{i,j}\right)+r_{LM\mathrm{HH}}({\mathrm{HH}}_{i,j-1}+{\mathrm{HH}}_{i,j+1}-2{\mathrm{HH}}_{i,j})$$ (39)

Appendix C: Simplified model, for large ν

Using the approximations (10) and (11) the right-hand side of the von Dassow et al. is simplified as follows:

$$f_{e{n}_i}=\frac{1}{H_{en}}({\theta }^{+}({\mathrm{EWG}}_{\underset{‒}{i}}{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}en}),{\kappa }_{\mathrm{WG}en})-e{n}_i)$$ (40)

$$f_{{\mathrm{EN}}_i}=\frac{1}{H_{\mathrm{EN}}}(e{n}_i-{\mathrm{EN}}_i)$$ (41)

$$f_{w{g}_i}=\frac{1}{H_{wg}}\left(\frac{{\alpha }_{\mathrm{CI}wg}{\theta }^{+}({\mathrm{CI}}_i{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}wg}),{\kappa }_{\mathrm{CI}wg})+{\alpha }_{\mathrm{WG}wg}{\theta }^{+}({\mathrm{IWG}}_i,{\kappa }_{\mathrm{WG}wg})}{1+{\alpha }_{\mathrm{CI}wg}{\theta }^{+}({\mathrm{CI}}_i{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}wg}),{\kappa }_{\mathrm{CI}wg})+{\alpha }_{\mathrm{WG}wg}{\theta }^{+}({\mathrm{IWG}}_i,{\kappa }_{\mathrm{WG}wg})}-w{g}_i\right)$$ (42)

$$f_{{\mathrm{IWG}}_i}=\frac{1}{H_{\mathrm{WG}}}(w{g}_i-{\mathrm{IWG}}_i+r_{\text{endo}}{H}_{\mathrm{IWG}}{\mathrm{EWG}}_{i,\mathrm{T}}-H_{\mathrm{WG}}{r}_{\text{exo}}{\mathrm{IWG}}_i)$$ (43)

$$f_{{\mathrm{EWG}}_{i,j}}=\frac{1}{4}{r}_{\text{exo}}{\mathrm{IWG}}_i-r_{\text{endo}}{\mathrm{EWG}}_{i,j}+r_M({\mathrm{EWG}}_{n(i,j),j+2}-{\mathrm{EWG}}_{i,j})+r_{LM}({\mathrm{EWG}}_{i,j-1}+{\mathrm{EWG}}_{i,j+1}-2{\mathrm{EWG}}_{i,j})-\frac{{\mathrm{EWG}}_{i,j}}{H_{\mathrm{WG}}}$$ (44)

$$f_{pt{c}_i}=\frac{1}{H_{ptc}}({\theta }^{+}({\mathrm{CI}}_i{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}ptc}),{\kappa }_{\mathrm{CI}ptc})-pt{c}_i)$$ (45)

$$f_{{\mathrm{PTC}}_{i,j}}=\frac{1}{H_{\mathrm{PTC}}}\left(\frac{1}{4}pt{c}_i-{\mathrm{PTC}}_{i,j}-{\kappa }_{\mathrm{PTCHH}}{H}_{\mathrm{PTC}}{\left[\mathrm{HH}\right]}_0{\mathrm{HH}}_{n(i,j),j+2}{\mathrm{PTC}}_{i,j}\right)+r_{LM\mathrm{PTC}}({\mathrm{PTC}}_{i,j-1}+{\mathrm{PTC}}_{i,j+1}-2{\mathrm{PTC}}_{i,j})$$ (46)

$$f_{c{i}_i}=\frac{1}{H_{ci}}(U_i{\theta }^{-}({\mathrm{EN}}_i,{\kappa }_{\mathrm{EN}ci})-c{i}_i)$$ (47)

$$f_{{\mathrm{CI}}_i}=\frac{1}{H_{\mathrm{CI}}}(c{i}_i-{\mathrm{CI}}_i-H_{\mathrm{CI}}{C}_{\mathrm{CI}}{\mathrm{CI}}_i{\theta }^{+}({\mathrm{PTC}}_{i,\mathrm{T}},{\kappa }_{\mathrm{PTCCI}}))$$ (48)

$$f_{{\mathrm{CN}}_i}=\frac{1}{H_{\mathrm{CI}}}(H_{\mathrm{CI}}{C}_{\mathrm{CI}}{\mathrm{CI}}_i{\theta }^{+}({\mathrm{PTC}}_{i,\mathrm{T}},{\kappa }_{\mathrm{PTCCI}})-{\mathrm{CN}}_i)$$ (49)

$$f_{h{h}_i}=\frac{1}{H_{hh}}({\theta }^{+}({\mathrm{EN}}_i{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}hh}),{\kappa }_{\mathrm{EN}hh})-h{h}_i)$$ (50)

$$f_{{\mathrm{HH}}_{i,j}}=\frac{1}{H_{\mathrm{HH}}}\left(\frac{1}{4}h{h}_i-{\mathrm{HH}}_{i,j}-{\kappa }_{\mathrm{PTCHH}}{H}_{\mathrm{HH}}{\left[\mathrm{PTC}\right]}_0{\mathrm{PTC}}_{n(i,j),j+2}{\mathrm{HH}}_{i,j}\right)+r_{LM\mathrm{HH}}({\mathrm{HH}}_{i,j-1}+{\mathrm{HH}}_{i,j+1}-2{\mathrm{HH}}_{i,j}).$$ (51)

To obtain Eq. (47) note that, in the original model of von Dassow et al., ci is written in terms of a constant forcing B_i:

$$c{i}_i={\theta }^{+}(B_i{\theta }^{-}({\mathrm{EN}}_i,{\kappa }_{\mathrm{EN}ci}),{\kappa }_{\mathrm{B}ci}),$$

where B_i is itself a parameter. Our definition of step function allows to merge the two parameters B_i and κ_Bci into one single parameter U_i by letting

$$U_i\in \{\begin{array}{cc}\left\{0\right\},\hfill & B_i<{\kappa }_{\mathrm{B}ci}\hfill \\ [0,1],\hfill & B_i={\kappa }_{\mathrm{B}ci}\hfill \\ \left\{1\right\},\hfill & B_i>{\kappa }_{\mathrm{B}ci}.\hfill \end{array}\phantom{\}}$$

More generality is obtained by assuming that B_i = κ_Bci, and always allowing U_i ∈ [0, 1].

Appendix D: Steady state equations

Solving equations (28)–(39) at steady state (dx/dt = f(x) = 0), and using the approximations (10), (11), yields the algebraic expressions:

$$e{n}_i={\theta }^{+}({\mathrm{EWG}}_{\underset{‒}{i}}{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}en}),{\kappa }_{\mathrm{WG}en})$$ (52)

$${\mathrm{EN}}_i=e{n}_i$$ (53)

$$w{g}_i=\frac{{\alpha }_{\mathrm{CI}wg}{\theta }^{+}({\mathrm{CI}}_i{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}wg}),{\kappa }_{\mathrm{CI}wg})+{\alpha }_{\mathrm{WG}wg}{\theta }^{+}({\mathrm{IWG}}_i,{\kappa }_{\mathrm{WG}wg})}{1+{\alpha }_{\mathrm{CI}wg}{\theta }^{+}({\mathrm{CI}}_i{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}wg}),{\kappa }_{\mathrm{CI}wg})+{\alpha }_{\mathrm{WG}wg}{\theta }^{+}({\mathrm{IWG}}_i,{\kappa }_{\mathrm{WG}wg})}$$ (54)

$${\mathrm{IWG}}_i=\frac{H_{\mathrm{IWG}}{r}_{\text{endo}}}{1+H_{\mathrm{IWG}}{r}_{\text{exo}}}{\mathrm{EWG}}_{i,T}+\frac{1}{1+H_{\mathrm{IWG}}{r}_{\text{exo}}}w{g}_i$$ (55)

$$M\mathrm{EWG}=-\frac{1}{4}\frac{r_{\text{exo}}}{1+H_{\mathrm{IWG}}{r}_{\text{exo}}}\widetilde{wg}$$ (56)

$$pt{c}_i={\theta }^{+}({\mathrm{CI}}_i{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}ptc}),{\kappa }_{\mathrm{CI}ptc})$$ (57)

$${\mathrm{PTC}}_{i,j}=\frac{1}{4}pt{c}_i-{\kappa }_{\mathrm{PTCHH}}{H}_{\mathrm{PTC}}{\left[\mathrm{HH}\right]}_0{\mathrm{HH}}_{n(i,j),j+2}{\mathrm{PTC}}_{i,j}+r_{LM\mathrm{PTC}}{H}_{\mathrm{PTC}}({\mathrm{PTC}}_{i,j-1}+{\mathrm{PTC}}_{i,j+1}-2{\mathrm{PTC}}_{i,j})$$ (58)

$$c{i}_i=U_i{\theta }^{-}({\mathrm{EN}}_i,{\kappa }_{\mathrm{EN}ci})$$ (59)

$${\mathrm{CI}}_i=U_i\frac{1}{1+H_{\mathrm{CI}}{C}_{\mathrm{CI}}{\theta }^{+}({\mathrm{PTC}}_{i,\mathrm{T}},{\kappa }_{\mathrm{PTCCI}})}{\theta }^{-}({\mathrm{EN}}_i,{\kappa }_{\mathrm{EN}ci})$$ (60)

$${\mathrm{CN}}_i=U_i\frac{H_{\mathrm{CI}}{C}_{\mathrm{CI}}{\theta }^{+}({\mathrm{PTC}}_{i,\mathrm{T}},{\kappa }_{\mathrm{PTCCI}})}{1+H_{\mathrm{CI}}{C}_{\mathrm{CI}}{\theta }^{+}({\mathrm{PTC}}_{i,\mathrm{T}},{\kappa }_{\mathrm{PTCCI}})}{\theta }^{-}({\mathrm{EN}}_i,{\kappa }_{\mathrm{EN}ci})$$ (61)

$$h{h}_i={\theta }^{+}({\mathrm{EN}}_i{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}hh}),{\kappa }_{\mathrm{EN}hh})$$ (62)

$${\mathrm{HH}}_{i,j}=\frac{1}{4}h{h}_i-{\kappa }_{\mathrm{PTCHH}}{H}_{\mathrm{HH}}{\left[\mathrm{PTC}\right]}_0{\mathrm{PTC}}_{n(i,j),j+2}{\mathrm{HH}}_{i,j}+r_{LM\mathrm{HH}}{H}_{\mathrm{HH}}({\mathrm{HH}}_{i,j-1}+{\mathrm{HH}}_{i,j+1}-2{\mathrm{HH}}_{i,j})$$ (63)

EWG is a vector in ${\mathbb{R}}^{16}$ with components:

$$\mathrm{EWG}={({\mathrm{EWG}}_{1,1},{\mathrm{EWG}}_{1,2},{\mathrm{EWG}}_{1,3},{\mathrm{EWG}}_{1,4},{\mathrm{EWG}}_{2,1},{\mathrm{EWG}}_{2,2},{\mathrm{EWG}}_{2,3},{\mathrm{EWG}}_{2,4},{\mathrm{EWG}}_{3,1},{\mathrm{EWG}}_{3,2},{\mathrm{EWG}}_{3,3},{\mathrm{EWG}}_{3,4},{\mathrm{EWG}}_{4,1},{\mathrm{EWG}}_{4,2},{\mathrm{EWG}}_{4,3},{\mathrm{EWG}}_{4,4})}^{\prime }$$

$\widetilde{wg}$ is also a vector in ${\mathbb{R}}^{16}$, given by the following Kronecker tensor product

$$\begin{array}{cc}\hfill \widetilde{wg}& ={(w{g}_1,w{g}_2,w{g}_3,w{g}_4)}^{\prime }{\times }_{\text{kron}}{(1,1,1,1)}^{\prime }\hfill \\ \hfill & ={(w{g}_1,w{g}_1,w{g}_1,w{g}_1,w{g}_2,w{g}_2,w{g}_2,w{g}_2,w{g}_3,w{g}_3,w{g}_3,w{g}_3,w{g}_4,w{g}_4,w{g}_4,w{g}_4)}^{\prime }.\hfill \end{array}$$

Putting together the 16 equations (56), and substituting IWG_i by its steady state expression (55), it is not difficult to see that the matrix $M\in {\mathbb{R}}^{16}\times {\mathbb{R}}^{16}$ is composed of various 4 × 4 blocks, as follows:

$$M=\left(\begin{array}{cccc}\hfill E\hfill & \hfill F_{24}\hfill & \hfill 0\hfill & \hfill F_{42}\hfill \\ \hfill F_{42}\hfill & \hfill E\hfill & \hfill F_{24}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill F_{42}\hfill & \hfill E\hfill & \hfill F_{24v}\hfill \\ \hfill F_{24}\hfill & \hfill 0\hfill & \hfill F_{42}\hfill & \hfill E\hfill \end{array}\right)$$ (64)

where

$$E=\left(\begin{array}{cccc}\hfill -d\hfill & \hfill r_{LM}\hfill & \hfill r_M\hfill & \hfill r_{LM}\hfill \\ \hfill r_{LM}\hfill & \hfill -d\hfill & \hfill r_{LM}\hfill & \hfill 0\hfill \\ \hfill r_M\hfill & \hfill r_{LM}\hfill & \hfill -d\hfill & \hfill r_{LM}\hfill \\ \hfill r_{LM}\hfill & \hfill 0\hfill & \hfill r_{LM}\hfill & \hfill -d\hfill \end{array}\right)+h\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right)$$

with

$$d=H_{\mathrm{IWG}}^{-1}+r_{\text{endo}}+r_M+2r_{LM},$$

$$h=\frac{1}{4}\frac{H_{\mathrm{IWG}}{r}_{\text{exo}}}{1+H_{\mathrm{IWG}}{r}_{\text{exo}}}{r}_{\text{endo}}$$

$$F_{24}=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill r_M\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),F_{42}=F_{24}^{\prime }=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill r_M\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Note that the steady state equations for EN, IWG, EWG and PTC are algebraic, and in fact exact solutions can be computed from the steady state values of wg and ptc. These are discussed in more detail in the Appendices E and F.

Remark The parameters are as in [3], except U_i, which represent the maximal values of ci, in each cell. These take values in the interval [0, 1] and generalize the possible ON values of ci (see explanation in Appendix C).

Appendix E: Analytically solving Wingless levels

The steady states of Wingless proteins (55) and (56) are given directly by algebraic equations, depending only on wingless mRNA (wg₂) and diffusion parameters for intracellular (membrane-to-membrane) and intercellular communication. Consider Eq. (56): it is easy to see that M is in fact always invertible (if all parameters are positive). First note that the matrix is diagonally dominant, by adding up the entries in any column:

$$-\left(H_{\mathrm{IWG}}^{-1}+r_{\text{endo}}+r_M+2r_{LM}\right)+2r_{LM}+r_M+4h=-H_{\mathrm{IWG}}^{-1}-r_{\text{endo}}\frac{1}{1+H_{\mathrm{IWG}}{r}_{\text{exo}}}$$

which is always a negative quantity. By Geršgorin's Theorem, all eigenvalues of M are contained in the disk centered at −d + h with radius 2r_LM + r_M + 3h, therefore all have negative real parts. Thus, the matrix M is symmetric and negative definite, and since the right-hand side vector is also non-positive, all solutions are real and positive, whatever the choice of parameters. As a fact, note that the vector $\overrightarrow{1}={(1,1,\dots ,1)}^{\prime }\in {\mathbb{R}}^{16}$ is an eigenvector of M, corresponding to the eigenvalue ${\lambda }_1=-H_{\mathrm{IWG}}^{-1}-r_{\text{endo}}\frac{1}{H_{\mathrm{IWG}}+r_{\text{exo}}}$.

The solution of Eqs. (56) and (55) can be written in terms of wg = (wg₁, wg₂, wg₃, wg₄) as described next.

Lemma E.1 There exist constants β_max > β_med > β_min > 0 and γ_max > γ_med > γ_min > 0 such that:

$$\begin{array}{cc}\hfill EW{G}_{\underset{‒}{1}}& =w{g}_1{\beta }_{\text{max}}+(w{g}_2+w{g}_4){\beta }_{\text{med}}+w{g}_3{\beta }_{\text{min}},\hfill \\ \hfill EW{G}_{\underset{‒}{2}}& =w{g}_2{\beta }_{\text{max}}+(w{g}_1+w{g}_3){\beta }_{\text{med}}+w{g}_4{\beta }_{\text{min}},\hfill \\ \hfill EW{G}_{\underset{‒}{3}}& =w{g}_3{\beta }_{\text{max}}+(w{g}_2+w{g}_4){\beta }_{\text{med}}+w{g}_1{\beta }_{\text{min}},\hfill \\ \hfill EW{G}_{\underset{‒}{4}}& =w{g}_4{\beta }_{\text{max}}+(w{g}_1+w{g}_3){\beta }_{\text{med}}+w{g}_2{\beta }_{\text{min}}.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill IW{G}_1& =w{g}_1{\gamma }_{\text{max}}+(w{g}_2+w{g}_4){\gamma }_{\text{med}}+w{g}_3{\gamma }_{\text{min}},\hfill \\ \hfill IW{G}_2& =w{g}_2{\gamma }_{\text{max}}+(w{g}_1+w{g}_3){\gamma }_{\text{med}}+w{g}_4{\gamma }_{\text{min}},\hfill \\ \hfill IW{G}_3& =w{g}_3{\gamma }_{\text{max}}+(w{g}_2+w{g}_4){\gamma }_{\text{med}}+w{g}_1{\gamma }_{\text{min}},\hfill \\ \hfill IW{G}_4& =w{g}_4{\gamma }_{\text{max}}+(w{g}_1+w{g}_3){\gamma }_{\text{med}}+w{g}_2{\gamma }_{\text{min}}.\hfill \end{array}$$

Proof Observe that (56) can be written:

$$\begin{array}{cc}\hfill \overrightarrow{E}& =-\frac{1}{4}\frac{r_{\text{exo}}}{1+H_{\mathrm{IWG}}{r}_{\text{exo}}}{M}^{-1}\widetilde{wg}\hfill \\ \hfill & =-w{g}_{1}c{M}^{-1}\left(\begin{array}{c}\hfill \overrightarrow{1}\hfill \\ \hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{0}\hfill \end{array}\right)-w{g}_{2}c{M}^{-1}\left(\begin{array}{c}\hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{1}\hfill \\ \hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{0}\hfill \end{array}\right)-w{g}_{3}c{M}^{-1}\left(\begin{array}{c}\hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{1}\hfill \\ \hfill \overrightarrow{0}\hfill \end{array}\right)-w{g}_{4}c{M}^{-1}\left(\begin{array}{c}\hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{1}\hfill \end{array}\right)\hfill \end{array}$$ (65)

where $c=\frac{1}{4}\frac{r_{\text{exo}}}{1+H_{\mathrm{IWG}}{r}_{\text{exo}}}$, $\overrightarrow{E}\in {\mathbb{R}}_{\ge 0}^{16}$, $\overrightarrow{1}={(1,1,1,1)}^{\prime }$, and $\overrightarrow{0}={(0,0,0,0)}^{\prime }$. Defining

$$L=\left(\begin{array}{cccccccccccccccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$$

one obtains:

$$\left(\begin{array}{c}\hfill {\mathrm{EWG}}_{\underset{‒}{1}}\hfill \\ \hfill {\mathrm{EWG}}_{\underset{‒}{2}}\hfill \\ \hfill {\mathrm{EWG}}_{\underset{‒}{3}}\hfill \\ \hfill {\mathrm{EWG}}_{\underset{‒}{4}}\hfill \end{array}\right)=L\overrightarrow{E}.$$

Theorem 1 (proved below) is equivalent to saying that:

$$X=-cL{M}^{-1}\left(\begin{array}{c}\hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{1}\hfill \\ \hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{0}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {\beta }_{\text{med}}\hfill \\ \hfill {\beta }_{\text{max}}\hfill \\ \hfill {\beta }_{\text{med}}\hfill \\ \hfill {\beta }_{\text{min}}\hfill \end{array}\right)$$

where: β_max > β_med > β_min > 0. In fact, X (EWG₁, . . ., EWG₄) corresponds to the solution of (56) when wg = (0, 1, 0, 0), in which case EWG₂ = β_max, EWG₁ = EWG₃ = β_med, and EWG₄ = β_min.

By symmetry of the system, it is easy to see that

$$-cL{M}^{-1}\left(\begin{array}{c}\hfill \overrightarrow{1}\hfill \\ \hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{0}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {\beta }_{\text{max}}\hfill \\ \hfill {\beta }_{\text{med}}\hfill \\ \hfill {\beta }_{\text{min}}\hfill \\ \hfill {\beta }_{\text{med}}\hfill \end{array}\right)$$

and so on (by circulating the positions of the β_*). Linear combination gives the general solution for the EWG_i.

Similarly, to compute the solution of (55), define

$$Q=\left(\begin{array}{cccccccccccccccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right)$$

and recall that

$$\left(\begin{array}{c}\hfill {\mathrm{EWG}}_{1,T}\hfill \\ \hfill {\mathrm{EWG}}_{2,T}\hfill \\ \hfill {\mathrm{EWG}}_{3,T}\hfill \\ \hfill {\mathrm{EWG}}_{4,T}\hfill \end{array}\right)=Q\overrightarrow{E}.$$

Now consider the solution for the case wg = (0, 1, 0, 0):

$$Y=-cQ{M}^{-1}\left(\begin{array}{c}\hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{1}\hfill \\ \hfill \overrightarrow{0}\hfill \\ \hfill \overrightarrow{0}\hfill \end{array}\right)$$

Using Facts E.1 and E.2, and then E.4 and E.6 (below), we define contains ${\widehat{\gamma }}_{\text{max}}>{\widehat{\gamma }}_{\text{med}}>{\widehat{\gamma }}_{\text{min}}>0$:

$$\begin{array}{cc}\hfill Y_1& =2E_{1,1}+E_{1,2}+E_{1,4}≔{\widehat{\gamma }}_{\text{med}},\hfill \\ \hfill Y_2& =2E_{2,1}+2E_{2,2}≔{\widehat{\gamma }}_{\text{max}},\hfill \\ \hfill Y_3& =2E_{1,1}+E_{1,2}+E_{1,4}≔{\widehat{\gamma }}_{\text{med}},\hfill \\ \hfill Y_4& =2E_{4,1}+2E_{4,2}≔{\widehat{\gamma }}_{\text{min}},\hfill \end{array}$$

Substituion into (55) yields:

$$\left(\begin{array}{c}\hfill Z_1\hfill \\ \hfill Z_2\hfill \\ \hfill Z_3\hfill \\ \hfill Z_4\hfill \end{array}\right)=\frac{H_{\mathrm{IWG}}{r}_{\text{endo}}}{1+H_{\mathrm{IWG}}{r}_{\text{exo}}}\left(\begin{array}{c}\hfill Y_1\hfill \\ \hfill Y_2\hfill \\ \hfill Y_3\hfill \\ \hfill Y_4\hfill \end{array}\right)+\frac{1}{1+H_{\mathrm{IWG}}{r}_{\text{exo}}}\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \widehat{c}{\widehat{\gamma }}_{\text{med}}\hfill \\ \hfill \tilde{c}+\widehat{c}{\widehat{\gamma }}_{\text{max}}\hfill \\ \hfill \widehat{c}{\widehat{\gamma }}_{\text{med}}\hfill \\ \hfill \widehat{c}{\widehat{\gamma }}_{\text{min}}\hfill \end{array}\right)$$

where $\widehat{c}=\frac{H_{\mathrm{IWG}}{r}_{\text{endo}}}{1+H_{\mathrm{IWG}}{r}_{\text{exo}}}$ and $\tilde{c}=\frac{1}{1+H_{\mathrm{IWG}}{r}_{\text{exo}}}$. Define:

$${\gamma }_{\text{max}}=\tilde{c}+\widehat{c}{\widehat{\gamma }}_{\text{max}},{\gamma }_{\text{med}}=\widehat{c}{\widehat{\gamma }}_{\text{med}},{\gamma }_{\text{min}}=\widehat{c}{\widehat{\gamma }}_{\text{min}}.$$

The solutions for the cases wg = (1, 0, 0, 0), wg = (0, 0, 1, 0), wg = (0, 0, 0, 1) are analogous (circulating the positions of the γ_*). An appropriate linear combination yields the desired result for IWG. □

Proof of Theorem 1 First, we state Facts E.1 and E.2, which hold for any wg.

Fact E.1 For all i = 1, 2, 3, 4 it holds that

$${\mathrm{EWG}}_{i,1}={\mathrm{EWG}}_{i,3}.$$

Proof This is easy to see from the respective equations:

$$(-d+h){\mathrm{EWG}}_{i,1}+(r_M+h){\mathrm{EWG}}_{i,2}+(r_M+h){\mathrm{EWG}}_{i,3}+(r_M+h){\mathrm{EWG}}_{i,4}=-\frac{h}{r_{\text{endo}}{H}_{\mathrm{IWG}}}w{g}_i$$

$$(r_M+h){\mathrm{EWG}}_{i,1}+(r_M+h){\mathrm{EWG}}_{i,2}+(-d+h){\mathrm{EWG}}_{i,3}+(r_M+h){\mathrm{EWG}}_{i,4}=-\frac{h}{r_{\text{endo}}{H}_{\mathrm{IWG}}}w{g}_i$$

which can be rearranged to

$$\begin{array}{cc}\hfill & -(d+r_M){\mathrm{EWG}}_{i,1}+(r_M+h)({\mathrm{EWG}}_{i,2}+{\mathrm{EWG}}_{i,4})+(r_M+h)\times ({\mathrm{EWG}}_{i,3}+{\mathrm{EWG}}_{i,1})=-\frac{h}{r_{\text{endo}}{H}_{\mathrm{IWG}}}w{g}_i\hfill \\ \hfill & -(d+r_M){\mathrm{EWG}}_{i,3}+(r_M+h)({\mathrm{EWG}}_{i,2}+{\mathrm{EWG}}_{i,4})+(r_M+h)\times ({\mathrm{EWG}}_{i,3}+{\mathrm{EWG}}_{i,1})=-\frac{h}{r_{\text{endo}}{H}_{\mathrm{IWG}}}w{g}_i.\hfill \end{array}$$ (66)

Subtracting these two equations yields the desired result. □

Fact E.2 It holds that

$${\mathrm{EWG}}_{2,2}={\mathrm{EWG}}_{2,4},{\mathrm{EWG}}_{4,2}={\mathrm{EWG}}_{4,4},{\mathrm{EWG}}_{1,2}={\mathrm{EWG}}_{3,4},{\mathrm{EWG}}_{1,4}={\mathrm{EWG}}_{3,2}.$$

Proof Exchanging the indexes:

$$2,2\leftrightarrow 2,44,2\leftrightarrow 4,41,2\leftrightarrow 3,41,4\leftrightarrow 3,2$$

it is easy to see that the system remains unchanged (see also Fig. 6). □

Assume now that wg = (0, w, 0, 0), for any w > 0. The equality part in (13) is now clear:

Fact E.3 EWG₁ = EWG₃.

Proof We first show that EWG_1,1 = EWG_3,3. Writing Eq. (66) for i = 1 and i = 3:

$$\begin{array}{cc}\hfill & -(d+r_M){\mathrm{EWG}}_{1,1}+(r_M+h)({\mathrm{EWG}}_{1,2}+{\mathrm{EWG}}_{1,4})+(r_M+h)({\mathrm{EWG}}_{1,3}+{\mathrm{EWG}}_{1,1})=0\hfill \\ \hfill & -(d+r_M){\mathrm{EWG}}_{3,3}+(r_M+h)({\mathrm{EWG}}_{3,2}+{\mathrm{EWG}}_{3,4})+(r_M+h)({\mathrm{EWG}}_{3,3}+{\mathrm{EWG}}_{3,1})=0\hfill \end{array}$$

Using Fact E.1 one has EWG_1,1 = EWG_1,3 and EWG_3,1 = EWG_3,3, and then using Fact E.2 obtains:

$$\begin{array}{cc}\hfill & -(d+r_M-2r_M-2h){\mathrm{EWG}}_{1,1}+(r_M+h)({\mathrm{EWG}}_{3,4}+{\mathrm{EWG}}_{3,2})=0\hfill \\ \hfill & -(d+r_M-2r_M-2h){\mathrm{EWG}}_{3,3}+(r_M+h)({\mathrm{EWG}}_{3,2}+{\mathrm{EWG}}_{3,4})=0.\hfill \end{array}$$

Subtracting these two equations shows that EWG_1,1 = EWG_3,3. Now recalling the notation for X_i from Appendix A

$$\begin{array}{cc}\hfill {\mathrm{EWG}}_{\underset{‒}{1}}& ={\mathrm{EWG}}_{1,1}+{\mathrm{EWG}}_{2,4}+{\mathrm{EWG}}_{1,3}+{\mathrm{EWG}}_{4,2}\hfill \\ \hfill {\mathrm{EWG}}_{\underset{‒}{3}}& ={\mathrm{EWG}}_{3,1}+{\mathrm{EWG}}_{4,4}+{\mathrm{EWG}}_{3,3}+{\mathrm{EWG}}_{2,2}.\hfill \end{array}$$

Using EWG_1,1 = EWG_3,3, Facts E.1 and E.2 obtains:

$${\mathrm{EWG}}_{\underset{‒}{1}}={\mathrm{EWG}}_{3,1}+{\mathrm{EWG}}_{2,2}+{\mathrm{EWG}}_{3,3}+{\mathrm{EWG}}_{4,4}={\mathrm{EWG}}_{\underset{‒}{3}}.$$

as we wanted to prove. □

To show the other inequalities, note first that the 16 variables EWG_i,j are thus reduced to only seven:

$$\begin{array}{cc}\hfill E_{1,1}& ={\mathrm{EWG}}_{1,1}={\mathrm{EWG}}_{1,3}={\mathrm{EWG}}_{3,1}={\mathrm{EWG}}_{3,3}\hfill \\ \hfill E_{1,2}& ={\mathrm{EWG}}_{1,2}={\mathrm{EWG}}_{3,4}\hfill \\ \hfill E_{1,4}& ={\mathrm{EWG}}_{1,4}={\mathrm{EWG}}_{3,2}\hfill \\ \hfill E_{2,1}& ={\mathrm{EWG}}_{2,1}={\mathrm{EWG}}_{2,3}\hfill \\ \hfill E_{2,2}& ={\mathrm{EWG}}_{2,2}={\mathrm{EWG}}_{2,4}\hfill \\ \hfill E_{4,1}& ={\mathrm{EWG}}_{4,1}={\mathrm{EWG}}_{4,3}\hfill \\ \hfill E_{4,2}& ={\mathrm{EWG}}_{4,2}={\mathrm{EWG}}_{4,4}\hfill \end{array}$$

and satisfy the equations:

$$-(d-r_M-2h)E_{1,1}+(r_{LM}+h)E_{1,2}+(r_{LM}+h)E_{1,4}=0$$ (67)

$$2(r_{LM}+h)E_{1,1}-(d-h)E_{1,2}+h{E}_{1,4}+r_M{E}_{2,2}=0$$ (68)

$$2(r_{LM}+h)E_{1,1}+h{E}_{1,2}-(d-h)E_{1,4}+r_M{E}_{4,2}=0$$ (69)

$$-(d-r_M-2h)E_{2,1}+2(r_{LM}+h)E_{2,2}=-\frac{h}{r_{\text{endo}}{H}_{\mathrm{IWG}}}w{g}_2$$ (70)

$$2(r_{LM}+h)E_{2,1}-(d-2h)E_{2,2}+r_M{E}_{1,2}=-\frac{h}{r_{\text{endo}}{H}_{\mathrm{IWG}}}w{g}_2$$ (71)

$$-(d-r_M-2h)E_{4,1}+2(r_{LM}+h)E_{4,2}=0$$ (72)

$$2(r_{LM}+h)E_{4,1}-(d-2h)E_{4,2}+r_M{E}_{1,4}=0.$$ (73)

To simplify notation, set:

$$A=d-r_M-2h,B=2(r_{LM}+h),\stackrel{‒}{w}=\frac{h}{r_{\text{endo}}{H}_{\mathrm{IWG}}}w{g}_2,$$

and note that A > B > 0.

Fact E.4 The following hold:

1. E_4,1 < E_4,2 < E_1,4 < E_1,2 < E_2,2;

2. E_4,1 < E_1,1 < E_1,2;

3. E_1,2 + E_1,4 < E_2,2 + E_4,2

Proof To prove part (a), from Eqs. (72) and (73) it holds that

$$E_{4,1}=\frac{B}{A}{E}_{4,2};E_{4,2}=\frac{r_{M}A}{A^2-B^2+r_{M}A}{E}_{1,4}$$

Because A > B > 0, it is clear that E_4,1 < E_4,2 < E_1,4. From Eqs. (70) and (71) it holds that

$$E_{2,1}=\frac{B}{A}{E}_{2,2}+\frac{1}{A}\stackrel{‒}{w};E_{2,2}=\frac{r_{M}A}{A^2-B^2+r_{M}A}{E}_{1,2}+\frac{A+B}{A^2-B^2+r_{M}A}\stackrel{‒}{w}$$

Then Eqs. (68) and (69) can be written in the form

$$\begin{array}{cc}\hfill \left(d-r_M\frac{r_{M}A}{A^2-B^2+r_{M}A}\right)E_{1,2}& =B{E}_{1,1}+h(E_{1,2}+E_{1,4})+r_M\frac{A+B}{A^2-B^2+r_{M}A}\stackrel{‒}{w}\hfill \\ \hfill \left(d-r_M\frac{r_{M}A}{A^2-B^2+r_{M}A}\right)E_{1,4}& =B{E}_{1,1}+h(E_{1,2}+E_{1,4})\hfill \end{array}$$

which implies that E_1,4 < E_1,2 (it is easy to see that the factor multiplying both E_1,2 and E_1,4 is positive, since d > r_M).

We still need to prove the last inequality in (a), but we can now prove (b). From Eq. (67)

$$E_{1,1}=\frac{1}{2}\frac{B}{A}(E_{1,2}+E_{1,4})<E_{1,2}$$

using (a) and because B < A. This proves the second inequality in (b). To prove (c), substitute this E_1,1 expression into the sum of Eqs. (68) and (69):

$$E_{2,2}+E_{4,2}=\frac{A^2-B^2+r_{M}A}{r_{M}A}(E_{1,2}+E_{1,4})>E_{1,2}+E_{1,4}.$$

The last part of (a) now follows from (c) together with E_4,2 < E_1,4, which implies E_1,2 < E_2,2.

Finally, the first part of (b) is easy to see from:

$$E_{1,1}-E_{4,1}=\frac{1}{2}\frac{B}{A}(E_{1,2}+E_{1,4})-\frac{B}{A}{E}_{4,2}>\frac{1}{2}\frac{B}{A}(E_{1,2}+E_{1,4}-E_{1,4})>0.$$

To prove the first inequality of Theorem 1 is now straighforward.

Fact E.5 EWG₄ < EWG₁

Proof Recall the notation for EWG_i and use Fact E.4

$$\begin{array}{cc}\hfill {\mathrm{EWG}}_{\underset{‒}{1}}-{\mathrm{EWG}}_{\underset{‒}{4}}& =2E_{1,1}+E_{2,2}+E_{4,2}-2E_{4,1}-2E_{1,4}\hfill \\ \hfill & =2(E_{1,1}-E_{4,1})+(E_{2,2}+E_{4,2}-E_{1,2}-E_{1,4})+(E_{1,2}-E_{1,4})>0.\hfill \end{array}$$

To prove the other inequality we need the next result.

Fact E.6 It holds that: E_2,1 > E_2,2.

Proof First notice that

$$E_{1,2}=r_M\frac{A+B}{d(A^2-B^2+r_{M}A)-r_M^{2}A}\stackrel{‒}{w}+\frac{1}{2}{r}_M\frac{A^2-B^2+r_{M}A}{d(A^2-B^2+r_{M}A)-r_M^{2}A}\frac{-A^2+B^2+(d-r_M)A}{(A-B)(A^2-B^2+2r_{M}A)}\stackrel{‒}{w}.$$

Now consider

$$E_{2,1}-E_{2,2}=-\frac{A-B}{A}\frac{r_{M}A}{A^2-B^2+r_{M}A}{E}_{1,2}-\frac{A-B}{A}\frac{A+B}{A^2-B^2+r_{M}A}\stackrel{‒}{w}+\frac{1}{A}\stackrel{‒}{w}.$$

The last two terms can be combined into

$$\frac{r_M}{A^2-B^2+r_{M}A}\stackrel{‒}{w},$$

and the two terms due to E_1,2 can be simplified to:

$$-r_M\frac{1}{A^2-B^2+r_{M}A}\frac{(A-B)(A+B)}{\frac{d}{r_M}(A^2-B^2+r_{M}A)-r_{M}A}\stackrel{‒}{w}$$

and

$$-\frac{r_M}{2}\frac{1}{A^2-B^2+2r_{M}A}\frac{-A^2+B^2+(d-r_M)A}{\frac{d}{r_M}(A^2-B^2+r_{M}A)-r_{M}A}\stackrel{‒}{w}.$$

Factoring out $r_M\stackrel{‒}{w}∕\left(\frac{d}{r_M}(A^2-B^2+r_{M}A)-r_{M}A\right)$, one obtains

$$\frac{1}{r_M\stackrel{‒}{w}}\left(\frac{d}{r_M}(A^2-B^2+r_{M}A)-r_{M}A\right)(E_{2,1}-E_{2,2})=\frac{1}{A^2-B^2+r_{M}A}\left(\frac{d}{r_M}(A^2-B^2+r_{M}A)-r_{M}A-(A^2-B^2)\right)-\frac{1}{2}\frac{-A^2+B^2+(d-r_M)A}{A^2-B^2+2r_{M}A}$$

which can be further simplified to

$$\frac{\left(\frac{d}{r_M}-1\right)(A^2-B^2)}{A^2-B^2+r_{M}A}+\frac{1}{2}\frac{A^2-B^2}{A^2-B^2+2r_{M}A}+\frac{(d-r_M)A}{A^2-B^2+r_{M}A}-\frac{1}{2}\frac{(d-r_M)A}{A^2-B^2+2r_{M}A}>0$$

because the first two terms are clearly positive, and the last two terms add up to a positive number. This shows that E_2,1 > E_2,2, as we wanted to prove. □

The proof of Theorem 1 can now be completed.

Fact E.7 EWG₁ < EWG₂.

Proof Consider:

$$\begin{array}{cc}\hfill {\mathrm{EWG}}_{\underset{‒}{2}}-{\mathrm{EWG}}_{\underset{‒}{1}}& =2E_{2,1}+2E_{1,2}-2E_{1,1}-E_{2,2}-E_{4,2}\hfill \\ \hfill & =2(E_{1,2}-E_{1,1})+(E_{2,1}-E_{2,2})+(E_{2,1}-E_{4,2}),\hfill \end{array}$$

which is positive because E_1,2 > E_1,1 [Lemma E.4(b)], and E_2,1 > E_2,2 > E_4,2 (Lemmas E.4(a) and E.6). □

Appendix F: Analytically solving PTC and HH levels

In this section, we prove uniqueness of solutions for PTC and HH given any set of parameters p_PTC−HH (see (16)), and hh = (0, 1, 0, 0) and ptc = (T₁, T₂, 0, T₄), with T₂ = T₄. The steady state levels of Patched and Hedgehog proteins are given by a system of nonlinear Eqs. (58) and (63). These equations can be solved explicitly and uniquely in the case ptc₂ = ptc₄ = T₂, which is true if the steady state is in $\mathcal{W}$. To simplify notation, we use

$$r_P=r_{LM\mathrm{PTC}},r_H=r_{LM\mathrm{HH}},{\kappa }_H={\kappa }_{\mathrm{PTCHH}}{\left[\mathrm{HH}\right]}_0,{\kappa }_P={\kappa }_{\mathrm{PTCHH}}{\left[\mathrm{PTC}\right]}_0,$$

and define

$$d_P=\frac{1}{H_{\mathrm{PTC}}}+2r_P,d_H=\frac{1}{H_{\mathrm{HH}}}+2r_H.$$

We introduce further notation:

$${\beta }_P=\frac{2r_P^2{d}_P}{d_P^2-2r_P^2},{\gamma }_P=\frac{1}{4H_{\mathrm{PTC}}}+\frac{1}{4H_{\mathrm{PTC}}}\frac{2r_P(r_P+d_P)}{d_P^2-2r_P^2}=\frac{1}{4H_{\mathrm{PTC}}}\frac{d_P(2r_P+d_P)}{d_P^2-2r_P^2}.$$

Lemma F.1 Let $x\in \mathcal{W}$. Then, the solution for HH is:

$$\begin{array}{cc}\hfill H{H}_{i,1}=H{H}_{i,2}=H{H}_{i,3}& =H{H}_{i,4}=0,i=1,2,4,\hfill \\ \hfill H{H}_{3,2}& =H{H}_{3,4}=Roo{t}_{+},\hfill \\ \hfill H{H}_{3,1}& =H{H}_{3,3}=\frac{1}{d_H}\left(\frac{1}{4}\frac{h{h}_3}{H_{HH}}+r_{H}H{H}_{3,2}+r_{H}H{H}_{3,4}\right),\hfill \end{array}$$

where Root₊ is the positive root of the quadratic equation:

$$k_H(d_H^2-4r_H^2)X^2+\left((d_P-{\beta }_P)(d_H^2-4r_H^2)-k_H(d_H+2r_H)\frac{h{h}_3}{4H_{HH}}+d_H{k}_P{\gamma }_{P}pt{c}_2\right)X-(d_P-{\beta }_P)(d_H+2r_H)\frac{h{h}_3}{4H_{HH}}=0.$$

And the solution for PTC is:

$$\begin{array}{cc}\hfill PT{C}_{3,1}=PT{C}_{3,2}=PT{C}_{3,3}& =PT{C}_{3,4}=0,\hfill \\ \hfill PT{C}_{2,2}& =PT{C}_{4,4}=\frac{{\gamma }_P{T}_2}{d_P-{\beta }_P+k_{H}H{H}_{3,4}},\hfill \\ \hfill PT{C}_{2,1}=PT{C}_{2,3}=PT{C}_{4,1}& =PT{C}_{4,3}\hfill \\ \hfill & =\frac{1}{d_P^2-2r_P^2}\left(r_P{d}_{P}PT{C}_{2,2}+\frac{1}{4H_{PTC}}(d_P+r_P)T_2\right),\hfill \\ \hfill PT{C}_{2,4}& =PT{C}_{4,2}=\frac{1}{d_P}\left(\frac{1}{4}\frac{T_2}{H_{PTC}}+2r_{P}PT{C}_{2,1}\right).\hfill \end{array}$$

Proof Let $x\in \mathcal{W}$ and h(x) be a vector defined by (3). Because hedgehog is not expressed in cells 1, 2 and 4, note that for i = 1, 2, 4

$$\begin{array}{cc}\hfill {\mathrm{HH}}_{i,\mathrm{T}}=\sum _{j=1}^4{\mathrm{HH}}_{i,j}& =h{h}_i-{\kappa }_P(\cdots )+r_H\sum _{j=1}^4({\mathrm{HH}}_{i,j+1}+{\mathrm{HH}}_{i,j-1}-2{\mathrm{HH}}_{i,j})\hfill \\ \hfill & =-{\kappa }_P(\cdots )\hfill \end{array}$$

since hh = (0, 0, 1, 0), and the sum that multiplies r_H cancels out. The terms in κ_P(· · ·) are all nonnegative, and therefore they can only be zero. We conclude that:

$${\mathrm{HH}}_{i,1}={\mathrm{HH}}_{i,2}={\mathrm{HH}}_{i,3}={\mathrm{HH}}_{i,4}=0,i=1,2,4.$$

A similar argument shows that ptc₃ = 0 implies:

$${\mathrm{PTC}}_{3,1}={\mathrm{PTC}}_{3,2}={\mathrm{PTC}}_{3,3}={\mathrm{PTC}}_{3,4}=0.$$

Therefore, the only nonlinear terms appear in the equations for PTC_2,2 and PTC_4,4:

$$\begin{array}{cc}\hfill & d_P{\mathrm{PTC}}_{2,2}-r_P{\mathrm{PTC}}_{2,1}-r_P{\mathrm{PTC}}_{2,3}+{\kappa }_H{\mathrm{PTC}}_{2,2}{\mathrm{HH}}_{3,4}=\frac{1}{4H_{\mathrm{PTC}}}pt{c}_2\hfill \\ \hfill & d_P{\mathrm{PTC}}_{4,4}-r_P{\mathrm{PTC}}_{4,1}-r_P{\mathrm{PTC}}_{4,3}+{\kappa }_H{\mathrm{PTC}}_{4,4}{\mathrm{HH}}_{3,2}=\frac{1}{4H_{\mathrm{PTC}}}pt{c}_4.\hfill \end{array}$$

Moreover, symmetry of the system shows that PTC_2,1 = PTC_2,3 and PTC_4,1 = PTC_4,3, because each pair satisfies exactly the same equation:

$$d_P{\mathrm{PTC}}_{2,1}-r_P{\mathrm{PTC}}_{2,2}-r_P{\mathrm{PTC}}_{2,4}=\frac{1}{4H_{\mathrm{PTC}}}pt{c}_2$$ (74)

$$d_P{\mathrm{PTC}}_{4,3}-r_P{\mathrm{PTC}}_{4,4}-r_P{\mathrm{PTC}}_{4,2}=\frac{1}{4H_{\mathrm{PTC}}}pt{c}_4.$$ (75)

We then have:

$$\begin{array}{cc}\hfill {\mathrm{PTC}}_{2,4}& =\frac{1}{d_P}\left(\frac{1}{4H_{\mathrm{PTC}}}pt{c}_2+2r_P{\mathrm{PTC}}_{2,1}\right)\hfill \\ \hfill {\mathrm{PTC}}_{4,2}& =\frac{1}{d_P}\left(\frac{1}{4H_{\mathrm{PTC}}}pt{c}_4+2r_P{\mathrm{PTC}}_{4,1}\right).\hfill \end{array}$$

Solving for PTC_2,1 as a function of PTC_2,2, and for PTC_4,1 as a function of PTC_4,4:

$$\begin{array}{cc}\hfill {\mathrm{PTC}}_{2,1}& =\frac{1}{d_P^2-2r_P^2}\left(r_P{d}_P{\mathrm{PTC}}_{2,2}+\frac{1}{4H_{\mathrm{PTC}}}(d_P+r_P)pt{c}_2\right)\hfill \\ \hfill {\mathrm{PTC}}_{4,1}& =\frac{1}{d_P^2-2r_P^2}\left(r_P{d}_P{\mathrm{PTC}}_{4,4}+\frac{1}{4H_{\mathrm{PTC}}}(d_P+r_P)pt{c}_4\right).\hfill \end{array}$$

Thus we get equations depending only on PTC_2,2 and HH_3,4, and on PTC_4,4 and HH_3,2:

$$d_P{\mathrm{PTC}}_{2,2}-\frac{2r_P}{d_P^2-2r_P^2}\left(r_P{d}_P{\mathrm{PTC}}_{2,2}+\frac{1}{4H_{\mathrm{PTC}}}(d_P+r_P)pt{c}_2\right)+{\kappa }_H{\mathrm{PTC}}_{2,2}{\mathrm{HH}}_{3,4}=\frac{1}{4H_{\mathrm{PTC}}}pt{c}_2$$ (76)

$$d_P{\mathrm{PTC}}_{4,4}-\frac{2r_P}{d_P^2-2r_P^2}\left(r_P{d}_P{\mathrm{PTC}}_{4,4}+\frac{1}{4H_{\mathrm{PTC}}}(d_P+r_P)pt{c}_4\right)+{\kappa }_H{\mathrm{PTC}}_{4,4}{\mathrm{HH}}_{3,2}=\frac{1}{4H_{\mathrm{PTC}}}pt{c}_4.$$ (77)

On the other hand, since PTC_3,j = 0 for all j, it follows that:

$${\mathrm{HH}}_{3,1}={\mathrm{HH}}_{3,3}=\frac{1}{d_H}\left(\frac{1}{4H_{\mathrm{HH}}}h{h}_3+r_H{\mathrm{HH}}_{3,2}+r_H{\mathrm{HH}}_{3,4}\right),$$

and substituting into the HH_3,4 and HH_3,2 equations:

$$d_H{\mathrm{HH}}_{3,4}-2\frac{r_H}{d_H}\left(\frac{1}{4H_{\mathrm{HH}}}h{h}_3+r_H{\mathrm{HH}}_{3,2}+r_H{\mathrm{HH}}_{3,4}\right)-{\kappa }_P{\mathrm{PTC}}_{2,2}{\mathrm{HH}}_{3,4}=\frac{1}{4H_{\mathrm{HH}}}h{h}_3$$ (78)

$$d_H{\mathrm{HH}}_{3,2}-2\frac{r_H}{d_H}\left(\frac{1}{4H_{\mathrm{HH}}}h{h}_3+r_H{\mathrm{HH}}_{3,2}+r_H{\mathrm{HH}}_{3,4}\right)-{\kappa }_P{\mathrm{PTC}}_{4,4}{\mathrm{HH}}_{3,2}=\frac{1}{4H_{\mathrm{HH}}}h{h}_3.$$ (79)

The last four equations may be solved for the four variables PTC_2,2, PTC_4,4, HH_3,2 and HH_3,4, and the remaining PTC, HH will then follow. Recalling the notation introduced above, one can write

$${\mathrm{PTC}}_{2,2}=\frac{{\gamma }_{P}pt{c}_2}{d_P-{\beta }_P+k_H{\mathrm{HH}}_{3,4}},{\mathrm{PTC}}_{4,4}=\frac{{\gamma }_{P}pt{c}_4}{d_P-{\beta }_P+k_H{\mathrm{HH}}_{3,2}}.$$ (80)

This leads to

$$\begin{array}{cc}\hfill & d_H{\mathrm{HH}}_{3,4}-2\frac{r_H}{d_H}\left(\frac{1}{4H_{\mathrm{HH}}}h{h}_3+r_H{\mathrm{HH}}_{3,2}+r_H{\mathrm{HH}}_{3,4}\right)-{\kappa }_P{\gamma }_{P}pt{c}_2\frac{{\mathrm{HH}}_{3,4}}{d_P-{\beta }_P+k_H{\mathrm{HH}}_{3,4}}=\frac{1}{4H_{\mathrm{HH}}}h{h}_3\hfill \\ \hfill & d_H{\mathrm{HH}}_{3,2}-2\frac{r_H}{d_H}\left(\frac{1}{4H_{\mathrm{HH}}}h{h}_3+r_H{\mathrm{HH}}_{3,2}+r_H{\mathrm{HH}}_{3,4}\right)-{\kappa }_P{\gamma }_{P}pt{c}_4\frac{{\mathrm{HH}}_{3,2}}{d_P-{\beta }_P+k_H{\mathrm{HH}}_{3,2}}=\frac{1}{4H_{\mathrm{HH}}}h{h}_3.\hfill \end{array}$$

From the symmetry of these equations, it is easy to see that

$$pt{c}_2=pt{c}_4\Rightarrow {\mathrm{HH}}_{3,4}={\mathrm{HH}}_{3,2}.$$

and thus have the following equation for HH_3,4 = HH_3,2 = X (after some simple algebra steps):

$$k_H(d_H^2-4r_H^2)X^2+\left((d_P-{\beta }_P)(d_H^2-4r_H^2)-k_H(d_H+2r_H)\frac{h{h}_3}{4H_{\mathrm{HH}}}+d_H{k}_P{\gamma }_{P}pt{c}_2\right)X-(d_P-{\beta }_P)(d_H+2r_H)\frac{h{h}_3}{4H_{\mathrm{HH}}}=0.$$ (81)

We next show that only one of the two roots of this second order polynomial is positive and hence the unique solution to HH_3,2, HH_3,4. Let the polynomial be of the form c₂X² + c₁X + c₀ = 0. The term inside the square root will be $c_1^2-4c_0{c}_2$ where:

$$-4c_0{c}_2=4k_H(d_H^2-4r_H^2)(d_P-{\beta }_P)(d_H+2r_H)\frac{h{h}_3}{4H_{\mathrm{HH}}}.$$

The factor $d_H^2-4r_H^2$ is positive, by definition of d_H. The factor

$$d_P-{\beta }_P=d_P-\frac{2r_P^2{d}_P}{d_P^2-2r_P^2}=d_P(d_P^2-4r_P^2)$$

is also positive, again by definition of d_P. This means that $c_1^2-4c_0{c}_2>c_1^2$, so whatever the sign of c₁, $-c_1-\sqrt{c_1^2-4c_0{c}_2}<0$, which leaves us with:

$${\mathrm{HH}}_{3,2}={\mathrm{HH}}_{3,4}=\frac{-c_1+\sqrt{c_1^2-4c_0{c}_2}}{2c_2}$$

(the coefficients are as in (81)).

Appendix G: The regulation of cubitus proteins

The cubitus interruptus activator and repressor proteins may only be expressed whenever engrailed is absent. In addition, they auto-regulate themselves through patched mRNA and protein. Using EN = en note that CI can be written in terms of CN:

$${\mathrm{CI}}_i=c{i}_i-{\mathrm{CN}}_i=U_i{\theta }^{-}(e{n}_i,{\kappa }_{\mathrm{EN}ci})-{\mathrm{CN}}_i$$

Note that, if Y = U − X:

$${\theta }^{+}(X,{\kappa }_a)={\theta }^{-}(Y,U-{\kappa }_a).$$

Furthermore,

$${\theta }^{-}(Y,U-{\kappa }_a){\theta }^{-}(Y,{\kappa }_b)={\theta }^{-}(Y,\text{min}\{U-{\kappa }_a,{\kappa }_b\}).$$

Then ptc can be simplified as follows:

$$\begin{array}{cc}\hfill pt{c}_i& ={\theta }^{+}({\mathrm{CI}}_i{\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}ptc}),{\kappa }_{\mathrm{CN}ptc}))\hfill \\ \hfill & =\{\begin{array}{cc}{\theta }^{+}({\mathrm{CI}}_i,{\kappa }_{\mathrm{CI}ptc}){\theta }^{-}({\mathrm{CN}}_i,{\kappa }_{\mathrm{CN}ptc}),\hfill & {\mathrm{CN}}_i\ne {\kappa }_{\mathrm{CN}ptc}\hfill \\ {\theta }^{+}({\mathrm{CI}}_i,{\kappa }_{\mathrm{CI}ptc}),\hfill & {\mathrm{CN}}_i={\kappa }_{\mathrm{CN}ptc}\hfill \end{array}\phantom{\}}\hfill \\ \hfill & =\{\begin{array}{cc}{\theta }^{-}({\mathrm{CN}}_i,\text{min}\{c{i}_i-{\kappa }_{\mathrm{CI}ptc},{\kappa }_{\mathrm{CN}ptc}\}),\hfill & {\mathrm{CN}}_i\ne {\kappa }_{\mathrm{CN}ptc}\hfill \\ {\theta }^{-}({\mathrm{CN}}_i,c{i}_i-{\kappa }_{\mathrm{CI}ptc}),\hfill & {\mathrm{CN}}_i={\kappa }_{\mathrm{CN}ptc}\hfill \end{array}\phantom{\}}\hfill \end{array}$$ (82)

ptc is used to compute PTC explicitly (as in Sect. F), which feeds back into CN:

$$\begin{array}{cc}\hfill {\mathrm{CN}}_i& =c{i}_i\frac{H_{\mathrm{CI}}{C}_{\mathrm{CI}}{\theta }^{+}({\mathrm{PTC}}_{i,T},{\kappa }_{\mathrm{PTCCI}})}{1+H_{\mathrm{CI}}{C}_{\mathrm{CI}}{\theta }^{+}({\mathrm{PTC}}_{i,T},{\kappa }_{\mathrm{PTCCI}})}\hfill \\ \hfill & =U_i{\theta }^{-}(e{n}_i,{\kappa }_{\mathrm{EN}ci})\frac{H_{\mathrm{CI}}{C}_{\mathrm{CI}}{\theta }^{+}({\mathrm{PTC}}_{i,T},{\kappa }_{\mathrm{PTCCI}})}{1+H_{\mathrm{CI}}{C}_{\mathrm{CI}}{\theta }^{+}({\mathrm{PTC}}_{i,T},{\kappa }_{\mathrm{PTCCI}})}.\hfill \end{array}$$

Throughout this paper, we will consider that θ⁻(en₃, κ_ENci) = 0 and θ⁻(en_i, κ_ENci) = 1 for i = 1, 2, 4. In addition, defining Q_i = 1 if PTCC_i,T > κ_PTCCI, and Q_i = 0 if PTC_i,T < κ_PTCCI, and Q_i ∈ [0, 1] if PTC_i,T = κ_PTCCI, we have:

$${\mathrm{CN}}_i=U_i\frac{Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}{1+Q_i{H}_{\mathrm{CI}}{C}_{\mathrm{CI}}}.$$ (83)

Appendix H: Computing the cylindrical algebraic decomposition

A CAD for the parameter space G can be computed from Eqs. (52)–(63), by imposing the conditions $x\in \mathcal{W}$, as given by (7). Following von Dassow et al.'s criteria, conditions on wild type expression of en, hh, and wg at steady state, are obtained from a threshold expression:

$$e{n}_3^{\mathrm{WT}}\ge \epsilon \text{and}e{n}_i^{\mathrm{WT}}<\epsilon ,i=1,2,4$$ (84)

$$h{h}_3^{\mathrm{WT}}\ge \epsilon \text{and}h{h}_i^{\mathrm{WT}}<\epsilon ,i=1,2,4$$ (85)

$$w{g}_3^{\mathrm{WT}}\ge \epsilon \text{and}w{g}_i^{\mathrm{WT}}<\epsilon ,i=1,3,4.$$ (86)

Throughout this paper, we will consider also that

$$c{i}_3^{\mathrm{WT}}<\epsilon \text{and}c{i}_i^{\mathrm{WT}}\ge \epsilon ,i=1,2,4,$$ (87)

$$pt{c}_3^{\mathrm{WT}}<\epsilon \text{and}pt{c}_i^{\mathrm{WT}}\ge \epsilon ,i=1,2,4,$$ (88)

thus incorporating further experimental data [18,19] into the analysis.

We will make four assumptions, only slightly stronger then (85), (87)–(89), but which allow to obtain clean analytical results for a large part of the problem of finding the feasible parameter space. These analytical results include Theorem 1 (for establishing symmetry of EWG) and explicit computation of PTC and HH. The modified assumptions are:

$$e{n}_3^{\mathrm{WT}}\ge \epsilon \text{and}e{n}_i^{\mathrm{WT}}=0,i=1,2,4$$ (89)

$$w{g}_2^{\mathrm{WT}}\ge \epsilon \text{and}w{g}_4\le \underset{i=1,3}{\text{min}}w{g}_i^{\mathrm{WT}}<\epsilon$$ (90)

$$c{i}_3^{\mathrm{WT}}=0\text{and}c{i}_i^{\mathrm{WT}}\ge \epsilon ,i=1,2,4$$ (91)

$$pt{c}_3^{\mathrm{WT}}<\epsilon \text{and}pt{c}_2^{\mathrm{WT}}=pt{c}_4^{\mathrm{WT}},pt{c}_1^{\mathrm{WT}}\ge \epsilon ,$$ (92)

Then, from (82) and (91), it is clear that:

$${\mathrm{CI}}_3={\mathrm{CN}}_3\equiv 0\text{and}pt{c}_3\equiv 0.$$

H.1 p_CI, p_PTC−HH, p_WG, α_CIwg

As already remarked, a CAD is not unique, and the choice of the “free” parameters, or those that come at the top of the hierarchy have to be chosen. In this case, we choose the parameters and intervals already listed in (16), which form S₁. von Dassow et al. chose the interval [5, 100] for half-lifes (H_X) and (0, 1] for ON/OFF parameters (κ_XY) to denote the range of physiological values. This is reasonable since, by model construction, all variables are “normalized” to 1. Assuming ON/OFF constants κ_XY > 1 would amount to assume that Y is permanently activated (or inhibited) by X.

Lemma H.1 Assume conditions (85), (89), and (91) hold. Then κ_ENci, κ_ENhh ∈ (0, σ_en3], κ_CNhh ∈ (0, 1]. Also U₃ ∈ [0, 1] U_{1, 2, 4} ∈ [ε, 1].

Proof Recalling (59) and EN = en, conditions (91) can be re-written:

$$e{n}_3>{\kappa }_{\mathrm{EN}ci}\text{and}e{n}_{1,2,4}\le {\kappa }_{\mathrm{EN}ci}.$$

These are equivalent to κ_ENci ∈ [max{en_i, i = 1, 2, 4}, en₃). From (89), max{en_i, i = 1, 2, 4} = 0 and en₃ = σ_en3 ∈ [ε, 1] denotes the possible values of θ⁻(EWG₃, κ_WGwg) when EWG₃ = κ_WGwg. Next, since ci₃ = 0 due to κ⁻EN₃, κ_ENci) = 0, U₃ can take any value in [0, 1], and U_{1, 2, 4} have to be larger than the thresdold ε.

Conditions (85) translate to:

$$\begin{array}{cc}\hfill & e{n}_3\ge {\kappa }_{\mathrm{EN}hh}\text{and}{\mathrm{CN}}_3\le {\kappa }_{\mathrm{CN}hh}\hfill \\ \hfill & e{n}_i\le {\kappa }_{\mathrm{EN}hh}\text{or}{\mathrm{CN}}_i\ge {\kappa }_{\mathrm{CN}hh},i=1,2,4.\hfill \end{array}$$ (93)

Since en_1,2,4 = 0, no restrictions on κ_CNhh are needed, so κ_CNhh ∈ (0, 1]. Since CN₃ = 0, all conditions are verified iff κ_ENhh ∈ (0, σ_en3].

H.2 κ_PTCCI, κ_CIptc, and κ_CNptc

For simplicity, here we will treat the case wg₂ > 0, wg_i = 0 (i = 1, 3, 4). As shown below (Sect. H.3), this is true for almost all feasible sets of parameters. It also implies that EWG₁ = EWG₃.

Lemma H.2 For system (1) with steady state set (7), the following hold:

1. ${\kappa }_{PTCCI}\in (0,\text{min}\{T_1,PT{C}_{2,T}^{WT}\}]$;

2. ${\kappa }_{CNptc}\in \left[{\text{max}}_{i=1,2,4}{U}_i\frac{Q_i{H}_{CI}{C}_{CI}}{1+Q_i{H}_{CI}{C}_{CI}},1\right]$;

3. ${\kappa }_{CIptc}\in \left(0,{\text{min}}_{i=1,2,4}\frac{U_i}{1+Q_i{H}_{CI}{C}_{CI}}\right]$.

Proof By assumption (92) (and definition of $\mathcal{W}$), $pt{c}_2^{\mathrm{WT}}=pt{c}_4^{\mathrm{WT}}$, By Lemma F.1 in Appendix F, we then have PTC_2,T = PTC_4,T. By Theorem 1 and (89) it follows that ${\mathrm{CN}}_2^{\mathrm{WT}}>0$ is always required for a nonemtpy parameter set. In the case EWG₁EWG₃, Theorem 1 also shows that ${\mathrm{CN}}_1^{\mathrm{WT}}<0$ is necessary. From Appendix G this can only be achieved iff κ_PTCCI ≤ {PTC_1,T, PTC_2,T} where PTC_1,T = T₁. We can write, for i = 1, 2, 4:

$$pt{c}_i=\{\begin{array}{cc}{\theta }^{-}({\mathrm{CN}}_i,\text{min}\{U_i-{\kappa }_{\mathrm{CI}ptc},{\kappa }_{\mathrm{CN}ptc}\}),\hfill & {\mathrm{CN}}_i\ne {\kappa }_{\mathrm{CN}ptc}\hfill \\ {\theta }^{-}({\mathrm{CN}}_i,U_i-{\kappa }_{\mathrm{CI}ptc}),\hfill & {\mathrm{CN}}_i={\kappa }_{\mathrm{CN}ptc}\hfill \end{array}\phantom{\}}$$

and (from $\mathcal{W}$) ptc_i ≥ ε, for i = 1, 2, 4. This implies CN_i ≤ min{U_i − κ_CIptc, κ_CNptc} which translates into intervals (b) and (c). □

H.3 κ_WGen, κ_CNen, κ_WGwg, κ_CNwg, and κ_CIwg

Recall the expression of wg at steady state, given in (12). For most sets of parameters, it will happen that R_CI, R_WG ∈ {0, 1}, yielding four possible values for wg_i:

$$w{g}_i\in \left\{0,\frac{{\alpha }_{\mathrm{WG}}}{1+{\alpha }_{\mathrm{WG}}},\frac{{\alpha }_{\mathrm{CI}}}{1+{\alpha }_{\mathrm{CI}}},\frac{{\alpha }_{\mathrm{CI}}+{\alpha }_{\mathrm{WG}}}{1+{\alpha }_{\mathrm{CI}}+{\alpha }_{\mathrm{WG}}}\right\}.$$

For the cases when variables CI, CN or IWG fall on their threshold values, CI_i = κ_CIwg, CN_i = κ_CNwg and/or IWG_i = κ_WGwg, it will happen that R_CI, R_WG ∈ [0, 1]. Since these cases imply a fine tuning of κ_CIwg, κ_CNwg, or κ_WGwg, they will happen for a much smaller family of parameters. Therefore, in this Section, we analyze only the case R_CI, R_WG ∈ {0, 1}. (However, note that the product “R_CIα_CI” can be treated as a new “α_CI”, and hence the cases R_{CI, WG} ∈ (0, 1) may be analyzed in a similar way.)

According to the choice of α_CI and α_WG in (16), we have:

$$\epsilon \le \text{min}\left\{\frac{{\alpha }_{\mathrm{WG}}}{1+{\alpha }_{\mathrm{WG}}},\frac{{\alpha }_{\mathrm{CI}}}{1+{\alpha }_{\mathrm{CI}}}\right\}$$

In this case, either the cubitus pathway or the wingless pathway may maintain wingless in the second cell, and both pathways need to be shutdown on the other cells:

$$w{g}_2\in \left\{\frac{{\alpha }_{\mathrm{WG}}}{1+{\alpha }_{\mathrm{WG}}},\frac{{\alpha }_{\mathrm{CI}}}{1+{\alpha }_{\mathrm{CI}}},\frac{{\alpha }_{\mathrm{CI}}+{\alpha }_{\mathrm{WG}}}{1+{\alpha }_{\mathrm{CI}}+{\alpha }_{\mathrm{WG}}}\right\}\text{and}w{g}_i=0,i=1,3,4,$$

Conditions (89) can then be written as inequalities on wg and CN:

$${\mathrm{EWG}}_{\underset{‒}{3}}=w{g}_2{\beta }_{\text{med}}\ge {\kappa }_{\mathrm{WG}en}\text{and}{\mathrm{CN}}_3\le {\kappa }_{\mathrm{CN}en},$$ (94a)

$${\mathrm{EWG}}_{\underset{‒}{1}}=w{g}_2{\beta }_{\text{med}}<{\kappa }_{\mathrm{WG}en}\text{or}{\mathrm{CN}}_1>{\kappa }_{\mathrm{CN}en},$$ (94b)

$${\mathrm{EWG}}_{\underset{‒}{2}}=w{g}_2{\beta }_{\text{max}}<{\kappa }_{\mathrm{WG}en}\text{or}{\mathrm{CN}}_2>{\kappa }_{\mathrm{CN}en},$$ (94c)

$${\mathrm{EWG}}_{\underset{‒}{4}}=w{g}_2{\beta }_{\text{min}}<{\kappa }_{\mathrm{WG}en}\text{or}{\mathrm{CN}}_4>{\kappa }_{\mathrm{CN}en}.$$ (94d)

Since β_max > β_med, it is clear that (94a–c) can only be satisfied if

$${\mathrm{CN}}_1,{\mathrm{CN}}_2>{\kappa }_{\mathrm{CN}en}.$$

Conditions (90) translate to:

$$\begin{array}{cc}\hfill w{g}_2{\gamma }_{\text{max}}\ge {\kappa }_{\mathrm{WG}wg}& \text{or}{\mathrm{CN}}_2\le \text{min}\{{\kappa }_{\mathrm{CN}wg},U_2-{\kappa }_{\mathrm{CI}wg}\},\hfill \\ \hfill w{g}_2{\gamma }_{\text{med}}\le {\kappa }_{\mathrm{WG}wg}& \text{and}{\mathrm{CN}}_1\ge \text{min}\{{\kappa }_{\mathrm{CN}wg},U_1-{\kappa }_{\mathrm{CI}wg}\},\hfill \\ \hfill w{g}_2{\gamma }_{\text{med}}\le {\kappa }_{\mathrm{WG}wg}& \text{and}{\mathrm{CN}}_3\ge \text{min}\{{\kappa }_{\mathrm{CN}wg},0-{\kappa }_{\mathrm{CI}wg}\},\hfill \\ \hfill w{g}_2{\gamma }_{\text{min}}\le {\kappa }_{\mathrm{WG}wg}& \text{and}{\mathrm{CN}}_4\ge \text{min}\{{\kappa }_{\mathrm{CN}wg},U_4-{\kappa }_{\mathrm{CI}wg}\}.\hfill \end{array}$$ (95)

From (91) it follows that CN₃ = 0. So the inequalities that represent en^WT and wg^WT are reduced to:

$$\begin{array}{cc}\hfill & {\kappa }_{\mathrm{WG}en}\le w{g}_2{\beta }_{\text{med}},\hfill \\ \hfill & {\kappa }_{\mathrm{CN}en}<\text{min}\{{\mathrm{CN}}_1,{\mathrm{CN}}_2\},\hfill \\ \hfill & {\kappa }_{\mathrm{WG}en}>w{g}_2{\beta }_{\text{min}}\text{or}{\kappa }_{\mathrm{CN}en}<{\mathrm{CN}}_4,\hfill \\ \hfill & {\kappa }_{\mathrm{WG}wg}\le w{g}_2{\gamma }_{\text{max}}\text{or}\text{min}\{{\kappa }_{\mathrm{CN}wg},U_2-{\kappa }_{\mathrm{CI}wg}\}\ge {\mathrm{CN}}_2,\hfill \\ \hfill & {\kappa }_{\mathrm{WG}wg}>w{g}_2{\gamma }_{\text{med}},\hfill \\ \hfill & \text{min}\{{\kappa }_{\mathrm{CN}wg},U_1-{\kappa }_{\mathrm{CI}wg}\}<{\mathrm{CN}}_1,\hfill \\ \hfill & \text{min}\{{\kappa }_{\mathrm{CN}wg},U_4-{\kappa }_{\mathrm{CI}wg}\}<{\mathrm{CN}}_4,\hfill \end{array}$$

There are now three possibilities according to the value of wg₂, yielding the intervals listed in Tables 3, 4, and 5.