Existence and stability analysis of spiky solutions for the Gierer–Meinhardt system with large reaction rates

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Abstract

We study the Gierer–Meinhardt system in one dimension in the limit of large reaction rates. First we construct three types of solution: (i) an interior spike; (ii) a boundary spike and (iii) two boundary spikes. Second we prove results on their stability. It is found that an interior spike is always unstable; a boundary spike is always stable. The two boundary spike configuration can be either stable or unstable, depending on the parameters. We fully classify the stability in this case. We characterize the destabilizing eigenfunctions in all cases. Numerical simulations are shown which are in full agreement with the analytical results.

Introduction

In this paper, we study the Gierer–Meinhardt system in the limit of large reaction rates. Let us first put it in the context of Turing’s diffusion-driven instability. Since the work of Turing [1] in 1952, many models have been established and investigated to explore the so-called Turing instability [2]. One of the most famous models in biological pattern formation is the Gierer–Meinhardt system [3], [4], [5], which in one dimension can be stated as follows: At=DAΔAA+ApHq,x(1,1),t>0,τHt=DHΔHH+AmHsx(1,1),t>0,Ax(±1,t)=Hx(±1,t)=0,t>0, where all of the parameters are positive and (p,q,m,s) satisfy 1<qm(s+1)(p1)<+,1<p<+.

In all of the recent mathematical investigations it was assumed that the activator diffuses much slower than the inhibitor, that is DHDA, a condition which is related to those required for Turing instability [1]. See Chapter 2 of [2] for a thorough investigation. Mathematically, this assumption allows perturbation techniques to be employed, since the inhibitor can be assumed to be a slow-varying variable, and the system becomes only weakly coupled, as first observed in [6]. The activator then exhibits spikes–regions of steep gradients–separated by regions where the activator is nearly zero. For any given number N, a steady state containing N such spikes can be constructed. In [7], [8] it was shown that there exists a sequence of numbers D1>D2> (which has been given explicitly) such that if DH<DN the symmetric N-spike solution is stable, while for DH>DN the symmetric N-spike solution is unstable.

We now introduce the setting of this paper. In contrast with the above-mentioned works, we do not assume the large diffusivity ratio (2). Instead, we study the the limit of large reaction rates of the activator. More precisely, we assume that p,m1with O(pm)=1. To simplify our analysis, we make the following assumptions: q=1,s=0. Further, we require that τ=0. Assumption (5) will be important for the stability analysis. (We expect that the analysis in this paper can be extended to the case τ small enough with minor changes. When τ is sufficiently large, Hopf bifurcations can occur, see for example [9]. The analysis of the case τ large is more complicated, and is left for future work.) For convenience, we rewrite m=(p1)r,r=O(1). By rescaling the space as xDAx and introducing D=DHDA,L=DA1/2, the system becomes At=AxxA+ApH;τHt=DHxxH+A(p1)r,x(L,L),Ax(±L)=0=Hx(±L),L,D=O(1);p1;r>1andr=O(1).

Now we describe some previous work and motivate the study of large reaction rates from biological applications.

Hunding and Engelhardt [10] considered the effect of large reaction rates on Turing’s instability for several well-known reaction-diffusion systems (the Sel’kov model, Brusselator, Schnakenberg model, Gierer–Meinhardt system, Lengyel–Epstein model). By increasing the reaction rate (or the so-called Hill constant for Hill-type kinetics) which models cooperativity in the system, they showed, through a linearized stability analysis, that pattern formation by Turing’s mechanism is facilitated by large reaction rates, even when the ratio of the diffusion constants is close to one.

The Hill equation assumes that many molecules can interact simultaneously, which is not a very realistic assumption [11]. Instead, it is a more realistic physical assumption that one ligand molecule after another is being bound to a receptor. This can happen in basically two different ways: By a sequential binding mechanism, for which the order in which the sites are filled is determined, or an independent binding mechanism, for which the sites can be occupied independently. Although for these two processes the Hill number is smaller than the number of sites, they can still lead to high Hill numbers.

A interesting case study about the molecular basis of cooperative interactions has recently been given in [12]. The mechanism of the binding of Calcium ions to Calmodulin, a multi-site and multi-functional protein, has been modeled quantitatively and the theoretical results have been confirmed by experimental observations.

In [13] evidence has been found for the fact that protein subunits can degrade less rapidly when associated in multimeric complexes, an effect which is called cooperative stability.

The assumption of large reaction rates is reasonable for models of pattern formation induced by gene hierarchy due to their high degree of cooperativity [10]. High cooperativity plays an important role even for rather primitive animals and plants like flatworm, ciliates, fungi and has been well investigated in Drosophila. In the latter example the homeobox genes are known to play a major role [14], [15] in facilitating a high degree of cooperativity. Key ingredients of the gene hierarchy have been identified such as the maternal gene bicoid, the gap gene hunchback and the primary pair-rule genes, which are expressed in a series of seven equally spaced and precisely phase shifted stripes. The occurrence of these stripes can be explained by a Turing mechanism in combination with maternal and gap gene interactions. These mechanisms have been reviewed in [16], [17], and [18].

The reason why cooperativity for homeobox genes is high is their ability to create proteins which can bind to several other genes, in this process activating or inhibiting them. Experimentally reaction rates exceeding 8 have been found for several different gene control systems. (Note that even for p=8 the steady state solution u constructed below is already very well approximated by a spike on a real line since its spatial decay rate is of the order e14|x|.) An explicit example is the pair rule gene hairy which was originally connected to the nervous system but plays a role in the initial body plan of Drosophila as well.

A high degree of cooperativity leads to a whole class of control systems with large reaction rates which can explain the emergence of a variety of complex patterns. These systems can read out and remember gradients in the positional information, a fact which is important since this information must often be used repeatedly for example in the anterior–posterior or dorsal–ventral gradients in Drosophila. These systems further have the ability to react in an almost on–off manner to very shallow gradients in positional information, a property which plays a major role for example in controlling the cell cycle governing mitosis, where the properties of the system must change qualitatively if its size is increased by a factor two. Further behaviors of solutions to the resulting nonlinear systems include time oscillations and multi-stability, the latter being important for modeling cell differentiation.

In this paper, we give a first analysis on Turing’s nonlinear patterns in the case of large reaction rates. The model we consider is the Gierer–Meinhardt system, although our analysis can be extended to other reaction-diffusion systems with large reaction rates (as in [10]).

We now present the main results of the paper. The first result is about the existence of steady states with one or two spikes which is formulated using the Sobolev space with Neumann boundary conditions defined by HN2(L,L)={uH2(L,L):uz(L)=uz(L)=0}.

Theorem 1

(a) Consider the system0=AxxA+ApH;0=DHxxH+A(p1)r,wherex(L,L),Ax(±L)=0=Hx(±L).We assume that D,r,L are positive and fixed and setα1p1.If p is large enough (i.e. if α is small enough), then(7)admits a solution (A,H)(HN2(L,L))2 such thatA(x)={(H0η3α)αwα(ηαx)(1+O(α)),|x|1,1ααcosh(|x|L)cosh(L)(1+O(α)),|x|O(α),H(x)=H0cosh(|x|LD)cosh(LD)(1+O(α)),whereH0=αηr1/2r1[2β1D1/2tanh(LD)]1/(r1)(1+O(α)),η=tanh2(L),β=(12)rsech2r(y2)dy,and w(y)=32sech2(y2) is the unique ground state solution towyyw+w2=0,w>0,wy(0)=0,w(y)Ce|y|,|y|.Further, the error is estimated in the following space:αA1/α(αz)H0η3w(ηz)H2(L/α,L/α)=O(α).

(b) The restriction of the solution constructed in Part (a) to the interval (0,L) is a solution (A,H)(HN2(0,L))2the system(7), whereAx(0)=Ax(L)=0=Hx(0)=Hx(L).

(c) The extension of the solution constructed in Part (b) from the interval (0,L) to interval (0,2L) by even reflection at x=L is a solution (A,H)(HN2(0,2L))2 to the system(7), whereAx(0)=Ax(2L)=0=Hx(0)=Hx(2L).

Remarks

1. The steady state in (a) has an interior spike for Ap located in the center x=0 of the interval. The solution in (b) has a boundary spike for Ap located at the left boundary x=0. The solution in (c) has two boundary spikes for Ap located at the boundaries x=0 and x=2L.

2. The key observation is that, unlike in the case of a slowly diffusing activator (DADH), the activator A does not look like a spike; nonetheless, its power Ap does. This is illustrated in Fig. 1, where both A and Ap are plotted. Note that Ap is localized near the origin, while A is not.

3. A remarkable fact is that in the above theorem the ratio of the two diffusivities D can be any finite number.

4. We will construct these steady-state solutions in Section 2. The solutions each consist of an inner and outer region, and its construction is done by the method of matched asymptotics.

The second main result of this paper is the stability analysis for these solutions. We summarize it as follows.

Theorem 2

Suppose p is large enough. Then we have the following results for the stability of the steady states constructed inTheorem 1:

(a) The interior spike is unstable. The eigenvalue problem has an eigenvalue with positive real part of exact order O(1) which is given in(67). The corresponding eigenfunction is an odd function.

(b) The boundary spike is stable.

(c) The steady state with two boundary spikes is stable if D<Dc and it is unstable if D>Dc , where Dc is given byDc=(Larctan(1/r))2.If D>Dcthere is an eigenvalue λ with Re(λ)=O(p2) . The corresponding eigenfunction is odd about x=L .

The two instabilities of Theorem 2 are shown in Fig. 2. The instability of the interior spike induces a spike motion towards the boundary and, due to the small eigenvalue, happens on a slow timescale O(1). On the other hand, the instability of the boundary spike occurs on a much faster timescale O(1p2), corresponding to a “large”eigenvalue. As a result, one of the two boundary spikes is annihilated.

Note that a multi interior spike solution can be constructed from an interior spike solution by even reflection. However, since a single interior spike is unstable, this multi-spike configuration is also automatically unstable. This is because an eigenfunction corresponding to one interior spike can be extended by even reflection to an eigenfunction for K interior spikes. So an unstable mode of a single spike automatically induces an instability for K spikes.

Finally, it should be mentioned that both Theorem 1, Theorem 2 can be made rigorous by the method of Liapunov–Schmidt reduction as used in [8]. We will give an outline of the proof of Theorem 1 in Remark 4 following the result.

We now summarize the contents of the paper. In Section 2 we use asymptotic analysis to construct the steady-state solution given in Theorem 1. In Section 3 we derive the eigenvalue problem. In Section 4 we consider the large eigenvalues and reduce the eigenvalue problem to a nonlocal eigenvalue problem. In Theorem 3 we fully classify its unstable eigenfunctions and their eigenvalues. When r=2 we are able to obtain necessary and sufficient conditions for stability. In Theorem 4 we state the corresponding result on a large bounded interval. In Section 5 we study the small eigenvalues. We show that all corresponding unstable eigenfunctions are odd. In Section 6 we apply the results of Section 4 and Section 5 to prove Theorem 2. Applying the results of Section 4, the stability of a boundary spike is established. Applying the results of Section 5 to the case of an interior spike, we show that there is a small eigenvalue with a positive real part, so that an interior spike is unstable. Applying the results of Section 4 to the case of a double boundary spike, we show that there is a large eigenvalue whose real part is positive or negative, depending on the condition on D<Dc or D>Dc for some Dc>0, respectively. All the other eigenvalues have a negative real part, so the double boundary spike can be stable or unstable. In Section 7 we discuss our results, emphasize their relevance for pattern formation in reaction-diffusion systems and biological application, and we conclude by stating some open problems.

Section snippets

Construction of the steady state

In this section we construct the steady state using asymptotic matching and prove Theorem 1. As a motivation, note that a solution to the ODEvxxv+vp=0,xR, is explicitly given by v(x)=[(p+12)sech2(p12x)]1p1. Taking into consideration the scalings of the spatial variable and of the amplitude of this solution, for p1, this motivates the following change of variables: A(x)=(u(z)α)α,z=xα, where α=1p11. With this rescaling, we anticipate that u will be independent of p at the leading order. We

Stability

We now study the linear stability of the non-homogeneous steady state. Linearize around the steady state as: A(x,t)=A(x)+eλtϕ(x),H(x,t)=H(x)+eλtψ(x), where (A(x),H(x)) is the steady state solution as given by Theorem 1. We obtain λϕ=ϕxxϕ+pAp1ϕHApH2ψ,0=Dψxxψ+r(p1)A(p1)r1ϕ. As before, we make the change of variables given in (15). Since A1 near x0 we have Ap=uαAuα. We obtain α2(λ+1)ϕα2ϕxx+uHϕ+αuHϕαuH2ψ,0Dψxxψ+rαr1urϕ. For an interior spike which is symmetric about the origin,

Large eigenvalues

We start by analyzing the large eigenvalues. Changing to inner variables, we have x=αηy;uH03ηw(y); and we obtainα2η(λ+1)ϕϕyy+13wϕα3ψ0H0w, where ψ0=ψ(0) and ψ(0) is determined by DψxxψC1δ(x);ψx(±L)=0,C1=(uα)rrαϕdx=rη(H0η3α)rwrϕdy. This implies thatψ(x)=C1G(0), where G(x)=cosh(L|x|D)2Dsinh(LD) is the Green’s function satisfyingDGxxG=δ(x),Gx(±L)=0. On the other hand, from (21) we haveH0=αη(H0η3α)rwrdyG(0). So the boundary conditions (25) lead to following dimensionless

Small eigenvalue

It remains to study the stability of small eigenvalues. In particular, we prove the following result.

Theorem 7

Consider the eigenvalue problem(22a), (22b)with the boundary conditions(26). If p1this problem admits a positive eigenvalue λthat satisfiesλ+1tanhLtanh(Lλ+1)=1+O(1p).

To start with, expand the inner region to two orders for both the eigenfunction and the steady state: x=αz;u=U0(z)+αU1(z)+H=H0+αH1(z)+ϕ=Φ0(z)+αΦ1(z)+Ψ=Ψ0+. The leading order equations are Φ0zz+U0H0Φ0=0;U0zU0z2U0+U02H0=0;H0

Proof of Theorem 2

We now complete the proof of Theorem 2. First, consider the case of a single boundary spike. In this case, the eigenfunction satisfies the boundary condition (25). The corresponding eigenvalue problem is given by (29); this is equivalent to (32) with γ=r31wrdy. But since we take r>1, we have γ>γ0 where γ0 is given by (33); hence Re(λ)c0<0 by Theorem 4. This proves the stability of a single boundary spike.

Next, consider the case of an interior spike centered at the origin. It admits two

Discussion

In this paper we have studied the Gierer–Meinhardt system with large reaction rates. Formal asymptotics were used to construct the steady state solutions; their stability was analyzed using a combination of formal computations and rigorous analysis. The main result, Theorem 2, is the classification of the stability of interior and boundary spike solutions. The behavior of the system differs significantly from the “standard”GM system (1). In particular, an interior spike is unstable with respect

Acknowledgements

We thank the anonymous referees. Their insightful comments and questions helped to strengthen the paper significantly. The work of JW is supported by an Earmarked Grant of RGC of Hong Kong. TK is supported by an NSERC discovery grant, Canada. MW and TK would like to thank the Department of Mathematics at CUHK for their kind hospitality, where a part of this paper was written.

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