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Reaction-diffusion equations
and biological pattern formation
Anna Marciniak-Czochra
Dr. Anna Marciniak-Czochra Group leader Professor (W3) of Applied Analysis and Modelling in Biosciences (accepted appointment) Interdisciplinary Center for Scientific Computing (IWR), Center for Modelling and Simulation in the Biosciences (BIOMS) Institute of Applied Mathematics and BIOQUANT
University of Heidelberg
E-mail: [email protected] URL: http://www.biostruct.uni-hd.de, http://www.marciniak-czochra.uni-hd.de
Anna Marciniak-Czochra is a head of a research group “Mathematics of self-organisation in cell systems” and recently appointed as professor of Applied Analysis and Modelling in Biosciences. She studied mathematics at the University of Warsaw, Poland, and received her PhD in 2004 at the University of Heidelberg under supervision of Prof. Willi Jäger. She worked also as a visiting member at Mathematical Biosciences Institute at Ohio-State-University in USA. Since 2008 she is a leader of a research group funded by the prestigious grant “Ideas” of European Research Council (ERC) and Emmy-Noether-Programme of German Research Council. She is also a member of the Junior Scientist Program WIN of Heidelberg Academy of Sciences in Humanities and a member of newly granted Collaborative Research Center, SFB 873 "Maintenance and Differentiation of Stem Cells in Development and Disease". She is a supervisor of PhD projects in the International Ph.D. Programme: Mathematical Methods in Natural Sciences established between University of Warsaw, University of Wroclaw, University of Pierre and Marie Curie in Paris, Charles University in Prague and University of Heidelberg and prinicipal investigator or consultant in several international projects.
Her transdisciplinary expertise lies in the areas of applied mathematics and mathematical biosciences. Specifically, her field of focus is the dynamics of pattern formation and self-regulation in developmental processes and cancer. Her mathematical areas of focus are nonlinear partial differential equations, dynamical systems, and multiscale analysis.
Anna Marciniak-Czochra (University of Heidelberg)
Reaction-diffusion equations and biological pattern formation
Partial differential equations of diffusion type have long served to model regulatory feedbacks and pattern formation in aggregates of living cells. Classical mathematical models of pattern formation in cell assemblies have been constructed and developed using reaction-diffusion equations. They have provided explanations of pattern formation for animal coat markings, bacterial and cellular growth patterns, angiogenesis (blood vessels), tumour growth and tissue development. Due to the recent development of new modelling approaches, reaction-diffusion equations are the subject of new mathematical interest concerning mechanisms of pattern formation and unbounded growth of solutions.
The lectures are devoted to the analysis of systems of reaction-diffusion equations, and their applications to describe processes of biological pattern formation. After presenting classical analytical results concerning existence, uniqueness and regularity of solutions,, we provide a set of tools allowing a comparison between the dynamics of reaction-diffusion models and their ODEs counterparts, such as comparison principle and theory of bounded invariant rectangles (Lectures 1 and 2). Then, we focus on the analysis of mechanisms of pattern formation based on Turing-type instability and hysteresis (Lecture 3). The last part of the course is devoted to the models of coupled reaction-diffusion equations and ordinary differential equations ( models with degenerated diffusion,). We show the derivation of such models based on the homogenisation techniques (Lecture 4) and, then, follow the analysis of homogeneous and nonhomogeneous stationary solutions and their stability(Lecture 5).
Analytical framework is illustrated by several examples of the applications, including classical Turing patters, spike patterns in the models with degenerated diffusion and transition layers arising from multistability effects.
References:
D. Henry, Geometric theory of semilinear parabolic equations. Springer-Verlag, New York, 1981.
A. Marciniak-Czochra, G. Karch and K. Suzuki. Unstable patterns in reaction-diffusion model of early carcinogenesis.
A. Marciniak-Czochra, M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenisation techniques. SIAM J. Mat. Anal. 40 (2008), 215-237.
J.D. Murray, Mathematical Biology. Springer-Verlag, 2003. M. Pierre, Global existence in reaction-diffusion systems with control of mass: a
survey, Milan J. Math. 78 (2010), 417--455. F. Rothe, Global solutions of reaction-diffusion systems. Lecture Notes in
Mathematics, 1072. Springer-Verlag, Berlin, 1984. J. Smoller, Shock waves and reaction-diffusion equations. Second edition.
Grundlehren der Mathematischen Wissenschaften, 258. Springer-Verlag, New York, 1994.
A.M. Turing The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. B 237 (1952), 37-72.
Reaction-diffusion equations and biological patternformation
Anna Marciniak-Czochra
Interdisciplinary Center for Scientific Computing (IWR)Institute of Applied Mathematics and BIOQUANT
University of HeidelbergIm Neuenheimer Feld 267, 69120 Heidelberg, Germany
June 7, 2011
1 Outline
1. Biological motivation and models of pattern formation
2. Derivation of receptor-based model (RD equations + ODEs) using homogenisation
3. Mechanisms of pattern formation in receptor-based models
4. Diffusion-driven instability and Turing type models
5. Multistability, hysteresis and transition layeres
1
2 Biological systems and their mathematical models
2.1 Challenges:
• Complexity of biological systems- multiple scales (tissues, cells, molecules)- nonlinear regulatory feedbacks- transport processes
• Enormous amount of biological data on molecular level
• Mathematical models needed to understand the processes
2.2 Research paradigm
and experimentsBiological theories
and their analysisMathematical models
Suggested new experiments
Inspiration
2.3 Pattern formation in developmental processes
Understanding the evolution of spatial patterns and the mechanisms which create them areamong the crucial issues of developmental biology. During the development of multicellularorganisms, embryonic cells choose differentiation programs based on positional information.This information is delivered by extracellular signalling molecules which are highly conservedand regulate the growth and differentiation of cells in all metazoans including humans [6]. Inother areas of biology, such as neurophysiology or ecology, mathematical modelling has led tomany discoveries and insights through a process of synthesis and integration of experimentaldata (see e.g. [15] and references therein). Also in developmental biology many differentmorphologies have been the subject of mathematical modelling, e.g., [4, 19]. Some biologicalsystems have attained the status of “paradigm” in theoretical work.
2
The embryo of Xenopus frog
2.4 Spatio-temporal models
• We model dynamics of biological process using mathematical variables describing con-centrations of chemicals, enzymes, cells densities, subspecies of populations, physicalquantities...
• We consider densities (concentrations) of some substances (molecules, morphogens) asfunctions of space and time u(x, t).
Rate of changeof u concentration
= local dynamicsof u interactions(de novo production,binding, dissocation,decay)
+ spatial effects(long and shortrange diffusion,convection,chemotaxis).
2.5 Mechanisms of pattern formation
• Reaction-diffusion process
– Diffusion Driven Instabilities (Turing-type mechanism)
– Multistability and transition layers
• Reaction-diffusion-taxis process
– Chemotaxis and aggregation
– Merging and emerging patterns
– Haptotaxis and movement of the front
3
2.6 Reaction-diffusion systems
ut =1
γD∆u+ f(u), in Ω×R+ (1)
∂nu = 0 on ∂Ω
u(0, x) = u0(x)
Here u ∈ Rn, D is the positive definite diagonal matrix and F : Rn → Rn is a smoothvector field, γ is a scaling parameter, γ = L2
dmax.
Stationary solutions
1
γD∆U(x) + f(U(x)) = 0,
∂nU(x) = 0,
• Spatially homogeneous steady states
• Spatially heterogeneous steady states (also discontinuous in case of RD-ODE models)
2.7 Test organism - a fresh-water polyp Hydra
Hydra, a fresh-water polyp, is a simple organism which can be treated as a model for axisformation and regeneration in higher organisms [5, 6]. It has a tubular body about 5 mm longwith a whorl of tentacles surrounding the mouth at the upper end and a disk-shaped organfor adhesion at the lower end. The longitudinal pattern is subdivided into a head, a gastricregion, a budding zone (where new animals are generated by a process of natural cloning), astalk and a foot. The developmental processes governing formation of the Hydra body planand its regeneration are well understood at the tissue level [14]. Experiments performedon Hydra suggest that the function of cells is determined by their location. Hydra retainsa population of stem cells which are activated when needed. Morphogenetic mechanismsactive in adult polyps are responsible for the regenerative ability and the establishment of anew body axis. Research on Hydra might reveal how to selectively reactivate the genes andproteins to regenerate human tissues.
Although much is known at the tissue level, the molecular basis for self-organisation ofthe body plan of Hydra is not well understood. Since cell differentiation and proliferation iscontrolled by signalling pathways, it is necessary to bridge the gap between observations atthe tissue level and at the cellular and subcellular level.
2.8 Objectives
• To understand mechanism of head formation and regeneration
4
• To bridge the gap between observations at the tissue level and at the cellular andsubcellular level
2.9 Concept of pattern generation
The body architecture of Hydra is relatively simple. Polyp has a tube-like body about 5 mmlong with a whorl of tentacles surrounding the mouth at the upper end and a disk-shapedorgan for adhesis at the lower end. The longitudinal pattern is subdivided into: a head, agastric region, a budding zone (where new animals are generated by a process of naturalcloning), a stalk and a foot.
Figure 1: Hydra
The function of cells is determined by their location. This can be shown by a simplecutting experiment (Fig. 2): after a transverse cut cells of the gastric region of the animalare located at the upper end of the lower fragment, with their former neighbours at the lowerend of the upper fragment. Within 2-3 days they form the head and the foot, respectively.Moreover, overlapping cut levels show that the same cells can form either the gastric region,or the head, or the foot, according to their position along the body axis. Experiments of thiskind suggest that the cells respond to local positional cues that are dynamically regulated.The hypothesis is that cells differentiate according to positional information (compare Fig.3). The question is how this information is supplied to the cells.
The same cells can produce different structures depending on their relative position alongthe body column. Experiments show (Fig. 2) that the head and foot regenerate when theyare cut off. If we cut at various levels, we will see that the entire body column between the
5
Figure 2: Cutting experiment. Hydra regenerates after a transverse cut of cells of the gastric region(from both upper and lower half of the body column). The experiment shows the availability ofpositional cues. A hydra, as shown in the middle, is cut in two ways. In one experiment (lefthand-side) the lower body column is removed, in the second experiment (right hand-side) the upperpart is removed. The cut levels are not identical but somewhat different to show that one and thesame group of cells (marked in grey) can form a foot ( left hand-side), or a head (right hand-side) ora gastric segment (original state in the middle). The function of the cells depends on the positionalong the body column. [Courtesy of W. Müller]
Figure 3: The illustration of the idea of “positional value”, which is supplied to the cells andinterpreted by them. The hypothesis is that the formation of the head is determined by the high“positional value” (which is above some threshold). The figure shows the “positional value” for asupernumerary head structure. The animal has been treated with activators of protein-kinase Cperiodically over about two weeks. This treatment caused a periodic increase in the positional valueand eventually the emergence of head structures. [Courtesy of W. Müller]
6
existing head and the foot has the potential to form both the head and the foot. An upperhalf of a transversally cut hydra will regenerate a foot at its lower end, the lower half willform a head at its upper end. By slightly varying the cut levels, as shown in the Fig. 2, itcan be demonstrated that one and the same group of cells forms a head, a gastric region ora foot, depending on their position in the body column.
Grafting experiments show how cells change their functions according to the local cues.Pieces of tissued are grafted from one animal to another and the outcome of such experimentsdepends on the change of position along the body axes. Transplantation of tissue from partsof the body column near to the head induces head formation while transplantation of tissuefrom the position near to the foot results in the foot formation (see Figure 4). If the positionof the tissue in host and donor organism is similar then the piece of tissue is integrated andnothing is observed. Grafting experiments show how the disparities between the positionalvalue of the transplant and the surrounding host tissue result in the head formation or thefoot formation and new organisms with multiple heads or feet.
Figure 4: Grafting experiment. Determination of relative positional values by transplantation.Pieces of tissue are grafted from one animal to another and one of three outcomes is observed. (1) Ifthe tissue is transplanted from the upper position along the body column to the lower position thena new head is formed. (2) If the former and new position is the same then the piece is integratedand nothing is observed. (3) If the tissue is grafted to the upper position a new foot is formed. Wecan see that the disparities between the positional value of the transplant and the surrounding hosttissue result in the head formation or the foot formation respectively. [Courtesy of W. Müller]
2.10 Diffusing morphogens and activator-inhibitor models
Transplantation and tissue manipulation experiments provided data for models of patterningin Hydra. In turn, theoretical models had a strong influence on experimental design, starting
7
with the positional information ideas of Wolpert [22] and the activator-inhibitor model ofGierer and Meinhardt [4, 12] and, finally, receptor-based models of Marciniak-Czochra [9, 10].
Wolpert [22] suggested a gradient model to account for head formation, in which at thehead end a morphogen S is emitted. The morphogen spreads by diffusion and is distributeddown the body. This diffusible chemical induces formation of the head.
A different model type, proposed by Meinhardt and Gierer [4], is based on local activa-tion and long-range inhibition. In this approach the positional value is interpreted as thedensity gradient of morphogen. Gradients of morphogens are formed by reaction-diffusionmechanism. Each of the various body parts is assumed to be under control of a separateactivator-inhibitor system (for details see [12]). The basic activator-inhibitor model takesthe form,
∂
∂tah = Dah
∂2
∂x2ah + µhρ
ah2 + ρ0h
hh− µhah,
∂
∂thh = Dhh
∂2
∂x2hh + µhρah
2 − νhhhh + ρ1h ,
where a, h - the concentrations of two substances (activator and inhibitor)
Figure 5: Simulations of the G-M model; Formation of a gradient-like pattern versus forma-tion of new peaks — the result of assumed domain growth.
Both models mentioned above operate with purely hypothetical long-range morphogens.The approach proposed by Sherrat, Maini, Jäger and Müller (the SMJM model) [?] is
based on the idea that both head and foot formation could be controlled by a receptor-ligand
8
binding. The model takes the form
∂
∂ta = Da
∂2
∂x2a+ sa(x)− µaa− keae− kaaf + kdb,
∂
∂tf = kdb− kaaf + ki[α(x) + βb− f ],
∂
∂tb = kaaf − (kd + ki)b,
∂
∂te = De
∂2
∂x2e+ se(x)− µee,
with zero-flux boundary conditions for a and e.The mechanism of pattern formation in this model is due to the presence of the x-
dependent source terms:
α(x) = α1[1− y(x)/L] + α2y(x)/L,
se(x) = s1[1− y(x)/L] + s2y(x)/L.
sa(x) is constant for y(x) ∈ [0, 4L5
] and decrease linearly to zero for y(x) ∈ [4L5, 1].
In [9, 10] we developed models of receptor-ligand dynamics with spatially homogeneousparameters. Following Müller [13] we assume that the positional value is determined by thedensity of bound receptors (Fig. 6). Our models are based on the idea that epithelial cellssecrete ligands (a regulatory biochemical), which diffuse locally within the interstitial spaceand bind to free receptors on the cell surface.
Figure 6: Bound receptors density determining “positional value”. The hypothesis is that a single“positional value” controls the formation of both the head and the foot: the head is formed if thedensity of bound receptors is high (above some threshold) and the foot is formed if it is low (belowanother threshold). Consequently, in normal development we expect a gradient-like distribution ofbound receptors. [Courtesy of W. Müller]
9
We consider the systems of reaction-diffusion equations coupled with the ordinary differ-ential equations and study what kind of phenomena rising from such systems (i.e. diffusion-driven instabilities, travelling waves, hysteresis and threshold behaviour) can explain theresults observed in experiments.
2.11 Receptor-based concept of pattern generation
Observations on the molecular level:
• Head formation is correlated with the overexpres-sion of Wnt gene (Hobmayer et al 2000)
Model of receptor-ligand dynamics:
• Diffusion of ligands (Wnt) in the intercellularspace
• Binding to the receptors (Frizzled) on the cellmembranes
• Nonlinear regulatory feebacks (intracellular sig-nalling)
3 Receptor-based models
In this section we present main lines of the derivation of the macroscopic receptor-basedmodel using homogenisation technique. The details of the proof can be found in [11].
In general, equations of such models can be represented by by the following initial-
10
boundary value problem
ut = D∆v + f(u, v),
vt = g(u, v) in Ω,
∂nu = 0 on ∂Ω,
u(x, 0) = uinit(x),
v(x, 0) = vinit(x),
where u is a vector of variables describing the dynamics of diffusing extracellular moleculesand enzymes, which provide cell-to-cell communication, while v is a vector of variables lo-calised on cells, describing cell surface receptors and intracellular signalling molecules, tran-scription factors, mRNA, etc. D is a diagonal matrix with positive coefficients on the diag-onal, the symbol ∂n denotes the normal derivative (no-flux condition), and Ω is a boundedregion.
A rigorous derivation, using methods of asymptotic analysis (homogenisation), of themacroscopic reaction-diffusion models describing the interplay between the nonhomogeneouscellular dynamics and the signaling molecules diffusing in the intercellular space has beenrecently published in [11]. It is shown that receptor-ligand binding processes can be modelledby reaction-diffusion equations coupled with ordinary differential equations in the case whenall membrane processes are homogeneous within the membrane, which seems to be the casein most of processes. If homogeneity of the processes on the membrane does not hold,equations with additional integral terms are obtained, see [11].
The receptor-based models give rise to interesting phenomena such as threshold behaviourand hysteresis when the steady state equation g(u, v) = 0 has multiple solutions vi = Hi(u).These concepts proved to be basis for the explanation of the morphogenesis of Hydra [10],dorso-ventral patterning in Drosophila [21] as well as for modelling of the formation of growthpatterns in populations of micro-organisms [7].
3.1 Geometry of the problem
11
• “Standard cell”, Z = [0, 1]3, repeated periodically over R3
• Y0 ⊂ Z - open subset with smooth boundary Γ, Y = Z − Y0,−→ν - outer normal of Y
• Let ε > 0 be a given scale factor
• Zk = Z +∑3
i=1 kiei, Y k0 = Y0 +
∑3i=1 kiei
• Ωε0 = ∪εY k
0 |εZk ⊂ Ω, k ∈ Z3, Ωε = Ω\Ωε0
• Γε = ∪εΓk|εZk ⊂ Ω, k ∈ Z3, Γk = Γ +∑3
i=1 kiei for k ∈ Z3
• Γ∗ = ∪Γk, k ∈ Z3.
3.2 Microscopic description
• Diffusion equations in the intercellular space:
∂
∂tlε = ∇ · (Dε(t, x)∇lε)− µεl (t, x)lε + pl(l
ε)
νε · ∇xlε = 0
lε(0, x) = lε0(x)
• Binding equation on the surfaces:
−Dε(t, x) νε · ∇xlε = ε(bε(t, x)lεrf
ε − dε(t, x)rεb)
• Reaction equations on the surfaces:
∂
∂trfε = −µεf (t, x)rf
ε + pεr(t, x, rεb)− bε(t, x)rf
εlε + dε(t, x)rbε
∂
∂trbε = −µεb(t, x)rb
ε + bε(t, x)rfεlε − dε(t, x)rb
ε
with initial conditions
rfε(0, x) = rf 0(x)
rbε(0, x) = rb0(x)
12
3.3 Weak solution
Definition 3.1. The tuple (lε, rεf , rεb) is a solution of the microscopic problem if
lε ∈ L2((0, T );H1(Ωε)), ∂tlε ∈ L2((0, T )× Ωε), rεf , rεb ∈ L∞((0, T )× Γε),∂tr
εf , ∂tr
εb ∈ L∞((0, T )× Γε) such that∫ T
0
∫Ωε
(∂tl
ε φ+Dε∇lε∇φ+ µεl lε φ)dx dt =
∫ T
0
∫Ωεpεl (t, x, l
ε)φ dxdt
+ε
∫ T
0
∫Γε
(dεrεb − bεrεf lε)φ dx dt.
for all φ ∈ L2((0, T );H1(Ωε)) and a.e. in (0, T )× Γε we have
∂trfε(x, t) = −µεfrf ε(x, t) + pεr(t, x, r
εb(x, t))− bεrf ε(x, t)lε(x, t) + dεrb
ε(x, t),
∂trbε(x, t) = −µεbrbε(x, t) + bεrf
ε(x, t)lε(x, t)− dεrbε(x, t),
where Dεi,j(t, x) = Di,j(t,
xε), µεl (t, x) = µl(t,
xε), pεl (t, x, ξ) = pl(t,
xε, ξ),
µεf (t, x) = µf (t,xε), µεb(t, x) = µb(t,
xε), bε(t, x) = b(t, x
ε), dε(t, x) = d(t, x
ε),
pεr(t, x, ξ) = pr(t,xε, ξ).
3.4 Assumptions
1. D ∈ L∞((0, T )× Ω), ∂tD ∈ L∞((0, T )× Ω),
2. D(t, x)ξξ ≥ d0|ξ|2, d0 > 0, for every ξ ∈ R3, for almost all (t, x) ∈ (0, T )× Ω,
3. µl ∈ L∞((0, T )× Ω), µl ≥ 0 a.e. in (0, T )× Ω,
4. |pl(t, x, ξ)| ≤ c1 + c2|ξ| for a.a. (t, x) ∈ (0, T )× Ω,
5. b ∈ C0,α([0, T ];C0,α(Γ∗)), b ≥ 0 in [0, T ]× Γ∗, ∂tb ∈ L∞((0, T )× Γ∗),
6. d ∈ C0,α([0, T ];C0,α(Γ∗)), d ≥ 0 in [0, T ]× Γ∗, ∂td ∈ L∞((0, T )× Γ∗),
7. µf , µb, pr(ξ) ∈ C0,α([0, T ];C0,α(Γ∗)) for a.a. ξ ∈ R, µf ≥ 0, µb ≥ 0 in [0, T ]× Γ∗, pr isLipschitz continuous in ξ.
3.5 Existence
Theorem 3.2. Let Assumption be satisfied and
l0 ∈ C0,α(Ω), l0 ∈ H1(Ω), l0 ≥ 0,
rf0, rb0 ∈ C0,α(Ω),
rf0 ≥ 0, rb0 ≥ 0.
13
Then, there exists an unique solution (lε, rεf , rεb) of the microscopic problem, such that
lε ∈ H1(0, T ;L2(Ωε)),
lε ∈ L2(0, T ;H1(Ωε)),
lε ∈ C0,β/2([0, T ];C0,β(Ωε ∪ Γε)),
rεf , rεb ∈ C1,β/2([0, T ], C0,β(Γε)), where β ∈ (0, α],
and lε ≥ 0, rεf ≥ 0, rεb ≥ 0.
3.6 A priori estimates
Lemma 3.3. For any solution of the microscopic problem hold
‖lε‖L2(0,T ;H1(Ωε)) ≤ C, ‖∂tlε‖L2(0,T ;L2(Ωε)) ≤ C,
‖rεf‖L∞((0,T )×Γε) ≤ C, ‖rεb‖L∞((0,T )×Γε) ≤ C,
‖∂trεf‖L2((0,T )×Γε) ≤ C, ‖∂trεb‖L2((0,T )×Γε) ≤ C,
||lε||L∞((0,T )×(Ωε∪Γε)) ≤ C,
Remark. To show the estimates for rεf and rεb we add up the equations for rεf and rεb andobtain
∂t(rεf + rεb) ≤ pεr(t, x, r
εb).
Since rεf and rεb are nonnegative, we obtain
||rεf ||L∞((0,T )×Γε) ≤ C, ||rεb||L∞((0,T )×Γε) ≤ C.
3.7 Extension of lε
Now we use the extension of lε from Ωε into Ω, (Hornung, Jäger, 1991, Acerbi et al. 1992,Cioranescu et al. 1979).
Lemma 3.4. 1. For l ∈ H1(Y ) exists an extension l to Z, such that
‖l‖L2(Z) ≤ c‖l‖L2(Y ) and ‖∇l‖L2(Z) ≤ c‖∇l‖L2(Y )
2. For lε ∈ H1(Ωε) exists an extension lε to Ω, such that
‖lε‖H1(Ω) ≤ c‖lε‖H1(Ωε).
Remark: For lε ∈ L2(0, T ;H1(Ωε)) we define
lε(·, t) := lε(·, t).
lε(·, t) ∈ H1(Ωε) for a.e. t. Since the extension operator is linear, lε ∈ L2(0, T ;H1(Ω)).
14
3.8 Convergence of oscillating functions - examples
• uε(x) = x− ε4πcos2πx
ε, uε → x
• ∇uε(x) = 1 + 12sin2πx
ε, ∇uε 1
• Oscillations are lost! We need a different concept of convergence.
3.9 Two-scale convergence
To show the convergence results we use the two scale convergence. We identify lε with theextension lε and obtain the following convergences.
Definition 3.5. Let uε be a sequence of functions in L2(Λ×Ω). uε is said to two-scaleconverge to a limit u ∈ L2(Λ× Ω× Y ) iff for any φ ∈ D(Λ× Ω, C∞per(Y )) we have
limε→0
∫Λ
∫Ω
uε(λ, x)φ(λ, x,x
ε)dxdλ =
∫Λ
∫Ω
∫Y
u(λ, x, y)φ(λ, x, y)dxdydλ.
Theorem 3.6. From each bounded sequence uε in L2(Λ×Ω) we can extract a subsequencewhich two-scale converges to u ∈ L2(Λ× Ω× Y ).
3.10 1-dimensional example
• uε(x) = x− ε4πcos2πx
ε, uε → x
• ∇uε(x) = 1 + 12sin2πx
ε, ∇uε 1 - oscillations are lost!
• ∇uε → 1 + 14πsin(2πy) two-scale.
• uε(x) = x− 14πcos2πx
ε
• uε → x− 14πcos(2πy) two-scale.
• ∇uε(x) = 1 + 12εsin2πx
ε, does not converge.
• ε∇uε 0 but ε∇uε → 12sin(2πy) two-scale.
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3.11 Compactness Theorem
Theorem 3.7. 1. Let uε be a bounded sequence in L2(Λ, H1(Ω)), which converges weaklyto a limit function u ∈ L2(Λ, H1(Ω)). Then there exists u1 ∈ L2(Λ × Ω, H1
per(Y )) such thatup to a subsequence, uε two-scale converges to u and ∇uε two-scale converges to ∇u(λ, x) +
∇yu1(λ, x, y).
2. Let uε and ε∇uε be bounded sequences in L2(Λ × Ω)). Then there exists u ∈L2(Λ×Ω, H1
per(Y )) such that up to a subsequence uε and ε∇uε, two-scale converge to u(λ, x, y)
and ∇yu(λ, x, y) respectively.
3.12 Two-scale convergence for n− 1-dimensional ε-periodic Γε ⊂ Ω
Let Γ ∈ Z be a smooth 2-dimensional manifold (in our application sphere). Then, Γε is theunion of all εΓ. For each Γε we consider the space L2(Γε) equipped with the scale scalarproduct (u, v)Γε := ε
∫Γεu(x)v(x)dx.
Definition 3.8. A sequence of functions wε ∈ L2(Λ × Γε), is said to two-scale convergeto a limit w ∈ L2(Λ× Ω× Γ) iff for any ψ ∈ D(Λ× Ω, C∞per(Γ)) we have
limε→0
ε
∫Λ
∫Γεwε(λ, x)ψ(λ, x,
x
ε)dγxdλ =
∫Λ
∫Ω
∫Γ
w(λ, x, y)ψ(λ, x, y)dxdγydλ.
3.13 Two-scale convergence for n− 1-dimensional ε-periodic Γε ⊂ Ω
Theorem 3.9 (Neuss-Radu, 1996, Allaire et al., 1996). From each bounded sequence wεin L2(Λ× Γε) we can extract a subsequence which two-scale converges to w ∈ L2(Λ×Ω× Γ)
Theorem 3.10. If the sequence sequence wε is bounded in L∞((0, T )×Γε), then the limitw ∈ L∞((0, T )× Ω× Γ).
3.14 Convergence of lε, rεf , and rεb
Lemma 3.11. There exist functions l, rf , and rb such that,
1. lε l in L2(0, T ;H1(Ω)), ∂tlε ∂tl in L2((0, T )× Ω),
2. lε → l in L2(0, T ;W β,2(Ω)) for 12< β < 1 and lim
ε→0||lε − l||L2((0,T )×Γε) = 0,
3. lε → l two-scale, ∇lε → ∇xl +∇yl1 two-scale, l1 ∈ L2((0, T )× Ω;H1per(Y )),
16
4. rεf → rf , rεb → rb two-scale and rf , rb ∈ L∞((0, T )× Ω× Γ),
5. ∂trεf → ∂trf , ∂trεb → ∂trb two-scale and ∂trf , ∂trb ∈ L2((0, T )× Ω× Γ).
Definition of W β,2: For β ∈ R, 0 < β define a Hilbert space W β,2 as the completion ofC∞(Ω) with respect to the norm
‖u‖Wβ,2(Ω) = ‖u‖Wk,2(Ω) +
∫Ω
∫Ω
|u(x)− u(y)||x− y|n−2(β−k)
dxdy12 ,
where k = [β].Proof.
A priori estimates =⇒ lε l in L2(0, T ;H1(Ω)), ∂tlε ∂tl in L2((0, T )× Ω).
Lions-Aubin Lemma =⇒ lε → l strong in L2((0, T ),W β,2(Ω)), 12< β < 1.
Therefore, ‖lε − l‖L2((0,T )×Γε) ≤ c‖lε − l‖2L2(0,T ;Wβ,2(Ωε))
≤ c‖lε − l‖2L2(0,T ;Wβ,2(Ω))
→ 0. forε→ 0. For lε on Γε we use the estimates of the type
ε
T∫0
∫Γε
|lε|2 dγ dt ≤ c
T∫0
∫Ωε
|lε|2 dx dt+ ε2T∫
0
∫Ωε
|∇lε|2 dx dt.
Weak convergence of lε =⇒ two-scale convergence of lε to l and the existence of l1 ∈L2((0, T )×Ω;H1
per(Y )) such that, up to a subsequence, ∇lε two-scale converges to ∇xl(x) +
∇yl1(x, y).
A priori estimates =⇒ weak convergence of rεf , rεb, ∂trεf , ∂trεb and two-scale convergence.
3.15 Macroscopic model
Theorem 3.12. The solution of the microscopic problem converges to the solution of∂tl = − 1
|Y |
∫Γ
(b rf l − d rb)dγy +∇(S∇l) + pl(l)− µl l, in (0, T )× Ω,
ν · ∇xl(t, x) = 0, on ΓN ,
∂trf = pr(t, y, rb)− b(t, y)rf l + d(t, y)rb − µf (t, y)rf , in (0, T )× Γ× Ω,
∂trb = b(t, y)rf l − d(t, y)rb − µb(t, y)rb, in (0, T )× Γ× Ω,
where µl = 1|Y |
∫Yµl(t, y) dy, p(l) = 1
|Y |
∫Yp(t, y, l) dy
and the matrix S is defined as
sij =1
|Y |
3∑k=1
∫Y
(Dij(t, y) +Dik(t, y)∂ykwj) dy
17
with wi being the solutions of the cell problem,
−∇y(D(t, y)∇ywi) =3∑
k=1
∂ykDki(y) in Y, −D∂wi∂ν
=3∑
k=1
Dkiνk on Γ.
Proof.We use the extension of lε to a function on the whole domain Ω. Choose a test function,
φ(t, x) = ψ0(t, x) + εψ1(t, x, xε), ψ0 ∈ L2(0, T ;H1(Ω)), ψ0 ∈ L2(0, T ;H1(Ω);C1
per(Y )), weobtain
T∫0
∫Ω
(∂tl
εχ(x
ε)(ψ0 + εψ1) +Dε∇lεχ(
x
ε)(∇xψ0 + ε∇xψ1 +∇yψ1)
)dx dt
+
T∫0
∫Ω
µεl lεχ(
x
ε)(ψ0 + εψ1) dx dt =
T∫0
∫Ω
pl(lε)χ(
x
ε)(ψ0 + εψ1) dx dt
+ε
T∫0
∫Γε
(dεrεb − bεrεf lε)(ψ0 + εψ1) dγx dt,
where χ is a characteristic function of Ωε.Let ε→ 0 and consider the limit separately in every term,
T∫0
∫Ω
∂tlεχ(
x
ε)(ψ0(t, x) + εψ1(t, x,
x
ε)) dx dt→ |Y |
T∫0
∫Ω
∂tlψ0(t, x) dx dt,
T∫0
∫Ω
Dε(t, x)∇lεχ(x
ε) (∇xψ0 + ε∇xψ1 +∇yψ1) dx dt →
T∫0
∫Ω
∫Y
D(t, y)(∇xl(t, x) +∇yl1(t, x, y))(∇xψ0(t, x) +∇yψ1(t, x, y)) dy dx dt,
T∫0
∫Ω
µεl lεχ(
x
ε)(ψ0 + εψ1) dx dt→ |Y |
T∫0
∫Ω
µl(t) l(t, x)ψ0(t, x) dx dt.
Since lε → l strong in L2((0, T )× Ω), we obtain the convergence of the nonlinear term,T∫
0
∫Ω
pl(lε)χ(
x
ε)(ψ0(t, x) + εψ1(t, x,
x
ε)) dx dt→ |Y |
T∫0
∫Ω
pl(t, l)ψ0(t, x) dx dt.
18
Now we show the convergence of the boundary integral,
ε
T∫0
∫Γε
(dεrεb − bεrεf lε)(ψ0 + εψ1) dγx dt = ε
T∫0
∫Γε
dεrεb(ψ0 + εψ1) dγx dt
−εT∫
0
∫Γε
bεrεf lε(ψ0 + εψ1) dγx dt.
We apply two-scale convergence of rεb in the first integral and obtain
ε
T∫0
∫Γε
dε(t, x)rεb(t, x)(ψ0 + εψ1)dγx dt→T∫
0
∫Ω
∫Γ
d(t, y)rb(t, x, y)ψ0(t, x) dγy dx dt.
The second integral can be rewritten as a sum of two integrals
ε
∫ T
0
∫Γεbε rεf l
ε (ψ0 + εψ1) dγx dt = ε
∫ T
0
∫Γεbε rεf l (ψ0 + εψ1)dγx dt
+ε
∫ T
0
∫Γεbε rεf (l
ε − l)(ψ0 + εψ1)dγx dt.
In the first integral we apply two-scale convergence of rεf and obtain
ε
T∫0
∫Γε
bεrεf l (ψ0 + εψ1)dγx dt→T∫
0
∫Ω
∫Γ
b(t, y)rf (t, x, y)l(t, x)ψ0(t, x) dγy dx dt.
Since ||lε − l||L2((0,T )×Γε) → 0 as ε→ 0, we obtain for the second integral
ε
T∫0
∫Γε
bεrεf (lε − l)(ψ0(t, x) + εψ1(t, x,
x
ε))dγx dt
≤ ε( T∫
0
∫Γε
|bεrεfψ0|2dγx dt)1/2( T∫
0
∫Γε
|lε − l|2dγx dt)1/2
+ε2( T∫
0
∫Γε
|bεrεfψ1|2dγx dt)1/2( T∫
0
∫Γε
|lε − l|2dγx dt)1/2 → 0 as ε→ 0.
3.16 How to find the unknown function l1 ∈ L2((0, T )×Ω;H1per(Y )) ?
Setting ψ0 = 0, we obtain the equation∫ T
0
∫Ω×Y
D(t, y)(∇xl(t, x) +∇yl1(t, x, y))∇yψ1(t, x, y) dt dx dy = 0
19
for all ψ1.From this follows that l1 depends linearly on ∇xl, and it can be written in the form
l1 = Σni=1
∂l
∂xi· wi,
where the functions wi are defined as solutions of the cell problem
−∇(D(t, y)∇wi) =3∑
k=1
∂ykDki(t, y) in Y, −D∂wi∂ν
=3∑
k=1
Dki νk on Γ.
Next, setting ψ1 = 0, we obtain
T∫0
∫Ω
∫Y
n∑i,j=1
Dij(t, y)(∂xil(t, x) +n∑k=1
∂yiwk∂xk l(t, x))∂xjψ0(t, x) dy dx dt
= |Y |T∫
0
∫Ω
n∑i,j=1
sij∂xiψ0(t, x)∂xj l(t, x) dy dx dt
with sij = 1|Y |
∫Y
(Dij(t, y) +∑3
k=1 Dik(t, y)∂ykwj) dy.
3.17 Convergence of the equations on the boundary
Now we take a limit of the terms defined on the boundary
ε
T∫0
∫Γε
∂trεfψ1(t, x,
x
ε)dγx dt = ε
T∫0
∫Γε
pεr(t, x, rεb(t, x))ψ1(t, x,
x
ε)dγx dt
+ε
T∫0
∫Γε
(−bεrεf (t, x)lε(t, x) + dε(t, x)rεb(t, x)− µεfrεf (t, x)
)ψ1(t, x,
x
ε) dγx dt,
ε
T∫0
∫Γε
∂trεb(t, x)ψ1(t, x,
x
ε) dγx dt = ε
T∫0
∫Γε
bεrεf (t, x)lε(t, x)ψ1 dγx dt
+ε
T∫0
∫Γε
(−dεrεb(t, x)− µεbrεb(t, x)
)ψ1(t, x,
x
ε) dγx dt.
20
3.18 Convergence of nonlinear pr on the boundary Γ
• To be able to take the two-scale limit in the equations on the boundary we have toshow that pεr(t, x, rεb(t, x))→ pr(t, y, rb(t, x, y)) in the two-scale sense.
• We use periodic modulation (Arbogast et al., 1990) or method of unfolding (Cioranescuet al., 2002) or localisation method (Jäger and Neuss-Radu, 2007).
We define a dilation operator rεb, rεf : (0, T )× Ω× Γ→ R such thatrεb(t, x, y) = rεb(t, εy + ε[x
ε]) and
rεf (t, x, y) = rεf (t, εy + ε[xε]) for x ∈ Ω, y ∈ Γ, t ∈ (0, T ).
We extend rεb, rεb from Γ to⋃k(Γ + k) periodically.
Lemma 3.13 (Bourgeat, Luckhaus and Mikelic, 1996). If rεb → r∗b weakly in L2((0, T ) ×Ω;L2
per(Γ)) and rεb → rb two-scale, then r∗b = rb almost everywhere in (0, T )× Ω× Γ.
Changing variables, Γε 3 x→ εy + cε(x) we obtain equations on (0, T )× Ω× Γ
∂
∂trεf (t, x, y) = −µf (t, y)rεf (t, x, y) + pr(t, y, r
εb(t, x, y))
−b(t, y)rεf (t, x, y)lε(t, x, y) + d(t, y)rεb(t, x, y),
∂
∂trεb(t, x, y) = −µb(t, y)rεb(t, x, y) + b(t, y)rεf (t, x, y)lε(t, x, y)
−d(t, y)rεb(t, x, y).
Applying the estimates for rεb we obtain the estimates for rεb and the weak convergence of rεb torb in L2((0, T )×Ω;L2
per(Γ)). Since sup[0,T ]×(Ωε∪Γε) |lε| ≤ C we conclude that sup[0,T ]×Ω×Γ |lε| ≤C.
21
3.19 Strong convergence of rεf and rεb in L2((0, T )× Ω;L2per(Γ)).
We show that rεf , rεb is a Cauchy sequence.
∂
∂t
∫Ω×Γ
|rε1f − rε2f |
2dxdγ = −∫
Ω×Γ
µf (t, y)|rε1f − rε2f |
2dxdγ
+
∫Ω×Γ
(pε1r (t, x, y, rε1b )− pε2r (t, x, y, rε2b ))(rε1f − rε2f )dxdγ
−∫
Ω×Γ
(b(t, y)(rε1f l
ε1 − rε2f lε2)(rε1f − r
ε2f ) + d(t, y)(rε1b − r
ε2b )(rε1f − r
ε2f ))dxdγ,
∂
∂t
∫Ω×Γ
|rε1b − rε2b |
2dxdγ = −∫
Ω×Γ
µb(t, y)|rε1b − rε2b |
2dxdγ
+
∫Ω×Γ
b(t, y)(rε1f lε1 − rε2f l
ε2)(rε1b − rε2b )dxdγ −
∫Ω×Γ
d(t, y)|rε1b − rε2b |
2 dxdγ.
We add these two equations and integrate with respect to time. Using strong convergenceof lε and boundedness of lε on Ω× Γ we obtain
||rε1f − rε2f ||
2 + ||rε1b − rε2b ||
2 ≤ C
∫ t
0
(||rε1f − rε2f ||
2 + ||rε1b − rε2b ||
2)dt+ C1σ(ε)
where C = C(||lε||L∞(Ωε∪Γε), sup |µf |, sup |µb|, sup |b|, sup |d|, sup |rεf |). Then the GronwallInequality yields
||rε1f − rε2f ||L2(Ω×Γ) ≤ Cσ(ε),
||rε1b − rε2b ||L2(Ω×Γ) ≤ Cσ(ε).
Using strong convergence of rεb, continuity of pr and weak convergence of pr(t, y, rεb), weobtain that pr(t, y, rεb) weakly converges to pr(t, y, rb(t, x, y)) in L2((0, T )×Ω;L2
per(Γ)). Then,acording to Lemma pεr(t, x, rεb) converges two-scale to pr(t, y, rb(t, x, y)).
3.20 Model reduction
We notice that the macroscopic receptor based model derived above is equivalent to thehomogeneous model
∂
∂trf = −µfrf + pr(rb)− brf l + drb,
∂
∂trb = −µbrb + brf l − drb, (2)
∂
∂tl =
1
γ
∂2
∂x2l − µll − brf l + pl(l) + drb,
22
defined on the macroscopic domain Ω, in the case when neither the model parameters northe initial conditions for rf and rb depend on the surface variable y and cells are symmetric.It means that the processes described are homogeneous within each cell and there is noheterogenity in the dissociation or binding processes on the cell surfaces. For non-adherentcells one can consider receptor production, binding, dissociation or decay to be uniformlydistributed on the cell surface, which results in model coefficients being constant in respectto the surface variable y. Under such assumptions we obtain a macroscopic model, in whichthe integral in the equation for the ligands disappears and the only difference with respectto model (2) is that the kinetics are multipled by a coefficient
∫Γdγy/|Y |. The diffusion
operator is diagonal in case of symmetric cells. Otherwise anisotropy appears.
3.21 SummaryMicroscopic models
Macroscopic receptor-based models
∂tu(x, t) = f(u(x, t), v(x, t))
∂tv(x, t) = D∆v(x, t) + g(u(x, t), v(x, t))
+ zero-flux boundary conditions+ initial conditions
x ∈ Ω ⊂ RN , t ∈ R+
homogenisation
(rigorous)
23
4 Mechanisms of pattern formation
Receptor-based models
∂tu = f(u, v)
∂tv = D∆v + g(u, v)
+ zero-flux boundary conditions+ initial conditions
x ∈ Ω ⊂ RN , t ∈ R+
u
v
f(u,v)=0
g(u,v
)=0
Mechanisms of pattern formation
• Diffusion-driven instabilities (DDI)
• Existence of multiple quasi-steady states (hysteresis)
4.1 Single reaction-diffusion equation
Theorem 4.1. There exists no stable spatial pattern (i.e. spatially inhomogeneous solution)for a scalar reaction-diffusion equation in one dimension with Neumann boundary conditions.
Proof. We consider a single reaction-diffusion system in one-dimensional space, i.e. Ω =
[0, 1] and
ut = u′′ + f(u),
u′(0) = u′(1) = 0, (3)
where u′ denotes the derivative of u in respect to x and ut derivative in respect to t.Stationary spatially inhomogeneous solution U satisfies
U ′′ + f(U) = 0,
U ′(0) = U ′(1) = 0. (4)
We will show that if U is a spatially nonuniform solution of (4) then U is unstable.The linearisation of (3) is
λy = y′′ + f ′(U(x))y,
y′(0) = y′(1) = 0. (5)
We realistically assume that f ′(U) is bounded by K > 0, which implies that the discreteeigenvalues λ of (5) must be bounded above so there must exist the largest eigenvalue, λ0
(see [20]). We have λ0 > λ1≥λ2...≥λi ≥ ...
24
Now consider the eigenvalue problem with fixed boundary values,
µy = y′′ + f ′(U(x))y,
y(0) = y(1) = 0. (6)
with eigenvalues µ0 > µ1≥µ2... From (4) we obtain
U ′′′ + f ′(U)U ′ = 0,
U ′(0) = U ′(1) = 0,
Now we notice that z = U ′(x) is an eigenfunction of (6) if µ = 0.This implies that µ = 0 is an eigenvalue of (6). Thus, we obtain that the largest eigenvalue
µ0 ≥ 0.Now, consider the two eigenvalue problems defined by (5) and (6), that is with zero flux
and zero boundary conditions on y. Using the Sturm comparison principle (see e.g. [8]) wecan show that λ0 > µ0. In fact, if λ0 ≤ µ0, then we would have f ′(U)− µ ≤ f ′(U)− λ. So,from the Sturm comparison principle (see e.g. [8]) we obtain that the principal eigenfunctioncorresponding to λ has a zero in (0, 1). This is impossible since the principal eigenfunctionis of one sign (see [20]). Now, since µ0 ≥ 0, we obtain also λ0 > 0. It means that the largesteigenvalue of (5) is positive and so w grows exponentially with time and hence U(x) is linearlyunstable. Thus there are no stable spatially patterned solutions a reaction-diffusion modelfor m = 1, and consequently the only stable solutions u(x, t) of (4) are homogeneous steadystates and satisfy f(U) = 0.
4.2 Diffusion-driven instabilities (Turing-type patterns)
Diffusion-driven instability (Turing-type instability) arises in a reaction-diffusion system. Itoccurs when there exists a spatially homogeneous solution, which is asymptotically stablein the sense of linearised stability in the space of constant functions, but it is unstablewith respect to spatially inhomogeneous perturbation. We study the linear instabilities ofthe homogeneous steady state to classify the patterns, which may grow, in terms of theirwavenumber.
We consider a dimensionless reaction-diffusion system with zero-flux boundary conditions.For such a system, we consider a bifurcation from the homogeneous (spatially uniform) steadystate by examining the response of the system to an initially small perturbation u from thesteady state u, |ui| 1.
Settingu = u+ u, (7)
we linearise the model in the variable u and obtain∂u
∂t= D4xu+Bu (8)
25
with zero-flux boundary conditions, where B is the Jacobian matrix of the kinetics systemf evaluated at the homogeneous state u.
We seek solutions to the system (8) of the the form,
u = exp(λt)Φ(x),
where Φ is a vector.Substituting into equation (8) we obtain
D4xΦ + (B − λI)Φ = 0. (9)
Eigenfunctions of the spatial eigenvalue problem (9) can be written as Φm = ymφm, whereym is a constant vector and φm are eigenfunctions of the Laplacian
4xφm = −µm2φm in Ω,
∂nu = 0 on ∂Ω
and µm is a wavenumber associated with φm.For nontrivial solutions of (9) we require
det(B − µm2D − λ(µm2)I) = 0. (10)
This condition yields the dispersion relation λ = λ(µm2) which is an algebraic equation
for the growth rate.The solution of the linear stability problem is then given by
u(x, t) = Σ∞m=0exp(λ(µm2)t)ymφm(x) (11)
with ym being vectors determined by the initial data. The labelm is called the mode number.For certain nonlinearities in the reaction term it can be shown that the amplitudes of growingmodes are bounded by a finite value.
The linear stability of the homogeneous steady state to spatially heterogeneous pertur-bation of mode m is determined by the sign of the real part of λ(µm
2). The steady stateis stable (asymptotically) for Reλ(µm
2) < 0 for all m and unstable for Reλ(µm2) > 0 for
some m. When it is unstable a perturbing mode of appropriate wavenumber may grow. Therequirement about the stability of the steady state in the absence of diffusion is equivalentto the nonexistence of growing modes for the wavenumbers with µm
2 = 0. This providesconditions on the Jacobian matrix B.
There are two kinds of Turing-type patterns (depending on the imaginary part of theeigenvalue with positive real part).• Stationary patterns, when a single eigenvalue of the Jacobian matrix B becomes positive
and the bifurcating solution is a nonconstant steady state. In such case long-time solutionsare stationary and spatially heterogeneous structures.
26
• Wave patterns, when 2 complex conjugate eigenvalues of B cross the imaginary axis.It is a supercritical Hopf bifurcation from a homogeneous solution to a stable periodic andnonconstant solution. The result is a pattern which oscillates in time.
The majority of theoretical studies in theory of pattern formation due to the diffusion-driven instability focus on the analysis of the systems of only 2 reaction-diffusion equations.They are activator-inhibitor systems and involve kinetics chosen in the way necessary for adiffusion-driven instability. One variable describes the density of chemical (activator) whichactivates its own production and the production of the other one (inhibitor) which in turnsinhibits the production of activator.
4.3 Summary:
• It may happen that
– the kinetics system is asymptotically stable
– the complete system unstable for spatially non-homogeneous perturbations.
• Such mechanism is called Diffusion Driven Instabilities or Turing-type instabilities(Turing A. (1952) The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. B, 237:37–72).
• Turing suggested that, under certain conditions, chemicals can react and diffuse insuch a way as to produce steady state heterogeneous spatial patterns of chemicalconcentration.
• Diffusion is usually considered a stabilising process which is why it was such a novelconcept.
To find conditions for DDI:
• Use linear stability analysis to derive the conditions for DDI
A(µm) = A− 1
γDµm
2,
A is the Jacobian matrix at (rf , rb, l), D- matrix of diffusion coefficients.µm
2 is a wavenumber obtained from the Laplacian’s eigenproblem:
4xφm = −µm2φm in Ω,
∂nφm = 0
The linear stability of the homogeneous steady state to spatially heterogeneous perturbationsis determined by the sign of Reλ(µm
2), where λ(µm2) belongs to the spectrum of A(µm
2).
27
4.4 The simplest scenarios with DDI
• The majority of theoretical studies in theory of pattern formation due to the diffusion-driven instability focus on the analysis of the systems of only 2 reaction-diffusionequations,
∂u
∂t=
d1
γ∆u+ f(u, v),
∂v
∂t=
d2
γ∆v + g(u, v).
with homogeneous Neumann (zero flux) boundary conditions in x.
• Stability of the kinetics system:
trA = a11 + a22 < 0 and det(A) > 0
A is the Jacobian matrix at the spatially homogeneous steady state.
4.4.1 Conditions for DDI
• Destabilisation: tr(A−Dµm2) > 0 or det(A−Dµm2) < 0.
• From the linear stability analysis we obtain the dispersion relation λ = λ(µm2) as a
solution of the characteristic polynomial,
det(A− λI) = det(A−Dµm2
γ− λI) = 0.
0 0.2 0.4 0.6 0.8 1!0.1
!0.05
0
0.05
0.1
0.15
0.2
0.25
det A
µm2/!
~
• We can see that detA changes sign in someinterval of µm2.
• For every γ we have a finite range of unstablemodes which can grow.
• We can isolate a specific mode to be excitedby choosing γ.
4.4.2 How do patterns grow?
4.4.3 How the patterns look like?
28
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.99
0.995
1
1.005
1.01
1.015
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.995
1
1.005
1.01
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.996
0.998
1
1.002
1.004
1.006
1.008
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.95
1
1.05
1.1
1.15
1.2
1.25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 7: Numerical simulation of a reaction-diffusion system with Schnakenberg kinetics on theunit square (Neumann boundary conditions).
29
4.5 Stationary and oscillatory Turing patterns
There are two kinds of Turing type patterns.
• Stationary patterns, when a single eigenvalue becomes possitive and the bifurcatingsolution is a nonconstant steady state. Long-time solutions are stationary and spatiallyheterogeneous structures.
• Wave patterns, when 2 complex conjugate eigenvalues cross the imaginary axis. It isa supercritical Hopf bifurcation from a homogeneous solution to a stable periodic andnonconstant solution.
Remark : Stationary Turing patterns are already possible for the 2-equations system (1diffusion) while for wave biffurcation 3 variables are neccessary.
4.6 Problems
• There are several general properties of Turing-type systems that limit their applicabil-ity.
• The parameters must be tightly controlled to obtain the instability at the desired pointin parameter space.
• In particular, the scaling parameter (corresponding to the domain size and diffusioncoefficients) must have the proper relative magnitudes.
• It is difficult to obtain a scale-invariance of the degree which is observed in biologicalsystems.
• Another important problem is that of pattern selection. It arises in the models withmultiple stable solutions. Tight control of the initial conditions is needed to select thedesired pattern.
4.7 What happens if there is only one diffusion operator?
∂u
∂t= f(u, v),
∂v
∂t=
1
γ∆v + g(u, v).
• Stability of the kinetics system:
trA = a11 + a22 < 0 and det(A) > 0
• Destabilization: since tr(A− µ2m
γD) < 0, then det(A− µ2
m
γD) < 0.
30
4.7.1 Dispersion relation
det(A)− µ2m
γa11 < 0
• For every γ there exist infinitely many different integer µm for which the above in-equality is fulfiled
• From the dispersion relation, λ = λ(µ2m/γ), we cannot decide which eigenfunctions,
that is, which spatial patterns, are linearly unstable and grow with time
• The index of the growing mode depends on initial conditions and on the scaling pa-rameter γ
4.7.2 Spatial profile of solutions
• The shape of the final pattern depends on γ.
• For γ small enough, it resembles the initial conditions.
• When γ is growing, more humps appear.
• The shape of the final pattern depends on the initial function
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8185
0.8186
0.8186
0.8187
0.8187
0.8188
c(x,
0)
x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
14x 1014
x
c(x,
1000
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8184
0.8186
0.8188
x
c(x,
0)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.75
0.8
0.85
c(x,
450)
x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6
0.8
1
x
c(x,
500)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
c(x,
600)
x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2x 1083
x
c(x,
5000
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8185
0.8186
0.8187
0.8188
c(x,
0)
x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8186
0.8187
0.8187
x
c(x,
600)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.7
0.8
0.9
1
c(x,
1000
)
x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6x 1017
x
c(x,
2000
)
Figure 8: Spatial profile of the solution with γ = 1, γ = 103, γ = 104 respectively.
31
5 Receptor-based model with DDI
∂
∂trf = −µfrf + pr(rf )− brf l + drb,
∂
∂trb = −µbrb + brf l − drb,
∂
∂tl = dl
∂2
∂x2l − µll − brf l + pl(rb) + drb,
• Autocatalysis of rf needed for DDI, i.e. pr = m1rαf with α > 1
• Patterns localised in points; strongly dependent on initial conditions.
5.1 Conditions for DDI in 3-equation model
• Characteristic polynomial:
χA(λ) = λ3 − trAλ2 +∑i<j
|Aij|λ− detA = 0.
• The Hurwitz matrix is
H =
−σ1 −σ3 0
1 σ2 0
0 −σ1 −σ3
=
−trA −detA 0
1∑
i<j |Aij| 0
0 −trA −detA
.
• From the Routh-Hurwitz theorem: The number k of roots of the real polynomial whichlie in the right half-plane is given by the formula
k = V (1,41) + V (41,43) + V (1,42),
where V is the number of changes of sign of adjacent numbers in a sequence and 4i
be the successive principal minors of matrix H.
5.2 DDI in the model with 1 diffusion operator
Lemma 5.1. Let A be the Jacobian matrix computed at a positive spatially homogeneoussteady state and such that aii < 0 for i = 1, 2, 3 and a12a21 > 0. There is a diffusion-driveninstability for the considered system if and only if the following conditions are fulfiled,
−tr(A) > 0,
−tr(A)∑i<j
|Aij|+ |A| > 0,
−|A| > 0,
|A12| < 0,
33
where Aij is a submatrix of A consisting of the i-th and j-th column and i-th and j-th row,and |A| and |Aij| denote the determinants of matrices A and Aij, respectively.
6 Dispersion relation
|A| = |A| − |A12|µ2m
γ> 0.
• For every γ there exist infinitely many different integer µm for which the above in-equality is fulfiled
• From the dispersion relation, λ = λ(µ2m/γ), we cannot decide which eigenfunctions,
that is, which spatial patterns, are linearly unstable and grow with time
• The index of the growing mode depends on initial conditions and on the scaling pa-rameter γ
6.1 DDI model with 2 diffusions
∂
∂trf = −µfrf + pr(rb)− brf l + drb,
∂
∂trb = −µbrb + brf l − drb,
∂
∂tl = dl
∂2
∂x2l − µll − brf l + pl(rb) + drb − bele,
∂
∂te = de
∂2
∂x2e− µee+ pe(l, rb),
7 Application to Hydra pattern formation
• Spatially uniform initial data evolves into a gradient pattern
• Models with DDI cannot explain the outcome of transplantation experiments
34
Stable gradient-like pattern
Grafting experiment:
- Initial data corresponding to thehead transplantation
- Final distribution showing thetransplant disappearance
35
7.1 Self-organisation
• Initial data (dashed line) corresponding toa surgical removal of the lower part (half)of the body column.
• Reorganisation of the “gradient” on asmaller domain corresponds to the forma-tion of a new “foot”.
• Initial data (dashed line) corresponding toa surgical removal of the upper part (half)of the body column.
• Reorganisation of the “gradient” on asmaller domain corresponds to the forma-tion of a new “head”.
8 Models with multistability and hysteresis
Transplantation experiments indicate co-existence of stable patterns of Wnt expression. Thepatterns may have multiple peaks, which depend on the local cues induced by the graftedtissue. The experiments suggest a mechanism of pattern formation based on multistability
in intracellular signaling.• New model: hysteresis in signalling pathway
∂
∂tpl = −δl
pl1 + p2
l
+m2lrb
(1 + σlp2l − βlpl)(1 + αlrb)
pl - rate of ligand synthesisl - ligandsrb - receptors
36
• Multistability leads to the formation of transition-layer patterns
Figure 9: Simulations of the receptor-based model with hysteresis: formation of a gradient-like pattern corresponding to a normal development and head formation in Hydra (left panel)and formation of two heads pattern for the initial conditions corresponding the transplanta-tion experiment (right panel).
8.1 Stationary solutions
∂
∂trf = −µfrf − brf l + drb + pr(rb),
∂
∂trb = −µbrb + brf l − drb,
∂
∂tl =
1
γ
∂2
∂x2l − µll − brf l + drb + pl,
∂
∂tpl = −δl
pl1 + p2
l
+m2lrb
(1 + σlp2l − βlpl)(1 + αlrb)
Quasi-steady states:
ut =1
γ∆u+ f(u, v),
0 = g(u, v),
37
with u := l and v := pl and
f(u, v) = v − b(µb + d)m1u
bµbu+ µfµbµfd− µlu+
bdm1u
bµbu+ µfµb + µfd,
g(u, v) = −δlv
1 + v2+
m2m1bu2
(bµbu+ µfµbµfd)(1 + σlv2 − βlv)(1 + αlbm1ubµbu+µfµbµfd
).
9 The prototype model of hysteresis-based pattern for-mation
In this section we investigate the role of multiple steady states and switches in the dynam-ics of the ODE-subsystem on the example of a basic model of a reaction-diffusion equationcoupled to an ordinary differential equation.
We focus in the following on a basic reaction-diffusion model exhibiting the multistabilityand hysteresis-driven mechanism of pattern formation,
ut = uxx + f(u, v), (12)
vt = g(u, v), (13)
for x ∈ [0, L], with zero-flux boundary condition for u and the kinetic functions given by
f(u, v) = αv − βu+ γ,
g(u, v) = µu− p(v).
To model the hysteresis effect where p(v) is a non-monotone polynomial of degree 3 withonly one real zero at v = 0.
We assume that the local maximum H = (uH , vH) and the local minimum T = (uT , vT ) ofv → p(v) have positive coordinates. Furthermore, we assume that there are three intersectionpoints of f = 0 and g = 0 with nonnegative coordinates S0 = (u0, v0), S1 = (u1, v1), whichis between T and H, and S2 = (u2, v2). Without loss of generality, we can take γ = 0 andµ = 1 using the transformation u → u − u0 and v → 1
µ(v − v0). Consequently, we obtain
S0 = (0, 0).As v → p(v) is non-monotone, it can be inverted only locally. We call u → hH(u) the
branch of the inverse which is defined on (−∞, uH ] and similarly u → hT (u) is the branch,defined on [uT ,∞) (see Fig. 10).
The system has three spatially homogeneous steady states S0, S1, S2. Based on linearstability analysis we conclude that S0 and S2 are stable, while S1 is a saddle. Furthermore,the system cannot exhibit diffusion-driven instability (Turing-type instability).
38
Figure 10: A typical configuration of the kinetic functions.
The system of the two equations (12)- (13) can be interpreted as a basic model exhibitingthe dynamics of the receptor-based model in [10] with u describing a density of diffusingligands (signalling molecules) and v standing for the rate of their production. In fact, it isa minimal reaction-diffusion model which may describe formation of stable patterns. In thenext Section we show that replacing the nonlinearities so that g(u, v) has no singular points,i.e., ∂vg is always strictly positive, leads to instability of the nonhomogeneous stationarysolutions.
9.1 Instability of spatial patterns in models without hysteresis
In this section we show that the system with multistability but reversible quasi-steady statethe ODE subsystem cannot exhibit stable spatially heterogeneous patterns. We considersystem (12)- (13) with
f(u, v) = αv − βu,g(u, v) = u− p(v).
Again we assume, that there are three intersection points S0 = (0, 0), S1 = (u1, v1) andS2 = (u2, v2) with nonnegative coordinates of f = 0 and g = 0. But in contrast to thehysteresis case, we assume p(v) to be a monotone polynomial of degree three (see Fig.11 A).
39
Figure 11: The model with bistability but without hysteresis effect. Panel A: Kinetic func-tions in steady state in case of monotone function p(v). Panel B: Stationary solutions for uand v, which is always unstable as shown in Theorem 9.1.
Theorem 9.1. There exists no stable spatial pattern (i.e., spatially non-homogeneous solu-tions) of system (12)- (13) with a monotone p(v).
Proof. Since u = p(v) is monotone, we can invert it globally. We call the inverse v = h(u).Taking the quasi-steady state solution of equation (13), v = h(u), and inserting it intoequation (12) yields the following two point boundary value problem
0 = uxx + f(u, h(u)) = uxx + αh(u)− βu, (14)
with zero-flux boundary conditions. Along the trajectories it holds,
u2x(x)
2+Q(u(x)) = constant, (15)
where
Q(u) :=
∫ u
0
(αh(u)− βu)du = F (h(u)) with F (v) =
∫ v
0
(αv − βp(v))p′(v)dv. (16)
As p(v) is monotone, we obtain ddvF (v) = αv − βp(v))p′(v), which is equal to zero in 0, v1
and v2. Consequently, Q(u) has a minimum on the interval (0, u2), and it is possible to findvalues such that 0 < u0 < uend < u2 with Q(u0) = Q(uend). Furthermore we can calculatethe so-called time-map, which gives the length of the interval one needs to travel from u0 touend
L =1√2
∫ uend
u0
du√Q(u0)−Q(u)
. (17)
40
When u0 → 0 (resp. uend → u2) then L tends to infinity. Consequently, there existsexactly one solution, U , on each interval length L.
We now show that the solution, (U(x), V (x) = h(U(x))), is unstable. Therefore weconsider the linearisation L of the operator(
u
v
)7→(uxx0
)+
(f(u, v)
g(u, v)
)around the steady state solution (U(x), V (x)).
L(η
ψ
)=
(ηxx0
)+
(−β α
1 −p′(V (x))
)·(η
ψ
)
The solution of the eigenvalue problem L(η
ψ
)= λ
(η
ψ
)has to fulfill the equation
λη = ηxx − βη +α
λ+ p′(V )η
with zero-flux boundary condtions. To show, that this problem has a positive eigenvalue,we define the operator A(λ) : η → ηxx − βη + α
λ+p′(V )η and ηx(0) = ηx(L) = 0 and observe
that αλ+p′(V )
− β is bounded independently of λ. Denoting µ(λ), respectively ν(λ), thespectrum of A(λ) with zero-flux, respectively Dirichlet boundary conditions and using theSturm comparism principle, we obtain that
µ(λ) > ν(λ).
Since Ux(x) is an eigenfunction for the eigenvalue 0 of the Dirichlet problem,
0 = (Ux)xx + αh′(U)Ux − βUx = (Ux)xx + (α1
p′(V )− β)Ux = A(0)Ux,
we conclude that ν(0) ≥ 0 and, therefore, µ(0) > 0. Furthermore, µ(λ) depends continuouslyon λ and it can be calculated by
µ(λ) = supw∈L2,||η||L2
=1
〈A(λ)η, η〉. (18)
Finally,
〈A(λ)η, η〉 =
∫ L
0
ηxxηdx+
∫ L
0
( α
λ+ p′(V )− β
)η2dx
≤ −∫ L
0
ηxηxdx+ c
∫ L
0
η2dx
≤ c
(19)
41
The boundedness of µ(λ) yields the existence of λ > 0 fulfilling µ(λ) = λ. Consequently,there exists a function η 6= 0 with
A(λ)η = µ(λ)η = λη with ηx(0) = ηx(L) = 0.
This theoreme shows, that bistability in the reduced model is not sufficient for patternformation. One necessarly needs hysteresis to get stable patterns.
9.2 Hysteresis-driven pattern formation
In this Section we investigate spatially heterogeneous stationary solutions of system (12)-(13) with hysteresis effect. The equation g(u, v) = 0 has 3 local solutions, but only the outerbranches v = hT (u) and v = hH(u) can give rise to stable stationary solutions. We definefH(u) = f(u, hH(u)) for u ≤ uH and fT (u) = f(u, hT (u)) for uT ≤ u. Using phase planeanalysis, we investigate the solutions of the two point boundary value problem, uxx+fi(u) = 0
for i = H,T with zero-flux boundary conditions.
Proposition 9.2. Neither uxx + fH(u) = 0, nor uxx + fT (u) = 0 has a nonconstant solutionfulfilling zero-flux boundary conditions.
Proof. We rewrite the equations as systems of first order ODEs
ux = w
wx = −fi(u) for i = H,T.
By definition fH(0) = 0 and −fH(u) > 0 for u ∈ (0, uH). That means that the flux at thepoint (u0, 0) always points upwards and to the right. Therefore, a solution starting at (u0, 0)
for 0 < u0 will never reach the w-axis again. Similarly, fT (u2) = 0 and −fT (u) < 0 foru ∈ (uT , u2). Consequently, all orbits ending at (uend, 0) with uend < u2 have started at somepoint with positive w-component.
We may conclude that there exist no stable patterns, which are continuous in v. Next,we construct transition layer solutions by gluing the phase planes together at an arbitrarychosen u ∈ [uH , uT ] (see Fig. 12), by solving the problem
uxx + fu(u) = 0 with ux(0) = ux(L) = 0, (20)
with
fu(u) =
fH(u) for u ≤ u
fT (u) for u > u.
42
Here, we do not expect u to be a C2-function on the whole interval [0, L], but there is somevalue x, so that we have
u ∈ C2([0, x)
)∩ C2
((x, L]
)∩ C0
([0, L]
)and
v ∈ C0([0, x)
)∩ C0
((x, L]
)as v is given by
v(x) =
hH(u(x)) if u ≤ u
hT (u(x)) if u > u.
Crucial for the analysis of (20) is the behaviour of the potential
Qu(u) =
∫ u
0
fu(u)du,
which fulfills Q′u(u) = fu(u) and Qu(0) = 0. Along the trajectories, it holds
u2x(x)
2+Qu(u(x)) = constant for all x. (21)
To investigate the behaviour of Qu, we define (see Fig. 13 )
F (v) =
∫ v
0
f(p(v), v)p′(v)dv
and, after a change of variables u 7→ p(u) in the definition of Qu(u), we obtain
Qu(u) =
F (hH(u)) if u ≤ u
F (hH(u))− F (hT (u)) + F (hT (u)) if u > u.
In particular, (21) yields the following relationships for the solution of (20):
• Qu(u0) = Qu(uend), which is equivalent to F (hH(u0)) − F (hH(u)) = F (hT (uend)) −F (hT (u)),
• Qu(u) has a local minimum at u,
• p2
2+Qu(u) = Qu(u0), which is equivalent to p2
2= F (hH(u0))− F (hH(u)) and
• Qu has a local maximum at u2, so the possible range for u0 and uend depends on Qu(u2).
Additionally, we shall verify that a solution constructed above reaches uend at x = L. There-fore, we consider the so-called “timemaps” and define
43
Figure 12: An example of the solution u connecting u0 with uend and the corresponding v(u).
• T1(u0) as the “time” x such that the orbit starting at (u0, 0) needs to reach for the firsttime the u = u axis and
• T2(uend) as the “time” such that the orbit starting at the u = u axis needs to reach(uend, 0) for the first time.
Using (21) and the boundary condition at x = 0, we obtain, for all x ∈ [0, x],
ux(x)2
2+Qu(u) = Qu(u0)
and
x =1√2
∫ u(x)
u0
du√Qu(u0)−Qu(u)
.
In particular, it holds
T1(u0) =1√2
∫ u
u0
du√Qu(u0)−Qu(u)
=1√2
∫ u
u0
du√F (hH(u0))− F (hH(u))
.
Similarly, using the boundary condition at x = L, we obtain
T2(uend) =1√2
∫ uend
u
du√Qu(uend)−Qu(u)
=1√2
∫ uend
u
du√F (hT (uend))− F (hT (u))
.
It leads to the following condition, T1(u0)+T2(uend(u0)) = L, where the dependence uend(u0)
is given by Qu(u0) = Qu(uend). Furthermore we obtain a condition for the jump value x,which has to respect x = T1(u0).
44
Furthermore, the shape of the stationary solution depends on the symmetry propertiesof the potential Qu. In Fig. 14 we present 3 representative cases of the stationary solutionsobtained for different values of u and the same kinetics and corresponding potentials.
Figure 13: F (v) =∫ v
0 f(p(v), v)p′(v)dv of model (12)- (13) with α = 11.8, β = 1, γ = 0, µ = 1 andp(v) = 0.1v3 − 6− 3v2 + 100v. Corresponding potentials Qu(u) obtained for different u are plottedin Fig. 14.
Let us notice that here we obtain the value of L (or equivalently diffusion size) for achosen u and u0. However, in the time-dependent problems the size of the domain (or dif-fusion) is given and values u and u0 are established dynamically depending on the initialconditions. Numerical simulations of the time-dependent problem show that, for fixed pa-rameters of the model, pattern selection depends strongly on the initial conditions, see Fig.15. Consequently, the values of u and u0 are determined by these data. On the other hand,for a given initial data, the solutions may tend to different stationary solutions (spatiallyhomogeneous or heterogeneous) for different sizes of diffusion, see Fig. 16.
45
Figure 14: Left panel: Stationary solutions for u (transition layers) and v (discontinuous solution)of model (12)-(13) with α = 11.8, β = 1, γ = 0, µ = 1 and p(v) = 0.1v3 − 6.3v2 + 100v. Rightpanel: Corresponding potentials Qu(u). We observe that depending on the choice of u, shape ofQu(u) differs and, consequently, we obtain different shapes of the stationary solution. In case ofQu(u2) < 0, the pattern achieves near-maximum values on a long x- interval (upper panel); in thesymmetric case Qu(u2) = 0 both high and small values are expressed on similar subdomains (middlepanel); in case of Qu(u2) > 0, the pattern achieves near-minimum values on a long x- interval (lowerpanel). 46
Figure 15: Time-dependent simulations of model (12)- (13) with parameters given in Fig. 14,a fixed diffusion coefficient d = 1/(L2) = 0.01 and different initial conditions. The simulationsare performed for constant initial data u0 = 0.1 and v0 = 40 for x ∈ [0.8, 1] (the upper panel),x ∈ [0.5, 1] (the middle panel) and x ∈ [0.2, 1] (the lower panel), and v0 = 0.01 otherwise. Weobserve the dependence of emerging pattern on the initial perturbation.
47
Figure 16: Time-dependent simulations of model (12)- (13) with parameters given in Fig. 14 anddiffusion coefficients d = 1/(L2) = 0.01 in the first and second upper panel and d = 1/(L2) = 1in the two lower panels, respectively. The simulations were performed for the constant initial datav0 = 0.01 and u0 equal to 500, 1000, 1000 and 5000 for x ∈ [0.8, 1] and u0 = 0.1 otherwise. Weobserve that in case of a large diffusion coefficient the solutions tend, depending on the initial data,to the lower or higher constant steady state. However, in case of small diffusion and large enoughinitial data, the solution may tend to a gradient-like pattern.
48
10 Conclusions
• Proposed models show how the nonlinearities of signalling pathways may result inspatial patterning
• Multistability in the dynamics of the signalling molecules is necessary to explain thetransplantation experiments
• Models with multiple steady states seem very promising and lead to mathematical andmodelling open problems
References
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