Non-Turing patterns in models with Turing-type instability

Transcription

1 Non-Turing patterns in models with Turing-type instability Anna Marciniak-Czochra Interdisciplinary Center for Scientific Computing (IWR) Institute of Applied Mathematics and BIOQUANT University of Heidelberg Based on joint works with Marek Kimmel (Rice University), Kanako Suzuki (Tohoku University), Grzegorz Karch (University of Wroclaw) and Steffen Härting (University of Heidelberg) April 2011

2 Biological system Cell proliferation is influenced by growth factor Growth factor is externally supplied or produced by the cells Growth factor diffuses along the structure formed by the cells and binds to cell membrane receptors Hypothesis: The diffusion of this growth factor may significantly influence the dynamics of the whole cell population

3 Cancerisation field theory Concept of field cancerisation first introduced by Slaughter et al. [D. P. Slaughter et al., Cancer 1953] Stem cells acquire genetic alterations and forms a patch, a clonal unit of altered daughter cells A patch converts into an expanding field Ultimately, clonal divergence leads to the development of one or more tumours within a contiguous field of pre-neoplastic cells An important clinical implication is that fields often remain after surgery of the primary tumour and may lead to new cancers Question: Is signalling and cooperation needed to create a background for malignant transformation?

4 Cancerisation field theory Concept of field cancerisation first introduced by Slaughter et al. [D. P. Slaughter et al., Cancer 1953] Stem cells acquire genetic alterations and forms a patch, a clonal unit of altered daughter cells A patch converts into an expanding field Ultimately, clonal divergence leads to the development of one or more tumours within a contiguous field of pre-neoplastic cells An important clinical implication is that fields often remain after surgery of the primary tumour and may lead to new cancers Question: Is signalling and cooperation needed to create a background for malignant transformation?

5 Cancerisation field theory Concept of field cancerisation first introduced by Slaughter et al. [D. P. Slaughter et al., Cancer 1953] Stem cells acquire genetic alterations and forms a patch, a clonal unit of altered daughter cells A patch converts into an expanding field Ultimately, clonal divergence leads to the development of one or more tumours within a contiguous field of pre-neoplastic cells An important clinical implication is that fields often remain after surgery of the primary tumour and may lead to new cancers Question: Is signalling and cooperation needed to create a background for malignant transformation?

6 Cancerisation field theory Concept of field cancerisation first introduced by Slaughter et al. [D. P. Slaughter et al., Cancer 1953] Stem cells acquire genetic alterations and forms a patch, a clonal unit of altered daughter cells A patch converts into an expanding field Ultimately, clonal divergence leads to the development of one or more tumours within a contiguous field of pre-neoplastic cells An important clinical implication is that fields often remain after surgery of the primary tumour and may lead to new cancers Question: Is signalling and cooperation needed to create a background for malignant transformation?

7 Example: Early peripheral lung tumours Example: Atypical Adenomatous Hyperplasia (AAH) Important finding: precursor of Bronchoalveolar Carcinoma (BAC). one of the major subtypes of lung cancer, which is usually located peripherally with respect to the bronchial tree in the lung, From a morphological perspective there seems little doubt that AAH might progress and develop, through a stage of bronchoalveolar adenocarcinoma (BAC), into invasive adenocarcinoma. (Kerr 2001)

8 Atypical Adenomatous Hyperplasia (AAH) Histological specimens of AAH - low grade lesions - more cellular lesions - very cellular lessions, very difficult to distinguish from bronchioloalveolar cell carcinoma (Kerr 2001)

9 Atypical Adenomatous Hyperplasia (AAH) Histological specimens of AAH - low grade lesions - more cellular lesions - very cellular lessions, very difficult to distinguish from bronchioloalveolar cell carcinoma (Kerr 2001) Objectives To elucidate the influence of the diffusion of growth factors on the cell dynamics To test how cooperation between partially transformed mutant cells influences the growth of the cell population

10 Model of cell proliferation The rate of change of u(t) is equal to du dt = 2pa(t)u(t) a(t)u(t) d cu(t) F (u, v) We assume that cell proliferation is regulated by the bound growth factor molecules Model 1. Proliferation rate depends on the density of the v(x,t) bound growth factor molecules per cell, i.e. u(x,t) and that this dependence is an increasing function with saturation, a(u, v) = a 0( v u )m 1 + ( v u )m Model 2. Proliferation process is modeled by a logistic growth function, a(u, v) = a 0 (v Ku)

11 Model of cell proliferation The rate of change of u(t) is equal to du dt = 2pa(t)u(t) a(t)u(t) d cu(t) F (u, v) We assume that cell proliferation is regulated by the bound growth factor molecules Model 1. Proliferation rate depends on the density of the v(x,t) bound growth factor molecules per cell, i.e. u(x,t) and that this dependence is an increasing function with saturation, a(u, v) = a 0( v u )m 1 + ( v u )m Model 2. Proliferation process is modeled by a logistic growth function, a(u, v) = a 0 (v Ku)

12 Growth factor regulation Macroscopic model: v t w t = α(u)w d b v dv, = D g w α(u)w d g w + κ(u) + dv, α(u) = α 0u s, α 0 0, s 1, u κ(u) = κ 1 + κ0, κ1, κ0 0 u + 1

13 Model Geometry 1. Line of cells, occupying the interval x [0, L] u t = (a p (u, v) d c ) u, v t = α(u)w dv d b v w t = D w d g w α(u)w + dv + κ(u). with homogeneous Neumann (zero flux) boundary conditions for g, x w(0, t) = x w(1, t) = 0; x [0, 1]. Geometry 2. Thin sheet of cells, occupying the square (x, y) [0, L 1 ] [0, L 2 ], rolled to form a cylindrical tube such that the intervals {(x, 0), x [0, L 1 ]}, and {(x, L 2 ), x [0, L 1 ]} coincide.

14 Mechanism for pattern formation Diffusion-driven instabilities (DDI) - Turing-type patterns DDI takes place when the kinetics system is asymptotically stable the complete system unstable for spatially non-homogeneous perturbations Find positive spatially homogeneous steady states Use linear stability analysis to derive the conditions for DDI Interpretation: Spatial dimension alters cancer dynamics

15 Mechanism for pattern formation Diffusion-driven instabilities (DDI) - Turing-type patterns DDI takes place when the kinetics system is asymptotically stable the complete system unstable for spatially non-homogeneous perturbations Find positive spatially homogeneous steady states Use linear stability analysis to derive the conditions for DDI Interpretation: Spatial dimension alters cancer dynamics

16 Mechanism for pattern formation Diffusion-driven instabilities (DDI) - Turing-type patterns DDI takes place when the kinetics system is asymptotically stable the complete system unstable for spatially non-homogeneous perturbations Find positive spatially homogeneous steady states Use linear stability analysis to derive the conditions for DDI Interpretation: Spatial dimension alters cancer dynamics

17 Diffusion-driven instabilities (DDI) Linear stability analysis Ã(µ m ) = A 1 γ Dµ m 2. A is the Jacobian matrix at a steady state, D- matrix of diffusion coefficients. µ m 2 is a wavenumber obtained from the Laplacian s eigenproblem: x φ m = µ m 2 φ m in Ω, n φ m = 0 The linear stability of the homogeneous steady state to spatially heterogeneous perturbations is determined by the sign of Reλ(µ m 2 ), where λ(µ m 2 ) belongs to the spectrum of Ã(µ m 2 ).

18 DDI in the model with 1 diffusion operator Lemma Let A be the Jacobian matrix computed at a positive spatially homogeneous steady state and such that a ii < 0 for i = 1, 2, 3 and a 12a 21 > 0. There is a diffusion-driven instability for the considered system if and only if the following conditions are fulfiled, tr(a) X i<j tr(a) > 0, A ij + A > 0, A > 0, A 12 < 0, where A ij is a submatrix of A consisting of the i-th and j-th column and i-th and j-th row, and A and A ij denote the determinants of matrices A and A ij, respectively.

19 Dispersion relation à = A A12 µ2 m γ > 0. For every γ there exist infinitely many different integer µ m for which the above inequality is fulfiled From the dispersion relation, λ = λ(µ 2 m/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 parameter γ

20 Space-homogeneous steady states Theorem The diffusion-driven instability for the considered system occurs if and only if the following conditions for the model parameters are satisfied, (i) a 1 > d c, (ii) κ 0, κ 1 such that the equation d c d c c κ( c) = d b c + d g (d + d b ) a 1 d c a 1 d c α( c) has two positive solutions. (iii) α(c) = α 1 c s+1, s > 0,

21 Asymptotic behaviour of the solutions Complete system exhibits a period of near-equilibrium After this period solutions start growing exponentially at some x-coordinates The rate of the exponential growth and the length of the dormancy period t = (1/λ 1 )ln(1/ɛ) can be calculated from the linearised system Behaviour broadly consistent with that of precancerous lesions

22 Asymptotic behaviour of the solutions Complete system exhibits a period of near-equilibrium After this period solutions start growing exponentially at some x-coordinates The rate of the exponential growth and the length of the dormancy period t = (1/λ 1 )ln(1/ɛ) can be calculated from the linearised system Behaviour broadly consistent with that of precancerous lesions

23 Spatial profiles of the solutions

24 Persistence of positive solutions Asymptotic bounds on the solutions depending on growth factor production If κ(0) = 0, then the system destabilised by diffusion eventually tends to 0 Interpretation: External supply of growth factor needed for stability of patterns It can be replaced by the constant influx of cells (flux from mutation)

25 Basic model ( av ) u t = u + v d c u, v t = d b v + u 2 w dv, (RD) w t = 1 γ w xx d g w u 2 w + dv + κ 0 for x (0, 1), t > 0 with the homogeneous Neumann boundary conditions for the function w = w(x, t) w x (0, t) = w x (1, t) = 0 for all t > 0, and with positive initial conditions u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), w(x, 0) = w 0 (x).

26 Kinetic system Solutions are nonnegative and uniformly bounded (tricky change of variables (u, v u, uw)). ( ) The trivial steady state (u 0, v 0, w 0 ) 0, 0, κ0 d g is locally asymptotically stable. Assume a > d c and κ 2 0 Θ, where Θ = 4d g d b d 2 c (d b + d) (a d c ) 2. Then, the kinetic system has two positive constant stationary solutions (u ±, v ±, w ± ), where w ± = κ 0 ± κ 2 0 Θ 2d g, v ± = d 2 c (d b + d) (a d c ) 2 1 w ±, u ± = a d c d c v ±. (u, v, w ) is stable, and (u +, v +, w + ) is unstable.

27 Kinetic system Solutions are nonnegative and uniformly bounded (tricky change of variables (u, v u, uw)). ( ) The trivial steady state (u 0, v 0, w 0 ) 0, 0, κ0 d g is locally asymptotically stable. Assume a > d c and κ 2 0 Θ, where Θ = 4d g d b d 2 c (d b + d) (a d c ) 2. Then, the kinetic system has two positive constant stationary solutions (u ±, v ±, w ± ), where w ± = κ 0 ± κ 2 0 Θ 2d g, v ± = d 2 c (d b + d) (a d c ) 2 1 w ±, u ± = a d c d c v ±. (u, v, w ) is stable, and (u +, v +, w + ) is unstable.

28 Kinetic system Solutions are nonnegative and uniformly bounded (tricky change of variables (u, v u, uw)). ( ) The trivial steady state (u 0, v 0, w 0 ) 0, 0, κ0 d g is locally asymptotically stable. Assume a > d c and κ 2 0 Θ, where Θ = 4d g d b d 2 c (d b + d) (a d c ) 2. Then, the kinetic system has two positive constant stationary solutions (u ±, v ±, w ± ), where w ± = κ 0 ± κ 2 0 Θ 2d g, v ± = d 2 c (d b + d) (a d c ) 2 1 w ±, u ± = a d c d c v ±. (u, v, w ) is stable, and (u +, v +, w + ) is unstable.

29 Model with diffusion Existence A unique positive global-in-time solution for every positive initial datum u, v C([0, T ], L (0, 1)), w C([0, T ], W 1,2 (0, 1)). Boundedness Theorem Let κ 0 0. The solution (u, v, w) of (RD) satisfies lim sup t lim sup t lim sup t Here µ = min{d g, d b } > 0. 0 u(x, t) dx κ 0 µd c, v(x, t) dx κ 0 µ, w(t) κ 0 ( Cd µd 1/2 g + 1 d g ).

30 Model with diffusion Existence A unique positive global-in-time solution for every positive initial datum u, v C([0, T ], L (0, 1)), w C([0, T ], W 1,2 (0, 1)). Boundedness Theorem Let κ 0 0. The solution (u, v, w) of (RD) satisfies lim sup t lim sup t lim sup t Here µ = min{d g, d b } > 0. 0 u(x, t) dx κ 0 µd c, v(x, t) dx κ 0 µ, w(t) κ 0 ( Cd µd 1/2 g + 1 d g ).

31 Model with diffusion Week proliferation (a < d c ) implies the convergence towards the trivial steady state (0, 0, κ 0 /d g ) Proposition Let a < d c and (u, v, w) be the solution of (RD). Then, there exist positive constants C 1, C 2, σ 1, σ 2 dependent on the parameters in (RD) and C 2 dependent also on u 0 and v 0, such that 0 u(x, t) u 0 (x)e (dc a)t, (1) 0 v(x, t) v 0 (x)e (db+d)t + C 1 u0(x)te 2 σ1t (2) w(, t) κ 0 C 2 e σ2t, (3) L d g for all x [0, 1] and t 0.

32 Model with diffusion Week proliferation (a < d c ) implies the convergence towards the trivial steady state (0, 0, κ 0 /d g ) Proposition Let a < d c and (u, v, w) be the solution of (RD). Then, there exist positive constants C 1, C 2, σ 1, σ 2 dependent on the parameters in (RD) and C 2 dependent also on u 0 and v 0, such that 0 u(x, t) u 0 (x)e (dc a)t, (1) 0 v(x, t) v 0 (x)e (db+d)t + C 1 u0(x)te 2 σ1t (2) w(, t) κ 0 C 2 e σ2t, (3) L d g for all x [0, 1] and t 0.

33 The model with diffusion The case of a strong proliferation (a > d c ): kinetic effect Theorem Let a > d c. Fix x 0 [0, 1] and assume that there exists a constant K w > 0 such that w(x 0, t) K w for and all t > 0. Suppose that there exists M > 0 such that ( ) 2 dc 0 < u 0 (x 0 ) M and 0 < v 0 (x 0 ) < M (4) a satisfying, moreover, ( MK w 1 + d ) 2 ( ) 2 c dc (d b + d). (5) a a Then, (u(x 0, t), v(x 0, t)) (0, 0) as t. This is the uniform convergence for all x 0 [0, 1], for which inequalities (4) are satisfied.

34 The model with diffusion Idea for the pointwise convergence towards the trivial steady state: small invariant region Lemma Assume that 0 w 0 (x) K 0 for a constant K 0 > 0. Under the assumptions of the theorem, for all t > 0. ( 0 < u(x, t) < M 1 + d ) c 0 < v(x, t) < a and w(x, t) K 0 + d ( ) 2 dc M + κ 0 d g a d g ( ) 2 dc M a (6)

35 Construction of patterns ( ) av U + V d c U = 0, (7) d b V + U 2 W dv = 0, (8) 1 γ W xx d g W U 2 W + dv + κ 0 = 0 (9) and the boundary condition W x (0) = W x (1) = 0. We interested only in U(x) > 0 and V (x) > 0, Let a > d c, We obtain from (7), (8) that U(x) = a d c d c V (x) and V (x) = d 2 c (d b + d) (a d c ) 2 1 W (x). (10)

36 Construction of patterns The boundary value problem for W (x) 1 γ W d g W d b d 2 c (d b + d) (a d c ) 2 1 W + κ 0 = 0, (11) We find the explicit γ 0 such that W x (0) = W x (1) = 0. (12) for all γ (0, γ 0 ], problem (11) (12) has only constant solutions, for all γ > γ 0, we describe all positive solutions of problem (11) (12).

37 Construction of patterns The boundary value problem for W (x) 1 γ W d g W d b d 2 c (d b + d) (a d c ) 2 1 W + κ 0 = 0, (11) We find the explicit γ 0 such that W x (0) = W x (1) = 0. (12) for all γ (0, γ 0 ], problem (11) (12) has only constant solutions, for all γ > γ 0, we describe all positive solutions of problem (11) (12).

38 Construction of patterns Definition Let k N and k 2. We call a function W C([0, 1]) a periodic function on [0, 1] with k modes if W = W (x) is monotone on [ ] 0, 1 k and if ( W W (x) = W x 2j k ( 2j+2 k ) ) x for for for every j {0, 1, 2, 3,...} such that 2j + 2 k. [ ] x 2j k, 2j+1 k [ ] x 2j+1 k, 2j+2 k

39 Construction of patterns Theorem Assume that a > d c and κ 0 > Θ. Fix γ > γ 0 and consider the biggest n N such that γ > n 2 γ 0. Then, problem (11) (12) has the following solutions: a unique strictly increasing solution and a unique strictly decreasing solution, for each k {2,..., n}, a unique periodic solution W k with k modes that is increasing on [0, 1 k ] as well as its symmetric counterpart: W k (x) W k (1 x), the constant steady states w ±. There are no other positive solutions of problem (11) (12).

40 Construction of patterns All solutions with n = 3: All solutions with γ > 0 satisfying 3 2 γ 0 < γ < 4 2 γ 0.

41 Construction of patterns Idea of a proof of theorem By the change of variables x Tx, where T = γ, the boundary value problem becomes w + h(w) = 0 x (0, T ), (13) w (0) = w (T ) = 0. (14) A solution w = w(x) to problem (13) (14) satisfies the differential equation w (x) = ± 2(C H(w(x))) (15) for C R. Here H = h.

42 Construction of patterns Idea of a proof of theorem By the change of variables x Tx, where T = γ, the boundary value problem becomes w + h(w) = 0 x (0, T ), (13) w (0) = w (T ) = 0. (14) A solution w = w(x) to problem (13) (14) satisfies the differential equation w (x) = ± 2(C H(w(x))) (15) for C R. Here H = h.

43 Potential energy and phase trajectories Idea of a proof of theorem Energy equation 1 2 z 2 + H(w) = C All trajectories are symmetric with respect to w-axis. The condition z(0) = z(t ) is satisfied for a certain T < 0 if energy describes a close curve for a certain C R. Such closed curves exist only if the potential energy H(w) has a local minimum.

44 Which solutions are stable? Several nonconstant solutions for sufficiently large γ As long as γ > m 2 γ 0 there exists a solution with m peaks.

45 Random initial perturbation

46 Instability of patterns Let W (x) be one of the functions from the previous theorem, and (U(x), V (x), W (x)) be a stationary solution of our system, where U(x) = a d c d c V (x) and V (x) = d 2 c (d b + d) (a d c ) 2 1 W (x). This stationary solution appears to be unstable solution of the reaction-diffusion equations (RD). Let us be more precise.

47 Instability of patterns Let W (x) be one of the functions from the previous theorem, and (U(x), V (x), W (x)) be a stationary solution of our system, where U(x) = a d c d c V (x) and V (x) = d 2 c (d b + d) (a d c ) 2 1 W (x). This stationary solution appears to be unstable solution of the reaction-diffusion equations (RD). Let us be more precise.

48 Instability of patterns Let W (x) be one of the functions from the previous theorem, and (U(x), V (x), W (x)) be a stationary solution of our system, where U(x) = a d c d c V (x) and V (x) = d 2 c (d b + d) (a d c ) 2 1 W (x). This stationary solution appears to be unstable solution of the reaction-diffusion equations (RD). Let us be more precise.

49 Instability of patterns Linearized operator The linearization of system (RD) at the steady state (U, V, W ) where L = A(x) = (a ij ) i,j=1,2, γ 2 x with the constant K = U(x)W (x) = dc (d b+d) a d c. + A(x), ( d dc c a 1 ) (a d c ) 2 a 0 2K d b d K 2 W 2 (x) 2K d d g K 2 W 2 (x) Notice that only the coefficients a 23 and a 33 depend on x.,

50 Instability of patterns Linearized operator We consider L as an operator in the Hilbert space with the domain H = L 2 (0, 1) L 2 (0, 1) L 2 (0, 1) D(L) = L 2 (0, 1) L 2 (0, 1) W 2,2 (0, 1). L has infinitely many positive eigenvalues.

51 Instability of patterns Linearized operator We consider L as an operator in the Hilbert space with the domain H = L 2 (0, 1) L 2 (0, 1) L 2 (0, 1) D(L) = L 2 (0, 1) L 2 (0, 1) W 2,2 (0, 1). L has infinitely many positive eigenvalues.

52 Instability of patterns Spectrum of L Together with the matrix A(x) = (a ij ) i,j=1,2,3 = we consider its sub-matrix A 12 ( d dc c a 1 ) (a d c ) 2 a 0 2K d b d K 2 W 2 (x) 2K d d g K 2 W 2 (x) ( ) a11 a 12. a 21 a 22 Lemma Let λ be an eigenvalue of the matrix A 12. Then λ belongs to the continuous spectrum of the operator L. The matrix A 12 has a positive eigenvalue λ 0.,

53 Instability of patterns Spectrum of L Together with the matrix A(x) = (a ij ) i,j=1,2,3 = we consider its sub-matrix A 12 ( d dc c a 1 ) (a d c ) 2 a 0 2K d b d K 2 W 2 (x) 2K d d g K 2 W 2 (x) ( ) a11 a 12. a 21 a 22 Lemma Let λ be an eigenvalue of the matrix A 12. Then λ belongs to the continuous spectrum of the operator L. The matrix A 12 has a positive eigenvalue λ 0.,

54 Instability of patterns Spectrum of L - the crucial lemma Lemma A complex number λ is an eigenvalue of the operator L if and only if the following two conditions are satisfied λ is not an eigenvalue of the matrix A 12, the boundary value problem has a nontrivial solution: 1 det(a λi ) γ η + η = 0, x (0, 1) det(a 12 λi ) η (0) = η (1) = 0. Proof. Study the system (a 11 λ)ϕ + a 12ψ = 0 a 21ϕ + (a 22 λ)ψ + a 23η = 0 1 γ 2 x η + a 31ϕ + a 32ψ + (a 33 λ)η = 0, supplemented with the boundary condition η x(0) = η x(1) = 0

55 Instability of patterns Spectrum of L - the crucial lemma Lemma A complex number λ is an eigenvalue of the operator L if and only if the following two conditions are satisfied λ is not an eigenvalue of the matrix A 12, the boundary value problem has a nontrivial solution: 1 det(a λi ) γ η + η = 0, x (0, 1) det(a 12 λi ) η (0) = η (1) = 0. Proof. Study the system (a 11 λ)ϕ + a 12ψ = 0 a 21ϕ + (a 22 λ)ψ + a 23η = 0 1 γ 2 x η + a 31ϕ + a 32ψ + (a 33 λ)η = 0, supplemented with the boundary condition η x(0) = η x(1) = 0

56 Instability of patterns Spectrum of L - main result Theorem Denote by λ 0 the positive eigenvalue of the matrix A 12. There exists a sequence {λ n} n N of positive eigenvalues of the operator L that satisfy λ n λ 0 as n. Recall that λ 0 belongs to the continuous spectrum of the operator L. Idea of the proof. Analysis of solutions of the generalized Sturm-Liouville problem 1 γ η + q(x, λ)η = 0, x (0, 1) where η (0) = η (1) = 0, q(x, λ) = det(a(x) λi ) det(a 12 λi ).

57 Instability of patterns Spectrum of L - main result Theorem Denote by λ 0 the positive eigenvalue of the matrix A 12. There exists a sequence {λ n} n N of positive eigenvalues of the operator L that satisfy λ n λ 0 as n. Recall that λ 0 belongs to the continuous spectrum of the operator L. Idea of the proof. Analysis of solutions of the generalized Sturm-Liouville problem 1 γ η + q(x, λ)η = 0, x (0, 1) where η (0) = η (1) = 0, q(x, λ) = det(a(x) λi ) det(a 12 λi ).

58 Discontinuous patterns Idea of the construction We look for nonnegative solutions W such that U(x) = V (x) = 0 on a certain set I [0, 1], which we call a null set. Then, we obtain Wxx dg W + κ0 = 0 1 γ We do not expect W to be a C 2 -function. Definition (U, V, W ) L ([0, 1]) L ([0, 1]) C 1 ([0, 1]) is a weak solution of the stationary system, if the algebraic equations for U and V are satisfied for almost all x [0, 1] and for all ϕ C 1 ([0, 1]) 1 γ Z 1 0 W (x)ϕ (x) dx+ Z 1 0 d g W (x) U 2 (x)w (x) + dv (x) + κ 0 ϕ(x) dx = 0.

59 Existence of discontinuous patterns Theorem Assume that a > d c and κ 2 0 > Θ. There exists a continuum of weak solutions of the stationary system with some γ > 0. Each such solution (U, V, W ) L (0, 1) L (0, 1) C 1 ([0, 1]) has the following property: there exists a sequence 0 = x 0 < x 1 < x 2 <... < x N = 1 such that for each k {0, N 1} either or for all x (x k, x k+1 ), U(x) = V (x) = 0 and W (x) satisfies 1 γ W d g W + κ 0 = 0, for all x (x k, x k+1 ), U(x) > 0 and V (x) > 0 are given by relations (10), where the function W is a solution of the stationary equation.

60 Instability of discontinuous stationary solutions Theorem Every discontinuous weak stationary solution (U I, V I, W I ) with a null set I [0, 1], is an unstable solution of the nonlinear system considered in the Hilbert space H I. For a null set I, we define the associate L 2 -space L 2 I(0, 1) = {v L 2 (0, 1) : v(x) = 0 on I}, supplemented with the usual L 2 -scalar product, which is a Hilbert space as the closed subspace of L 2 (0, 1). If u 0 (x) = v 0 (x) = 0 for some x [0, 1] then u(x, t) = v(x, t) = 0 for all t 0. Hence, the space H I = L 2 I (0, 1) L2 I (0, 1) L2 (0, 1) is invariant for the flow generated by the system.

61 Instability of discontinuous stationary solutions Theorem Every discontinuous weak stationary solution (U I, V I, W I ) with a null set I [0, 1], is an unstable solution of the nonlinear system considered in the Hilbert space H I. For a null set I, we define the associate L 2 -space L 2 I(0, 1) = {v L 2 (0, 1) : v(x) = 0 on I}, supplemented with the usual L 2 -scalar product, which is a Hilbert space as the closed subspace of L 2 (0, 1). If u 0 (x) = v 0 (x) = 0 for some x [0, 1] then u(x, t) = v(x, t) = 0 for all t 0. Hence, the space H I = L 2 I (0, 1) L2 I (0, 1) L2 (0, 1) is invariant for the flow generated by the system.

62 Numerical simulations What are the patterns, which we see in numerical simulations? Our conjecture: there are singular patterns.

63 2D simulations

64 Conclusions System of a single reaction-diffusion equation coupled to ordinary differential equations may exhibit Turing-type instability, which, however, does not lead to the formation of Turing patterns. All stationary solutions (both regular and discontinuous) are unstable. Numerical simulations show emergence of growth patterns concentrated around discrete points along the spatial coordinate. Diffusion in the mixed/degenerated systems may lead not only to the destabilisation of the constant steady state (Turing-type mechanism) but also to the unbounded growth of solutions and concentration of mass. It is similar for the systems of 2 equations (1 reaction-diffusion + 1 ODE).

65 Conclusions System of a single reaction-diffusion equation coupled to ordinary differential equations may exhibit Turing-type instability, which, however, does not lead to the formation of Turing patterns. All stationary solutions (both regular and discontinuous) are unstable. Numerical simulations show emergence of growth patterns concentrated around discrete points along the spatial coordinate. Diffusion in the mixed/degenerated systems may lead not only to the destabilisation of the constant steady state (Turing-type mechanism) but also to the unbounded growth of solutions and concentration of mass. It is similar for the systems of 2 equations (1 reaction-diffusion + 1 ODE).

66 Conclusions System of a single reaction-diffusion equation coupled to ordinary differential equations may exhibit Turing-type instability, which, however, does not lead to the formation of Turing patterns. All stationary solutions (both regular and discontinuous) are unstable. Numerical simulations show emergence of growth patterns concentrated around discrete points along the spatial coordinate. Diffusion in the mixed/degenerated systems may lead not only to the destabilisation of the constant steady state (Turing-type mechanism) but also to the unbounded growth of solutions and concentration of mass. It is similar for the systems of 2 equations (1 reaction-diffusion + 1 ODE).

67 Conclusions System of a single reaction-diffusion equation coupled to ordinary differential equations may exhibit Turing-type instability, which, however, does not lead to the formation of Turing patterns. All stationary solutions (both regular and discontinuous) are unstable. Numerical simulations show emergence of growth patterns concentrated around discrete points along the spatial coordinate. Diffusion in the mixed/degenerated systems may lead not only to the destabilisation of the constant steady state (Turing-type mechanism) but also to the unbounded growth of solutions and concentration of mass. It is similar for the systems of 2 equations (1 reaction-diffusion + 1 ODE).

68 Conclusions System of a single reaction-diffusion equation coupled to ordinary differential equations may exhibit Turing-type instability, which, however, does not lead to the formation of Turing patterns. All stationary solutions (both regular and discontinuous) are unstable. Numerical simulations show emergence of growth patterns concentrated around discrete points along the spatial coordinate. Diffusion in the mixed/degenerated systems may lead not only to the destabilisation of the constant steady state (Turing-type mechanism) but also to the unbounded growth of solutions and concentration of mass. It is similar for the systems of 2 equations (1 reaction-diffusion + 1 ODE).

69 References A. Marciniak-Czochra, G. Karch and K. Suzuki. Unstable patterns in reaction-diffusion model of early carcinogenesis. Submitted. Preprint available at A. Marciniak-Czochra and M. Kimmel (2008) Reaction-difusion model of early carcinogenesis: The effects of influx of mutated cells. Math. Mod. Natural Phenomena 7: A. Marciniak-Czochra and M. Kimmel (2007) Modelling of early lung cancer progression: Influence of growth factor production and cooperation between partially transformed cells. Math. Mod. Meth. Appl. Sci. 17: A. Marciniak-Czochra and M. Kimmel (2006) Dynamics of growth and signalling along linear and surface structures in very early tumours. Comp. Math. Meth. Med. 7: A. Marciniak-Czochra and M. Kimmel (2006) Reaction-diffusion approach to modeling of the spread of early tumors along linear or tubular structures. J. Theor. Biol. 244: A. Marciniak-Czochra and M. Kimmel (2005) Mathematical model of tumour invasion along linear or tubular structures. Math. Comp. Modelling. 41:

70 Acknowledgements ERC Starting Grant No Multiscale mathematical modelling of dynamics of structure formation in cell systems. Emmy Noether Programme of German Research Council