Turing patterns with an external linear morphogen gradient - Western ...

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NONLINEARITY

Nonlinearity 22 (2009) 2541–2560

doi:10.1088/0951-7715/22/10/012

Interaction of Turing patterns with an external linear morphogen gradient Tilmann Glimm1,2,3 , Jianying Zhang1,3 and Yun-Qiu Shen1 1

Department of Mathematics, Western Washington University, Bellingham, WA 98225, USA

E-mail: [email protected], [email protected] and [email protected]

Received 3 April 2009, in final form 23 August 2009 Published 10 September 2009 Online at stacks.iop.org/Non/22/2541 Recommended by J A Glazier Abstract We investigate a simple generic model of a reaction–diffusion system consisting of an activator and an inhibitor molecule in the presence of a linear morphogen gradient. We assume that this morphogen gradient is established independently of the reaction–diffusion system and acts by increasing the production of the activator proportional to the morphogen concentration. The model is motivated by several existing models in developmental biology in which a Turing patterning mechanism is proposed and various chemical gradients are known to be important for development. Mathematically, this leads to reaction– diffusion equations with explicit spatial dependence. We investigate how the Turing pattern is affected, if it exists. We also show that in the parameter range where a Turing pattern is not possible, the system may nevertheless produce ‘Turing-like’ patterns. Mathematics Subject Classification: 92C15, 35K57, 35B32, 37B55 (Some figures in this article are in colour only in the electronic version)

1. General introduction One of the most fundamental problems in developmental biology is the question of morphogenesis, or pattern formation; that is, by what mechanisms cells self-organize into highly complex spatial distributions giving rise to tissues and organs during embryonic development. While tremendous advances have been made in the last 50 years, there is still debate over basic mechanisms and their relationships. 2

Author to whom any correspondence should be addressed.

3

Joint first authors of this article.

0951-7715/09/102541+20$30.00 © 2009 IOP Publishing Ltd and London Mathematical Society Printed in the UK

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One such mechanism whose significance is universally accepted is the action of gradients of signalling molecules [3, 7, 13, 19]. The idea is that cells respond to a special chemical— a morphogen—whose concentration increases in a certain direction, forming a chemical gradient. Different concentrations of the signalling molecule cause different responses in the cells. It has been shown experimentally that chemical gradients are indeed central to embryonic development in an abundance of cases [14, 19]. One of many examples is the protein Bicoid in Drosophila larvae, which is produced at the head, leading to an anterior– posterior Bicoid gradient [13]. There is still some debate as to how these gradients are set up. It is known that there are certain centres of production of special chemicals in the embryo; that is, localized clusters of cells that secrete a certain biomolecule. Recent research suggests that diffusion of this biomolecule throughout the extracellular matrix and its decay is sufficient to explain the observed concentration profiles which are gradually decreasing as the distance from the centre of production increases [9–11]. It is possible that some more elaborate process is required, so this question is not settled. An example of a model employing this mechanism is Dillon and Othmer’s model of morphogens in the embryonic avian limb bud [4, 5]. The authors showed that the action of the zone of polarizing activity, a cluster of cells secreting the protein sonic hedgehog (Shh), can lead to the observed Shh gradients under plausible assumptions. In the following, we are not concerned about the mechanism of how chemical gradients are set up, but rather take them as given. Another important, somewhat related, but more specialized model for morphogenesis is the Turing mechanism in reaction–diffusion systems, first proposed by Alan Turing in the 1950s [18]. Turing showed that if two or more chemicals are present in a substance, then under certain conditions, the interplay of diffusion and reaction can lead to the emergence of patterns in the concentration of the chemicals. This idea has since become a paradigm in explaining emergent properties in self-organizing systems [3, 12]. There has been relatively little research in the applied mathematics literature on how the two mechanisms described alone—Turing-type reaction–diffusion patterns and morphogen gradients set up by specialized clusters of cells—may interact, even though it is well known that in many systems for which Turing mechanisms have been proposed, chemical gradients of morphogens uninvolved in the Turing mechanism are present and important for pattern formation, for example in bone pattern formation in the embryonic chick limb [4, 5] or in the formation of hair follicles in mouse skin [17]. Sometimes it is argued that in these cases morphogen gradients may be a secondary patterning mechanism, which serves to stabilize the pattern formation process, or to alter or ‘shape’ the Turing patterns after they have formed through the primary reaction–diffusion mechanism [1, 2, 17]. While these arguments are certainly biologically sound, they arguably lack precise mathematical foundation; this is the point we would like to address in this paper. Clearly the question of the effects of the interaction between the two mechanisms is very complex and a precise answer will most probably depend on the exact details of the interaction in each special instance. However, it is not unreasonable to assume that there are probably some typical features that are largely independent of the exact interaction, and in this paper, we approach the problem by considering a generic model system consisting of a Turing activator and inhibitor in the presence of the gradient of a third chemical signal. We call this third chemical an ‘external’ signal since we assume that its concentration profile is formed independently of the Turing mechanism. We assume that the Turing activator and inhibitor are secreted by cells, react with each other and diffuse, and that the external chemical signal acts by increasing the rate at which the activator is secreted. We assume that the concentration of the external chemical decreases linearly from its source. We further assume that the cell

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density is essentially constant, so that the cells do not enter our model explicitly. For analytical convenience, we analyse a one-dimensional system. Mathematically, the resulting equations are reaction–diffusion equations with a spatially dependent term, i.e. with reaction kinetics that depend explicitly on the spatial position. More specifically, the equations are ∂U εc a 2 , = F (U ) + D∇ U + x 0 ∂t where U (x, t) = (u(x, t), v(x, t)) is a vector of chemical concentrations, F (U ) describes the reaction kinetics, D = diag(D1 , D2 ) is the matrix of diffusion coefficients and εc a is some constant, where εc is a ‘small’ parameter. We investigate the following key questions: (1) how does the external chemical gradient affect the Turing pattern, provided the system is in the parameter region where a Turing pattern is possible? (2) suppose the system is in the parameter region where a Turing pattern is not possible. Can the external chemical gradient create a ‘Turing-like’ pattern, i.e. can the presence of an external chemical gradient induce pattern formation in reaction–diffusion systems? The second point is especially interesting, since one of the most important criticisms of models which propose a Turing mechanism is that it typically requires greatly different diffusivities for activator and inhibitor, while in reality these molecules are typically very similar chemically and thus should have very similar diffusivities [3]. Based on our analytical results, we obtain the following answers to the above questions: (1) in the parameter region where a Turing pattern is possible, adding a ‘small’ chemical signal essentially does not change the wave number of the pattern. Rather, it changes the baseline of the pattern to a linear function which increases towards the source of the external signal, as one would expect. The amplitude of the pattern either increases or decreases towards the source, or, somewhat unexpectedly, it may have a minimum somewhere in the interval. See also figures 1 and 2. (2) maybe surprisingly, this question can be answered in the positive, with the caveat that the amplitude of the generated pattern appears to be very small compared with typical concentrations. We varied the ratio d = D1 /D2 of diffusivities of the activator and the inhibitor. It is well known that if this ratio is above a certain critical number d0 < 1, a Turing pattern is not possible. However, we found that there is a second critical value, d1 , with the property that an oscillatory pattern can still form for d0 < d < d1 , but no pattern can form for d > d1 . We can thus argue that in the presence of an external chemical signal, Turing-‘like’ patterning may be possible even if the diffusion coefficients of activator and inhibitor are too close together to allow for Turing pattern formation without the external chemical signal. See also figure 3. Our analysis thus lends plausibility to claims that a ‘small’ external chemical can alter the ‘shape’ of a Turing bifurcation; however, this only concerns the amplitude of the pattern but not the wave number, which cannot be changed. Interestingly, the external signal can generate a pattern with small amplitude even if the reaction–diffusion equations by themselves cannot generate a pattern. The paper is organized as follows. In sections 2 and 3, we introduce the model of interest and the regular perturbation method in two small parameters we use for the analysis. We then investigate the cases d < d0 (Turing patterns) and d > d0 (no Turing patterns) separately, presenting and discussing the analytical results and the corresponding numerical verifications in section 4. A derivation of the results is included in section 5. We summarize our results and point to future work in the last section.

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2. Introduction of the model equations We analyse reaction–diffusion equations with a spatially dependent term of the form ∂U εc a + AU + DUxx + Q(U, U ) + C(U, U, U ) + h.o.t., =x 0 ∂t where U (t, x) = (u(t, x) − u0 , v(t, x) − v0 ) is a vector of concentrations of two chemicals, offset by some constant concentrations u0 , v0 , respectively. A is a matrix, Q and C are quadratic and cubic terms, respectively, and D = diag(d, 1) is a diagonal matrix of diffusivities. The first term on the right hand side describes a spatially dependent kinetics term, where a is a constant and εc is a small parameter. Our theorems 4.1–4.3 are valid for this general model, but for the sake of exhibition, and to show how these equations may arise from models in development, we mostly concentrate on a special example, which is the following simple generic model based on Schnakenberg kinetics [12]. Suppose we have two chemicals C1 and C2 that are secreted by cells, decay, react with each other and diffuse according to the following scheme: we assume that the activator C1 decays at the rate k3 and that the inhibitor C2 is secreted by cells at the rate k2 . Here cell concentration is assumed to be constant, that is we are interested in the time scale of chemical interactions [2]. Additionally, we assume that there is a third chemical M which has a temporally constant concentration [M](x), and that the rate at which cells secrete chemical C1 depends on the concentration of M; more precisely, we assume that this production rate of C1 is given by k4 (1 + k5 [M](x)). We further assume that C1 and C2 obey Schnakenberg kinetics k1

2C1 + C2 → 3C1 . The resulting equations for the concentrations [C1 ] and [C2 ] are thus ∂[C1 ] = k1 [C1 ]2 [C2 ] − k3 [C1 ] + k4 (1 + k5 [M](x)) + D1 ∇ 2 [C1 ], ∂t ∂[C2 ] = −k1 [C1 ]2 [C2 ] + k2 + D2 ∇ 2 [C2 ], ∂t where D1 and D2 are the diffusion coefficients of C1 and C2 , respectively; t is the time and x

is the location. Defining the nondimensional quantities t ∗ = k3 t, u = kk13 [C1 ], v = kk13 [C2 ], x ∗ = Dk32 x, a = k4 kk13 , b = k2 kk13 , and dropping the stars for notational convenience, we 3

3

get the nondimensional equations  ∂u  = a(1 + k5 [M](x)) + u2 v − u + duxx ,  ∂t   ∂v = b − u2 v + v . xx ∂t Here d=

D1 D2

(2.1)

is the ratio of the diffusion coefficients of the activator and the inhibitor. We finally make the simplifying assumption that the chemical gradient [M](x) is linear: k5 [M](x) = εc x, where εc is a ‘small’ parameter. (On the modelling side, this means that the


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gradient is shallow, or that the effect of the chemical signal M on the production of activator C1 is small.) This finally gives us the equations we will analyse in this paper as follows:  ∂u  = a(1 + εc x) + u2 v − u + duxx ,  ∂t (2.2)   ∂v = b − u2 v + v . xx ∂t We consider the equations in the (one-dimensional) interval (0, L), with no-flux boundary condition ux (t, 0) = ux (t, L) = 0. 3. Regular perturbation for the model equations We now start the analysis of the model equations (3.1). The analysis in the case εc = 0 is standard, see, e.g., [6, 15, 16]. We are primarily interested in the steady state U (x) = (u(x), v(x)) given by 0 = a(1 + εc x) + u2 v − u + duxx , (3.1) 0 = b − u2 v + vxx . The system has a spatially homogeneous steady state solution U0 = (u0 , v0 ) corresponding to εc = 0. A simple computation yields that u0 = a + b and v0 = b/(a + b)2 . For convenience of notation, we expand the right hand sides in their Taylor expansions at the spatially homogeneous steady state yielding the vector equation a + h.o.t., 0 = AU + DUxx + Q(U, U ) + C(U, U, U ) + εc x 0 where U = (u − u0 , v − v0 ), the linearization matrix is   b−a

2 2 (a + b) u0 2u0 v0 − 1   +b =  a 2b A= , 2 −2u0 v0 −u0 − −(a + b)2 a+b and Q and C refer to the quadratic and cubic terms of the right hand side of (3.1). The matrix D is the diffusion matrix: d 0 D= . 0 1 3.1. Turing bifurcation at the critical value for diffusion d = d0 Suppose we keep εc = 0 for now and fix all parameters except d, the ratio of diffusion coefficients. For some critical value d = d0 , the steady state solution U0 = (u0 , v0 ) becomes unstable and a Turing bifurcation occurs for d < d0 . Our purpose is to understand how the Turing patterns are affected when we take the external chemical signal into consideration; that is when we let εc > 0. More precisely, we consider the following expansions. We introduce the small parameter εd by writing1 d = d0 + εd2 d2 + · · · The more general form is an expansion of the form d = d0 + εd d1 + εd2 d2 + · · ·, but as shown in section 5, the expansion (3.2) implies d1 = 0.

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and then expanding u(x) = u0 + εd u01 (x) + εc u10 (x) + εd2 u02 (x) + εc2 u20 (x) + εd εc u11 (x) + · · · v(x) = v0 + εd v01 (x) + εc v10 (x) + εd2 v02 (x) + εc2 v20 (x) + εd εc v11 (x) + · · · or in vector form with U = (u − u0 , v − v0 ), U (x) = εd U01 (x) + εc U10 (x) + εd2 U02 (x) + εc2 U20 (x) + εd εc U11 (x) + · · · .

(3.2)

The following two assumptions are made for the linearization matrix A, guaranteeing a Turing bifurcation [12]. (1) With no spatial variation, A corresponds to a stable dynamical system, i.e. both eigenvalues of A, γ1 and γ2 , satisfy Reγ1 < 0 and Reγ2 < 0. So traceA = γ1 + γ2 < 0,

detA = γ1 γ2 > 0.

(2) For d > d0 , A is stable with respect to any spatially varying perturbation with wave number k, i.e. det(A − k 2 D) > 0

for all k,

for d > d0 .

For d < d0 , A is unstable with respect to some spatially varying perturbation with wave number k, i.e. for some k > 0, A − k 2 D is unstable. The critical ratio of diffusivities d0 is given by the conditions det(A − k02 D0 ) = 0 for some critical wave number k0 and det(A − k 2 D0 ) > 0 for k 2 = k02 . Thus there is a Turing bifurcation at d = d0 ; a Turing pattern with wave number k0 appears. To match the boundary conditions we assume that k0 is of the form2 k0 =

2nπ , L

n ∈ N.

(3.3)

Note that D0−1 A has repeated eigenvalue k02 with only one corresponding eigenvector; thus D0−1 A is not diagonalizable. Let V1 be the eigenvector to k02 and V2 be a vector that satisfies (D0−1 A−k02 I )V2 = V1 . Since V1 and V2 are linearly independent, we have P −1 (D0−1 A)P = , k 2 1 where P = [V1 V2 ] and = 00 k2 . 0

3.2. Analysis for d > d0 (no Turing patterns) If we fix a ratio of diffusivities d > d0 , the system cannot produce Turing patterns by the previous discussion. To investigate the effect of the external chemical signal on the steady state in this case, it is thus sufficient to consider the expansion u(x) = u0 + εc u10 (x) + εc2 u20 (x) + · · · , v(x) = v0 + εc v10 (x) + εc2 v20 (x) + · · · or again in vector form U (x) = εc U10 (x) + εc2 U20 (x) + · · · .

(3.4)

Of course k0 = (2n+1)π with n ∈ N would also match the boundary conditions. However, as we show in section 5.2, L in this case the regular perturbation approach fails.

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Table 1. Error between the first order approximation uana (x) = u0 +εd u01 +εc u10 from theorems 4.1 and 4.2 and the numerical solution unum (x) of the full nonlinear system (2.2). Note that the error is O(εd2 + εc2 ), supporting the validity of the regular perturbation approach (3.2). Parameters used were as those in figure 1; the numerical method was Newton’s method for the steady state equations. εd

εc

Error (uana − unum 2 )

10−2

10−2

10−3

10−2

10−2 10−3 10−2 0

10−3 10−3 0 10−2

4.8706 × 10−4 3.8497 × 10−5 3.2760 × 10−4 4.16322 × 10−6 3.2254 × 10−4 9.6161 × 10−6

4. Results 4.1. Patterns generated by the external chemical gradient at the critical diffusion coefficient for Turing bifurcation: d d0 To determine the linear approximation of U in the form of (3.2), we compute U01 and U10 analytically and derive the following results. Theorem 4.1. Let V ⊥ be a vector that is perpendicular to D0 V1 . Then the linear term in the εd expansion is U01 = c˜1 cos(k0 x)V1 , α1 where c˜1 = ± − , with α1 = k02 D2 V1 , V ⊥ and α3 α3 = Q(V1 , A−1 Q(V1 , V1 )), V ⊥ + 21 Q(V1 , (A − 4k02 D0 )−1 Q(V1 , V1 )), V ⊥ − 43 C(V1 , V1 , V1 ), V ⊥ . Here ·, · denotes the inner product of two vectors. a = β1 V1 + β2 V2 , then the linear term in the εc expansion, Theorem 4.2. Let A−1 0 c4 U10 = c1 cos k0 x + c2 sin k0 x + x cos k0 x V1 + (c4 sin k0 x)V2 − x(β1 V1 + β2 V2 ), 2k0 β1 β2 β2 − 3 , c4 = and c1 solves some linear algebraic equation which will be k0 k0 2k0 stated in detail in the derivation of U10 in section 5.2. where c2 =

We tested these first order approximations by comparing with the numerical solutions of the full nonlinear equations for small εd and εc , see table 1. In the context of the model on which the equations (2.2) are based, these results may be interpreted as follows: up to first order, the steady state of the chemical concentrations, that is the pattern that will form spontaneously from random initial conditions, consists of the wellknown Turing pattern with wave number k0 , which is distorted by the pattern stemming from the presence of the external chemical signal. The Turing pattern is identified by the function U01 in theorem 4.1, and its distortion is determined by U10 in theorem 4.2. This distortion is essentially a cosine curve with the same wave number k0 , which is shifted by a linear function and whose amplitude is increasing or decreasing linearly in the interval [0, L].

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The resulting steady state is relatively hard to picture, but the following observation helps: using the expansions U01 and U10 above, we may write the first order approximation of the activator concentration u(x) as u(x) = u0 + εc κ0 x + (εd κ1 + εc (κ2 + κ3 x)) cos(k0 x) + εc κ4 sin(k0 x) + h.o.t., where κ0 , κ1 , . . . , κ4 are certain constants that can be read off the equations in theorems 4.1 and 4.2. This in turn can be rewritten as u(x) = u0 + εc κ0 x + A(εd , εc , x) cos(k0 x − φ(εd , εc , x)) + h.o.t., where the amplitude A(εd , εc , x) and the phase shift φ(εd , εc , x) are given by A(εd , εc , x) = (εd κ1 + εc (κ2 + κ3 x))2 + εc2 κ42 , ε c κ4 φ(εd , εc , x) = arctan . εd κ1 + εc (κ2 + κ3 x) Note that the amplitude has a minimum (as a function of x) where εd κ1 + εc (κ2 + κ3 x) = 0, that is using the notation from theorems 4.1 and 4.2 again, at 2k 2 εd c˜1 + c1 . xmin = − 0 β 2 εc The basic shape of the steady state can thus be described as follows: it is approximately a shifted cosine curve with wave number k0 over a linearly increasing base line. The amplitude has a minimum at xmin , and it is strictly increasing towards xmin and strictly increasing away from xmin . (Note that xmin may not lie in the interval [0, L].) Typical pictures of this behaviour are shown in figures 1 and 2. Hence the external chemical signal leaves the wave number of the Turing pattern more or less unaffected, but distorts its amplitude. This gives some support to the idea that the action of the external chemical signal can ‘shape’ the Turing pattern. It would be interesting to explore in which other situations similar patterns can be found in developmental biology. 4.2. Patterns generated by the external chemical gradient with no Turing patterns: d > d0 Let now d > d0 , and recall that D = diag(d, 1). Let µ1 and µ2 be the eigenvalues of D −1 A, and V1 and V2 the corresponding eigenvectors. Since 1 det(D −1 A) = · detA = µ1 µ2 > 0, d µ1 and µ2 are real with the same sign, or they are complex conjugates. If they are both positive, then det(A − µ1 D) = det(A − µ2 D) = 0 for µ1 > 0 and µ2 > 0, which contradicts the assumption of no Turing patterns for d > d0 . So there are only two possibilities for µ1 and µ2 , either µ1 and µ2 are both negative or µ1 and µ2 are complex conjugates. To determine the linear approximation of U in the form of (3.4), we compute U10 analytically and derive the following result. a = β1 V1 + β2 V2 . Then there are the following two possibilities Theorem 4.3. Let A−1 0 for the linear term in the εc expansion with no Turing patterns, depending on the eigenvalues µ1 , µ2 of D −1 A. (1) If µ1 and µ2 are both real and negative, then U10 = [c1 (e−αx + c21 eαx ) − β1 x)]V1 + [c3 (e−βx + c43 eβx ) − β2 x)]V2 , √ √ where α = −µ1 and β = −µ2 .


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Figure 1. Examples for steady state patterns in the case d d0 . Here d = d0 − 117εd2 , where d0 ≈ 0.1167. The six figures correspond to the following values of εd and εc : (A) εc = 0, εd = 0.01, (B) εc = 0.001, εd = 0.01, (C) εc = 0.01, εd = 0.01, (D) εc = 0.01, εd = 0.001, (E) εc = 0.01, εd = 0.0001, (F ) εc = 0.01, εd = 0. The solid line is the u-component of the first order approximation given in (3.2) with the first order terms from theorems 4.1 and 4.2. The circles show a numerical solution of the steady state of the full nonlinear equations (2.2). (For a table of errors of the approximation, see table 1.) The graphs illustrate that for εd εc , the Turing pattern dominates, but for εd εc , the external chemical signal leads to a pattern over a linear baseline. See also figure 2. (Other parameters were a = 0.1, b = 0.9, L = 8π/k0 .)

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Figure 2. Examples for steady state pattern in the case d d0 . Again d = d0 − 117εd2 , where d0 ≈ 0.1167. The six figures correspond to the same values for εd and εc as in figure 1, with the only difference that the domain [0, L] is larger; here L = 30π/k0 . ((A) εc = 0, εd = 0.01, (B) εc = 0.001, εd = 0.01, (C) εc = 0.01, εd = 0.01, (D) εc = 0.01, εd = 0.001, (E) εc = 0.01, εd = 0.0001, (F ) εc = 0.01, εd = 0). The graphs illustrate that for εd εc , the Turing pattern dominates, but when εd εc , the external chemical signal leads to a pattern that has a very similar wave number as the Turing pattern, but its base line is shifted by a linear curve, and its amplitude typically has a minimum in the interval [0, L]. See also the discussion after theorem 4.3. (Other parameters were a = 0.1, b = 0.9.)

(2) If µ1 and µ2 are both complex, then U10 = (c1 s1 (x) − β1 x)V1 + (c3 s2 (x) − β2 x)V2 ,

(4.1)


where

√

µ1 = a1 + ib1 and

√

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µ2 = a2 + ib2 , √

s1 (x) = ei µ1 x + c21 e−i µ1 x = (c21 eb1 x + e−b1 x ) cos a1 x − i(c21 eb1 x − e−b1 x ) sin a1 x and √

√

s2 (x) = ei µ2 x + c43 e−i µ2 x = (c43 eb2 x + e−b2 x ) cos a2 x − i(c43 eb2 x − e−b2 x ) sin a2 x. In both (1) and (2), c1 and c3 are constants determined by the boundary conditions (U10 )x = 0 at x = 0 and x = L, √

c21

1 − ei µ1 L √ = 1 − e−i µ1 L

√

and

c43

1 − ei µ2 L √ = . 1 − e−i µ2 L

In the context of the model on which the equations (2.2) are based, we may again interpret these results as follows. In this case, a Turing pattern is not possible. Up to first order in εc , the resulting concentration is given by U10 . There are two cases: in case (1) there is no oscillatory pattern. However, in case (2), the presence of the terms cos(a1 x), sin(a1 x), cos(a2 x) and sin(a2 x) means that the resulting pattern does have an oscillatory appearance. Judging from our numerical experiments, we may expect the amplitude of this oscillation to be small compared with the linear terms in formula (4.1) for U10 , so that the appearance is that of a ‘wavy line’. (See figure 3.) It is interesting to see for which parameter ranges cases (1) and (2) apply, respectively. For this, consider again the parameter d, the ratio of the activator and inhibitor diffusivities. By considering the characteristic equation of the matrix D −1 A as a function of d, it is straightforward to derive that there is a fixed value d1 > d0 with the following property: for d0 < d < d1 , we are in case (2); that is, the eigenvalues of D −1 A are complex and the steady state still has an oscillatory component. For d > d1 , we are in case (1); that is, the eigenvalues of D −1 A are negative, and the steady state does not show any oscillatory behaviour. This result is interesting because it shows that in the presence of an external chemical gradient, a ‘Turing-like’ pattern may appear even in the parameter range where a Turing pattern is not possible; in particular when the diffusivities of activator and inhibitor are ‘close’. 5. Derivation of theorems In this section, we show the derivations of theorems 4.1, 4.2 and 4.3. The methods make repeated use of the Fredholm alternative theorem. See also [6, 15, 16]. 5.1. Derivation of theorem 4.1: determine U01 Collecting the εd terms, we get D0−1 AU01 + (U01 )xx = 0 which has the general solution c3 c4 c1 cos k0 x + c2 sin k0 x − x sin k0 x + x cos k0 x V1 U01 = 2k0 2k0 +( c3 cos k0 x + c4 sin k0 x)V2

(5.1)

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location x

15 location x

Figure 3. Four examples for steady state pattern in the case d > d0 . Here d0 ≈ 0.1167, and the figures show the cases (A) d = 0.12, (B) d = 0.2, (C) d = 1.0 and (D) d = 6.0. In each case εc = 0.01. The solid line is the first order approximation given in theorem 4.3. (There is good agreement with the numerical solutions to the full nonlinear equations (3.2); numerical results are not shown.) The examples illustrate the transition from case (2) to case (1) in theorem 4.3, which occurs here at the value d1 ≈ 5.483. It is the transition from complex eigenvalues of D −1 A to negative real eigenvalues of D −1 A, showing up graphically as the transition from a ‘wavy’ appearance to a straight curve. (Other parameters were a = 0.1, b = 0.9, L = 8π/k0 .)

with arbitrary constants c1 , c2 , c3 and c4 . To determine the constants c1 , c2 , c3 and c4 , we use the boundary conditions (U01 )x |x=0,L = 0, which lead to  c4   c2 k0 + V1 + c4 k0 V2 = 0,  2k0 (5.2) c3 L c4   cos k0 L V1 + c2 k0 − c4 k0 V2 = 0. + 2 2k0 Therefore c2 = c3 = c4 = 0 and U01 = c1 cos(k0 x)V1 with c1 free. We further determine c1 using the quadratic terms in the εd expansion of U . Collecting the εd 2 terms, we get AU02 + D0 (U02 )xx + D1 (U01 )xx + Q(U01 , U01 ) = 0. The nonhomogeneous terms in equation (5.3) take the following form cos 2k0 x 2 2 1 c1 k0 cos(k0 x)D1 V1 − c1 −D1 (U01 )xx − Q(U01 , U01 ) = + Q(V1 , V1 ) 2 2

(5.3)


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and we can assume that equation (5.3) has a particular solution Up = B0 + B1 (x) + cos(2k0 x)B2 , where B0 and B2 are constant vectors to be determined, whereas B1 (x) is a vector valued function induced by the nonhomogeneous cos k0 x term. Substituting Up into equation (5.3), we can write B0 and B2 in terms of c1 as 1 2 −1 B0 = − B2 = − 21 c12 (A − 4k02 D0 )−1 Q(V1 , V1 ). c A Q(V1 , V1 ), 2 1 These are valid since A is invertible by the original assumption and A − 4k02 D0 is invertible due to the fact that zero is the repeated eigenvalue of D0−1 A − k02 I . In addition, B1 (x) is actually a particular solution to the system 2 −1 r1 ck d d V (B1 )xx + D0−1 AB1 = c1 k02 cos(k0 x)D0−1 D1 V1 = cos k0 x 1 0 0 1 11 := cos k0 x r2 0 and therefore r1 r2 r2 r2 B1 = c1 cos k0 x + − 3 x sin k0 x + 2 x 2 cos k0 x V1 + x sin(k0 x)V2 2k0 2k0 8k0 8k0 and U02

r1 r2 r2 2 = c1 cos k0 x + − 3 x sin k0 x + 2 x cos k0 x V1 2k0 8k0 8k0 r2 + x sin(k0 x)V2 + B0 + cos(2k0 x)B2 . 2k0 The boundary conditions lead to r2 r1 r2 r2 L cos k0 L k0 L − 3 + 2L 2 V1 + V2 = 0. 2k0 2 8k0 8k0

and hence r1 = r2 = 0. Therefore, U02 = c1 cos(k0 x)V1 + B0 + cos(2k0 x)B2 with c1 free. Recall that 2 −1 1 c1 k02 d0−1 d1 V11 V22 r1 c k d d V = P −1 1 0 0 1 11 = , r2 0 c1 k02 d0−1 d1 V11 V12 |P | −   b−a (a + b)2  d0  detA = (a + b)2 and D0−1 A =  d0 (a + b) , 2b 2 − −(a + b) a+b we then have the repeated eigenvalue of D0−1 A, that is k02 = det(D0−1 A).

Solving

2b T (D0−1 A − k02 I )V1 = 0, we can derive that V1 is a multiple of ((a + b)2 + k02 , − a+b ) , which has no zero components and hence the only possibility for r1 = r2 = 0 and c1 = 0 is d1 = 0. To determine c1 , we need to consider the cubic terms in the εd expansion of U . Collecting the εd 3 terms, we get

AU03 + D0 (U03 )xx + D2 (U01 )xx + 2Q(U01 , U02 ) + C(U01 , U01 , U01 ) = 0.

(5.4)

Using the trigonometry identities cos k0 x cos 2k0 x = 21 (cos k0 x + cos 3k0 x)

and

cos3 k0 x =

3 4

cos k0 x + 41 cos 3k0 x,

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the nonhomogeneous terms in equation (5.4) take the following form −D2 (U01 )xx − 2Q(U01 , U02 ) − C(U01 , U01 , U01 ) = T0 + cos(k0 x)T1 + cos(2k0 x)T2 + cos(3k0 x)T3 with T0 = − c1 ( c1 + A1 )Q(V1 , V1 ), T1 = c1 (k02 D2 V1 − 2Q(V1 , B0 ) − Q(V1 , B2 )) − 43 c13 C(V1 , V1 , V1 ), T2 = − c1 ( c1 + A1 )Q(V1 , V1 ), T3 = − c1 Q(V1 , B2 ) − 41 c13 C(V1 , V1 , V1 ). We assume that equation (5.4) has a particular solution Up = S0 + S1 (x) + cos(2k0 x)S2 + cos(3k0 x)S3 , where S0 , S2 and S3 are constant vectors to be determined, whereas S1 (x) is a vector valued function induced by the nonhomogeneous cos(k0 x)T1 term and can be solved using the similar method discussed above. The general solution to (5.4) takes the following form: U03 = general solution to the homogeneous part + F0 + cos(2k0 x)F2 + cos(3k0 x)F3 + particular solution induced by cos(k0 x)T1 , where F0 , F2 and F3 are constant vectors that can be determined using T0 , T2 and T3 . As derived conditions, the second component of P −1 D0−1 T1 must above, for U03 to satisfy the boundary

c1 . Since be 0. That is P −1 D0−1 T1 · 1 = 0, which leads to an algebraic equation of 1 P −1 V1 = , D0−1 T1 must be a multiple of V1 or T1 is a multiple of D0 V1 . In the case T1 = 0, 0 c1 can be determined as follows: let V ⊥ be a vector that is perpendicular to D0 V1 , then c1 solves the algebra equation T1 , V ⊥ = 0 which takes the form 0

α1 c1 + α3 c13 = 0 with α1 = k02 D2 V1 , V ⊥ and α3 = Q(V1 , A−1 Q(V1 , V1 )), V ⊥ + 21 Q(V1 , (A − 4k02 D0 )−1 Q(V1 , V1 )), V ⊥ − 43 C(V1 , V1 , V1 ), V ⊥ .

Therefore, the nonzero solutions are c1 = ± −

α1 . α3

5.2. Derivation of theorem 4.2: determine U10 Collecting the εc terms, we get AU10 + D0 (U10 )xx

a = −x 0

and its general solution reads c3 c4 U10 = c1 cos k0 x + c2 sin k0 x − x sin k0 x + x cos k0 x V1 2k0 2k0 +(c3 cos k0 x + c4 sin k0 x)V2 − x(β1 V1 + β2 V2 ) where c1 , c2 , c3 and c4 are arbitrary constants and we assume a A−1 = β1 V1 + β2 V2 . 0

(5.5)


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To determine the constants c1 , c2 , c3 and c4 , we use the boundary conditions (U10 )x |x=0,L = 0, which lead to  c   c2 k0 + 4 V1 + c4 k0 V2 = β1 V1 + β2 V2 ,  2k0 (5.6) c4 c3 L   cos k0 L c2 k0 − V 1 + c 4 k0 V 2 = β 1 V1 + β 2 V 2 . + 2 2k0 If cos k0 L = −1, adding the above two equations leads to c3 L V1 = 2(β1 V1 + β2 V2 ) 2 which does not hold when β2 = 0. We assume that cos k0 L = 1 (see (3.3)), subtracting the above two equations we get c3 = 0 and c4 = β1 , c2 k0 + 2k0 (5.7) c4 k0 = β2 .

U10

Therefore, c4 = c1 cos k0 x + c2 sin k0 x + x cos k0 x V1 + (c4 sin k0 x)V2 − x(β1 V1 + β2 V2 ) 2k0 β1 β2 β2 − 3 and c4 = . k0 k0 2k0 We further determine c1 using the quadratic terms of εc . Collecting the εc 2 terms, we get

with c1 free, c2 =

AU20 + D0 (U20 )xx + Q(U10 , U10 ) = 0.

(5.8)

The nonhomogeneous term −Q(U10 , U10 ) in equation (5.8) expands into many terms. With Gi s and γi s being constant vectors, we group them into the following two groups: Group 1. x cos(k0 x)G1 , x sin(k0 x)G2 , x 2 cos(k0 x)G3 ; Group 2. γ0 + xγ1 + x 2 γ2 , cos(2k0 x)γ3 , sin(2k0 x)γ4 , x cos(2k0 x)γ5 , x sin(2k0 x)γ6 , x 2 cos(2k0 x)γ7 where G2 = 2Q(c2 V1 + c4 V2 , β1 V1 + β2 V2 ), G1 = 2c1 Q(V1 , β1 V1 + β2 V2 ), c4 G3 = Q(V1 , β1 V1 + β2 V2 ). k0 c 1 c4 γ1 = γ5 = − Q (V1 , V1 ) := c1 γ1 = c1 γ5 , γ4 = −c1 Q(V1 , c2 V1 + c4 V2 ) := c1 γ4 , 2k0 1 c4 2 1 c4 2 γ2 = −Q(β1 V1 + β2 V2 , β1 V1 + β2 V2 ) − Q (V1 , V1 ) , γ7 = − Q (V1 , V1 ) , 2 2k0 2 2k0 c4 γ6 = − Q(c2 V1 + c4 V2 , V1 ). 2k0 Finding the particular solutions induced by terms in group 2 is quite straightforward. We next derive the particular solutions induced by terms in group 1. • The particular solution to AI1 + D0 (I1 )xx = x cos(k0 x)

G11 r1 := c1 x cos(k0 x)D0 P G12 r2

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is

I1 = c1 α11 sin k0 x + α21 x cos k0 x + α31 x 2 sin k0 x + α41 x 3 cos k0 x V1 +c1 α51 x cos k0 x + α61 x 2 sin k0 x V2 with a0 = −

1 , 8k04

a1 =

1 , 4k03

a2 =

1 , 4k02

a3 = −

1 6k0

(5.9)

and α51 =

r2 , 4k02

r2 r1 − α51 , α11 = −α61 a0 , α21 = − α61 a1 , 4k0 4k02 r1 − α51 = − α61 a2 , α41 = −α61 a3 . 4k0

α61 = α31

• The particular solution to

G21 r1 := x sin(k0 x)D0 P = x sin(k0 x) G22 r2

AI2 + D0 (I2 )xx is

I2 = α12 cos k0 x + α22 x sin k0 x + α32 x 2 cos k0 x + α42 x 3 sin k0 x V1 + α52 x sin k0 x + α62 x 2 cos k0 x V2 with a0 , a1 , a2 and a3 given in (5.9) and α52 =

r2 , 4k02

r2 r1 − α52 , α12 = α62 a0 , α22 = − α62 a1 , 4k0 4k02 r1 − α52 =− + α62 a2 , α42 = −α62 a3 . 4k0

α62 = − α32

• The particular solution to

AI3 + D0 (I3 )xx

G31 = x cos(k0 x) G32 2

r1 := x cos(k0 x)D0 P r2

2

is

I3 = α13 cos k0 x + α23 x sin k0 x + α33 x 2 cos k0 x + α43 x 3 sin k0 x + α53 x 4 cos k0 x V1 + α63 cos k0 x + α73 x sin k0 x + α83 x 2 cos k0 x + α93 x 3 sin k0 x V2 , where α63 = r2 a0 , α73 = −r2 a1 , α83 = r2 a2 and α93 = −r2 a3 , with a0 , a1 , a2 and a3 given in (5.9). And α63 α73 a0 + (r1 − α83 ) a0 , α23 = − a1 , − 2 − (r1 − α83 ) a1 − α93 α13 = −α93 2k0 4k0 α33 =

α73 + (r1 − α83 ) a2 − α93 a2 , α43 = − (r1 − α83 ) a3 − α93 a3 , α53 = −α93 a4 4k0

with a0 = −

3 , 16k05

a1 = −

3 , 8k04

a2 =

3 , 8k03

a3 =

1 , 4k02

a4 = −

1 . 8k0


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Terms in group 1 then contribute the following terms in the particular solution: c1 x cos k0 x(α21 V1 + α51 V2 ), x 2 cos k0 x ((α32 + α33 )V1 + (α62 + α83 )V2 ) , c1 x 3 cos(k0 x)α41 V1 , x 4 cos(k0 x)α53 V1 , c1 sin(k0 x)α11 V1 , x sin k0 x ((α22 + α23 )V1 + (α52 + α73 )V2 ) , c1 x 2 sin k0 x(α31 V1 + α61 V2 ), x 3 sin k0 x ((α42 + α43 )V1 + α93 V2 ) . On the other hand, terms in group 2 contribute the following terms in the particular solution. • Solving AU + D0 Uxx = c1 x γ1 , we get a particular solution c1 xA−1 γ1 := c1 x(β11 V1 + β21 V2 ). • Solving AU + D0 Uxx = x 2 γ2 , we get a particular solution x 2 A−1 γ2 − 2A−1 D0 A−1 γ2 := x 2 (β12 V1 + β22 V2 ) + C. • Solving AU + D0 Uxx = c1 sin(2k0 x) γ4 , we get a particular solution c1 sin(2k0 x)(A − 4k02 D0 )−1 γ4 . • Solving AU + D0 Uxx = c1 x cos(2k0 x) γ5 , we get a particular solution c1 (x cos(2k0 x)P11 − sin(2k0 x)P01 ), where P11 = (A − 4k02 D0 )−1 γ5 , P01 = −4k0 (A − 4k02 D0 )−1 D0 (A − 4k02 D0 )−1 γ5 . • Solving AU + D0 Uxx = x sin(2k0 x)γ6 , we get a particular solution x sin(2k0 x)P12 + cos(2k0 x)P02 , where P12 = (A − 4k02 D0 )−1 γ6 , P02 = −4k0 (A − 4k02 D0 )−1 D0 (A − 4k02 D0 )−1 γ6 , • Solving AU + D0 Uxx = x 2 cos(2k0 x)γ7 , we get a particular solution x 2 cos(2k0 x)P2 + cos(2k0 x)P03 − x sin(2k0 x)P13 , where P2 = (A − 4k02 D0 )−1 γ7 , P13 = −8k0 (A − 4k02 D0 )−1 D0 P2 , P03 = −2(A − 4k02 D0 )−1 D0 P2 + 4k0 (A − 4k02 D0 )−1 D0 P13 . Therefore, the effective terms (after taking the x derivative and evaluating at x = 0 and x = L) induced by terms in group 2 are c1 x(β11 V1 + β21 V2 ), x 2 (β12 V1 + β22 V2 ), c1 sin(2k0 x)(β13 V1 + β23 V2 ), c1 x cos(2k0 x)(β14 V1 + β24 V2 ), x sin(2k0 x)(β15 V1 + β25 V2 ), x 2 cos(2k0 x)(β16 V1 + β26 V2 ), where (A − 4k02 D0 )−1 γ4 − P01 = β13 V1 + β23 V2 , P11 = β14 V1 + β24 V2 , P12 − P13 = β15 V1 + β25 V2 , P2 = β16 V1 + β26 V2 .

U20

The general solution A3 A4 = A1 cos k0 x + A2 sin k0 x − x sin k0 x + x cos k0 x V1 2k0 2k0 1 2 +(A3 cos k0 x + A4 sin k0 x)V2 + U20 + U20 ,

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1 2 where A1 , A2 , A3 and A4 are arbitrary constants and U20 and U20 are the particular solutions generated by terms in groups 1 and 2, respectively. The boundary conditions (U20 )x |x=0,L = 0 lead to  A4  1 2  V1 + A4 k0 V2 + (U20 )x |x=0 + (U20 )x |x=0 = 0,  A2 k0 + 2k0 (5.10) A3 L A4  1 2  V1 + A4 k0 V2 + (U20 + )x |x=L + (U20 )x |x=L = 0.  A2 k0 − 2 2k0

Dividing the second equation by cos k0 L and subtracting it from the first equation, we get A3 L + L1 (c1 ) V1 + L2 (c1 )V2 = 0, 2 where L1 (c1 ) and L2 (c1 ) are linear functions of c1 . Hence c1 can be determined by solving the linear equation L2 (c1 ) = 0. By direct computation, L2 (c1 ) = l1 c1 − l0 , where l1 = −k0 L2 α61 , l0 = 2L(k0 β25 + β26 + β22 + α62 + α83 ) + k0 L(α52 + α73 ) + k0 L3 α93 . 5.3. Derivation of theorem 4.3 To derive theorem 4.3, we write the general solution to (5.5) as the linear combination of the general solution to the homogeneous part and a particular solution to the nonhomogeneous equation, i.e. U10 = c1 ei

√

µ1 x

V1 + c2 e−i

√

µ1 x

V1 + c 3 e i

√

µ2 x

V2 + c4 e−i

√

µ2 x

V2 − x(β1 V1 + β2 V2 )

with c1 , c2 , c3 and c4 being arbitrary constants. In addition, the boundary conditions (U10 )x = 0 at x = 0 and x = L imply that √ √ c1 − c2 = c1 ei µ1 L − c2 e−i µ1 L , √ √ c3 − c4 = c3 ei µ2 L − c4 e−i µ2 L . Therefore, U10 = c1 (ei

√

µ1 x

+ c21 e−i

√

µ1 x

)V1 + c3 (ei

√

µ2 x

+ c43 e−i

√

µ2 x

)V2 − x(β1 V1 + β2 V2 ),

where c1 and c3 are constants determined by the boundary conditions, and √

c21

1 − ei µ1 L √ = 1 − e−i µ1 L

√

and

c43

1 − ei µ2 L √ = . 1 − e−i µ2 L

(i) If µ1 and µ2 are both real and negative, then U10 = [c1 (e−αx + c21 eαx ) − β1 x)]V1 + [c3 (e−βx + c43 eβx ) − β2 x)]V2 , √ √ 1 − eαL 1 − eβL −µ1 and β = −µ2 , c21 = and c = . No oscillations 43 1 − e−αL 1 − e−βL are expected in the solution. (ii) If µ1 and µ2 are complex conjugates, assume that √ √ µ1 = a1 + ib1 , µ2 = a2 + ib2 . where α =

Then U10 = (c1 s1 (x) − β1 x)V1 + (c3 s2 (x) − β2 x)V2 ,


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where s1 (x) = ei

√

µ1 x

+ c21 e−i

√

µ1 x

= (c21 eb1 x + e−b1 x ) cos a1 x − i(c21 eb1 x − e−b1 x ) sin a1 x

and s2 (x) = ei

√

µ2 x

+ c43 e−i

√

µ2 x

= (c43 eb2 x + e−b2 x ) cos a2 x − i(c43 eb2 x − e−b2 x ) sin a2 x.

In this case, c1 , c3 , c21 and c43 are complex constants determined by the boundary √ √ conditions, both µ1 and µ2 have non-zero real parts, therefore sine and cosine functions are involved in the solution and oscillations may occur. 6. Conclusions and future investigations We investigated the effect of an external morphogen gradient on the reaction–diffusion equations in our model system, using regular perturbation methods in two small parameters. We first considered the case that the ratio of activator and inhibitor diffusivities is below a critical value and thus a Turing pattern can form. Then the effect of the external signal is that the wave number essentially remains unchanged, but the amplitude of the pattern is nonconstant, see figures 1 and 2. Surprisingly, if the ratio of activator and inhibitor diffusivities is above the critical value and thus a Turing pattern cannot form, the external signal nevertheless induces a ‘Turing-like’ pattern close to the critical value, see figure 3. Our analytical results were verified by numerical computation, see table 1. There are several further questions that are raised by our results. While it is very interesting that the system can produce Turing-like patterns in parameter ranges where a Turing pattern could not form without the external gradient, we noted from our numerical results that the amplitude of the wave pattern is typically very small, leading to a very ‘shallow’ wavy pattern. So at least for the examples in figure 3(B)–(D), the amplitude appears to be too small to be biologically relevant. It would be interesting to determine whether there exist any parameter ranges with a ratio of diffusivities significantly above the critical value which do, however, exhibit patterns with a large amplitude. In principle, determining whether this is the case should be possible with the explicit formula for the first order expansion of the pattern given in theorem 4.3. This analysis is, however, made difficult by the fact that the dependence of the amplitude on the parameters of the original equations is quite complicated. Another interesting question is how the presence of the external gradient influences the timing of pattern formation; that is, whether the full time dependent system with the external gradient takes longer or shorter to reach a steady state than the system without the external gradient. From our numerical experiments, it appears that the presence of the gradient slows down patterning, that is, our examples suggest that the larger the influence of the external gradient, the longer it takes to reach the steady state. This too is potentially relevant for biological applications, and it would be desirable to develop a general theory to determine whether these tentative findings from our numerical examples are true in general. One of the limitations of our work is the restriction to one spatial dimension. An extension to two dimensions would be interesting. Also, we did not analyse the stability of the steady state patterns in this paper, although from our numerical explorations, the examples for figures 1 and 3 appeared to be stable. It would be interesting to settle this question. Furthermore, other forms for the profile of the external chemical signal than the simple linear form are possible and motivated by biological models, such as a profile obtained by solving more realistic model equations for the spread of the morphogens with one or more centres of secretion. Our generic model, while interesting in its own right, points to certain general features about the interaction between reaction–diffusion Turing patterns and external morphogen

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gradients that are set up independent of the Turing pattern and influence the expression of activator and/or inhibitor molecules. It would be interesting to investigate more specific models in depth where such interaction is likely to play a role such as models of formation of hair follicles in mouse skin [17] and models of bone formation embryonic in the chick limb [2, 4, 5, 8]. References [1] Alber M, Glimm T, Hentschel H, Kazmierczak B and Newman S 2005 Stability of n-dimensional patterns in a generalized Turing system: implications for biological pattern formation Nonlinearity 18 125–38 [2] Alber M, Glimm T, Hentschel H G E, Kazmierczak B, Zhang Y-T, Zhu J and Newman S A 2008 The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb Bull. Math. Biol. 70 460–83 [3] Baker R E, Gaffney E A and Maini P K 2008 Partial differential equations for self-organization in cellular and developmental biology Nonlinearity 21 R251–90 [4] Dillon R, Gadgil C and Othmer H G 2003 Short- and long-range effects of sonic hedgehog in limb development Proc. Natl Acad. Sci. USA 100 10152–7 [5] Dillon R and Othmer H G 1999 A mathematical model for outgrowth and spatial patterning of the vertebrate limb bud J. Theor. Biol. 197 295–330 [6] Ermentrout B 1991 Stripes or spots? Nonlinear effects in bifurcation of reaction–diffusion equations on the square Proc. R. Soc. Lond. Ser. A 434 413–7 [7] Forgacs G and Newman S 2005 Biological Physics of the Developing Embryo (Cambridge: Cambridge University Press) [8] Hentschel H G E, Glimm T, Glazier J A and Newman S A 2004 Dynamical mechanisms for skeletal pattern formation in the vertebrate limb Proc. R. Soc. Lond. B Biol. Sci. 271 1713–22 [9] Knobloch E 2008 Spatially localized structures in dissipative systems: open problems Nonlinearity 21 T45–60 [10] Lander A D, Nie Q and Wan F Y 2002 Do morphogen gradients arise by diffusion? Dev. Cell 2 785–96 [11] Merkin J H and Sleeman B D 2005 On the spread of morphogens J. Math. Biol. 51 1–17 [12] Murray J D 2002 Mathematical Biology. I (Interdisciplinary Applied Mathematics vol 17) 3rd edn (New York: Springer) [13] N¨usslein-Vollhard C 1996 Gradients that organize embryo development Sci. Am. 56–61 [14] N¨usslein-Vollhard C 2006 Coming to Life: How Genes Drive Development (Carlsbad, CA: Kales Press) [15] Sattinger D H 1973 Topics in Stability and Bifurcation Theory (Lecture Notes in Mathematics vol 309) (Berlin: Springer) [16] Sattinger D H 1979 Group-Theoretic Methods in Bifurcation Theory (Lecture Notes in Mathematics vol 762) (Berlin: Springer) (With an appendix entitled ‘How to find the symmetry group of a differential equation’ by Peter Olver) [17] Sick S, Reinker S, Timmer J and Schlake T 2006 WNT and DKK determine hair follicle spacing through a reaction–diffusion mechanism Science 314 1447–50 [18] Turing A M 1952 The chemical basis of morphogenesis Phil. Trans. R. Soc. (B) 237 37–72 [19] Wolpert L, Beddington R, Brockes J, Jessell T, Lawrence P and Meyerowitz E 1998 Principles of Development (New York: Oxford University Press)

Turing patterns with an external linear morphogen gradient - Western ...

Turing patterns with an external linear morphogen gradient - Western ...

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