February  2017, 37(2): 757-800. doi: 10.3934/dcds.2017032

Stable patterns with jump discontinuity in systems with Turing instability and hysteresis

1. 

Institute of Applied Mathematics and BIOQUANT, Heidelberg University, Im Neuenheimer Feld 205,69120 Heidelberg, Germany

2. 

Institute of Applied Mathematics, IWR and BIOQUANT, Heidelberg University, Im Neuenheimer Feld 205,69120 Heidelberg, Germany

3. 

Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

* Corresponding author: takagi@m.tohoku.ac.jp.

Received  May 2015 Revised  November 2015 Published  November 2016

Fund Project: This work was undertaken in the framework of German-Japanese University Partnership Program (HeKKSaGOn Alliance). The first two authors were supported by European Research Council Starting Grant No 210680 'Multiscale mathematical modelling of dynamics of structure formation in cell systems' and Emmy Noether Program of DFG. The second author was supported by 'Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences' and 'Baden-Württemberg Stipendium plus' scholarship of Baden-Württemberg Stiftung. The third author was supported in part by JSPS Grant-in-Aid for Scientific Research (A) #22244010 'Theory of Differential Equations Applied to Biological Pattern Formation|from Analysis to Synthesis' and #26610027 'Control of Patterns by Multi-component Reaction-Diffusion Systems of Degenerate Type'

Classical models of pattern formation are based on diffusion-driven instability (DDI) of constant stationary solutions of reaction-diffusion equations, which leads to emergence of stable, regular Turing patterns formed around that equilibrium. In this paper we show that coupling reaction-diffusion equations with ordinary differential equations (ODE) may lead to a novel pattern formation phenomenon in which DDI causes destabilization of both constant solutions and Turing patterns. Bistability and hysteresis effects in the null sets of model nonlinearities yield formation of far from the equilibrium patterns with jump discontinuity. We derive conditions for stability of stationary solutions with jump discontinuity in a suitable topology which allows us to include the discontinuity points and leads to the definition of $(\varepsilon_0, A)$-stability. Additionally, we provide conditions on stability of patterns in a quasi-stationary model reduction. The analysis is illustrated on the example of three-component model of receptor-ligand binding. The proposed model provides an example of a mechanism of de novo formation of far from the equilibrium patterns in reaction-diffusion-ODE models involving co-existence of DDI and hysteresis.

Citation: Steffen Härting, Anna Marciniak-Czochra, Izumi Takagi. Stable patterns with jump discontinuity in systems with Turing instability and hysteresis. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 757-800. doi: 10.3934/dcds.2017032
References:
[1]

M. Akam, Making stripes inelegantly, Nature, 341 (1989), 282-283. doi: 10.1038/341282a0. Google Scholar

[2]

A. AnmaK. Sakamoto and T. Yoneda, Unstable subsystems cause Turing instability, Kodai Math. J., 35 (2012), 215-247. doi: 10.2996/kmj/1341401049. Google Scholar

[3]

D.G. AronsonA. Tesei and H. Weinberger, A density-dependent diffusion system with stable discontinuous stationary solutions, Ann. Mat. Pura Appl., 152 (1988), 259-280. doi: 10.1007/BF01766153. Google Scholar

[4]

W. Bangerth, R. Hartmann and G. Kanschat, deal. Ⅱ -a general purpose object oriented finite element library, ACM Trans. Math. Softw. , 33 (2007), Art. 24, 27 pp.Google Scholar

[5]

R. Casten and C. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations, 27 (1978), 266-273. doi: 10.1016/0022-0396(78)90033-5. Google Scholar

[6]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin), 12 (1972), 30-39. doi: 10.1007/BF00289234. Google Scholar

[7]

S. Härting and A. Marciniak-Czochra, Spike patterns in a reaction-diffusion-ode model with Turing instability, Math. Methods Appl. Sci., 37 (2013), 1377-1391. Google Scholar

[8]

S. HockY. NgJ. HasenauerD. WittmannD. LutterD. TrümbachW. WurstN. Prakash and F. J. Theis, Sharpening of expression domains induced by transcription and microRNA regulation within a spatio-temporal model of mid-hindbrain boundary formation, BMC Systems Biol., 7 (2013), p48. doi: 10.1186/1752-0509-7-48. Google Scholar

[9]

V. KlikaR. BakerD. Headon and E. Gaffney, The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organisation, Bull. Math. Biol., 74 (2012), 935-957. doi: 10.1007/s11538-011-9699-4. Google Scholar

[10]

A. Köthe and A. Marciniak-Czochra, Multistability and hysteresis-based mechanism of pattern formation in biology, in Pattern Formation in Morphogenesis-problems and their Mathematical Formalization (eds. V. Capasso, M. Gromov and N. Morozova), Springer Proceedings in Mathematics, 15 (2012), 153-173.Google Scholar

[11] D. A. Lauffenburger and J. J. Linderman, Receptors. Models for Binding, Trafficking, and Signaling, Oxford University Press, 1993. Google Scholar
[12]

A. Marciniak-Czochra, Receptor-based models with diffusion-driven instability for pattern formation in hydra, J. Biol. Systems, 11 (2003), 293-324. doi: 10.1142/S0218339003000889. Google Scholar

[13]

A. Marciniak-Czochra, Receptor-based models with hysteresis for pattern formation in Hydra, Math. Biosci., 199 (2006), 97-119. doi: 10.1016/j.mbs.2005.10.004. Google Scholar

[14]

A. Marciniak-Czochra, Strong two-scale convergence and corrector result for the receptor-based model of the intercellular communication, IMA J. Appl. Math., 77 (2012), 855-868. doi: 10.1093/imamat/hxs052. Google Scholar

[15]

A. Marciniak-CzochraG. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl., 99 (2013), 509-543. doi: 10.1016/j.matpur.2012.09.011. Google Scholar

[16]

A. Marciniak-CzochraG. Karch and K. Suzuki, Instability of turing patterns in reaction-diffusion-ODE systems, J. Math. Biol., (2016). doi: 10.1007/s00285-016-1035-z. Google Scholar

[17]

A. Marciniak-Czochra and M. Kimmel, Modelling of early lung cancer progression: Influence of growth factor production and cooperation between partially transformed cells, Math. Models Methods Appl. Sci., 17 (2007), 1693-1719. doi: 10.1142/S0218202507002443. Google Scholar

[18]

A. Marciniak-CzochraM. Nakayama and I. Takagi, Pattern formation in a diffusion-ODE model with hysteresis, Differential Integral Equations, 28 (2015), 655-694. Google Scholar

[19]

A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenisation techniques, SIAM J. Math. Anal., 40 (2008), 215-237. doi: 10.1137/050645269. Google Scholar

[20]

M. MimuraM. Tabata and Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal., 11 (1980), 613-631. doi: 10.1137/0511057. Google Scholar

[21]

J. D. Murray, Mathematical Biology. Ⅱ. Spatial Models and Biomedical Applications 3rd edition, Interdisciplinary Applied Mathematics, 18, 2003, Springer-Verlag, New York.Google Scholar

[22] W. A. Müller, Developmental Biology, Springer-Verlag, New York, 1997. Google Scholar
[23]

K. PhamA. ChauviereH. HatzikirouX. LiH.M. ByrneV. Cristini and J. Lowengrub, Density-dependent quiescence in glioma invasion: Instability in a simple reaction-diffusion model for the migration/proliferation dichotomy, J. Biol. Dyn., 6 (2012), 54-71. doi: 10.1080/17513758.2011.590610. Google Scholar

[24]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, 1984, Springer-Verlag, Berlin.Google Scholar

[25] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2 edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar
[26]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012. Google Scholar

[27]

D. M. UmulisM. SerpeM. B. O'Connor and H. G. Othmer, Robust, bistable patterning of the dorsal surface of the Drosophila embryo, PNAS, 103 (2006), 11613-11618. doi: 10.1073/pnas.0510398103. Google Scholar

[28]

H. F. Weinberger, A simple system with a continuum of stable inhomogeneous steady states, Nonlinear Partial Differential Equations in Applied Science; Proceedings of the U.S.-Japan Seminar, North-Holland Math. Stud., 81 (1983), 345-359. Google Scholar

[29] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Heidelberg/Dordrecht/London/New York, 2010. doi: 10.1007/978-3-642-04631-5. Google Scholar

show all references

References:
[1]

M. Akam, Making stripes inelegantly, Nature, 341 (1989), 282-283. doi: 10.1038/341282a0. Google Scholar

[2]

A. AnmaK. Sakamoto and T. Yoneda, Unstable subsystems cause Turing instability, Kodai Math. J., 35 (2012), 215-247. doi: 10.2996/kmj/1341401049. Google Scholar

[3]

D.G. AronsonA. Tesei and H. Weinberger, A density-dependent diffusion system with stable discontinuous stationary solutions, Ann. Mat. Pura Appl., 152 (1988), 259-280. doi: 10.1007/BF01766153. Google Scholar

[4]

W. Bangerth, R. Hartmann and G. Kanschat, deal. Ⅱ -a general purpose object oriented finite element library, ACM Trans. Math. Softw. , 33 (2007), Art. 24, 27 pp.Google Scholar

[5]

R. Casten and C. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations, 27 (1978), 266-273. doi: 10.1016/0022-0396(78)90033-5. Google Scholar

[6]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin), 12 (1972), 30-39. doi: 10.1007/BF00289234. Google Scholar

[7]

S. Härting and A. Marciniak-Czochra, Spike patterns in a reaction-diffusion-ode model with Turing instability, Math. Methods Appl. Sci., 37 (2013), 1377-1391. Google Scholar

[8]

S. HockY. NgJ. HasenauerD. WittmannD. LutterD. TrümbachW. WurstN. Prakash and F. J. Theis, Sharpening of expression domains induced by transcription and microRNA regulation within a spatio-temporal model of mid-hindbrain boundary formation, BMC Systems Biol., 7 (2013), p48. doi: 10.1186/1752-0509-7-48. Google Scholar

[9]

V. KlikaR. BakerD. Headon and E. Gaffney, The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organisation, Bull. Math. Biol., 74 (2012), 935-957. doi: 10.1007/s11538-011-9699-4. Google Scholar

[10]

A. Köthe and A. Marciniak-Czochra, Multistability and hysteresis-based mechanism of pattern formation in biology, in Pattern Formation in Morphogenesis-problems and their Mathematical Formalization (eds. V. Capasso, M. Gromov and N. Morozova), Springer Proceedings in Mathematics, 15 (2012), 153-173.Google Scholar

[11] D. A. Lauffenburger and J. J. Linderman, Receptors. Models for Binding, Trafficking, and Signaling, Oxford University Press, 1993. Google Scholar
[12]

A. Marciniak-Czochra, Receptor-based models with diffusion-driven instability for pattern formation in hydra, J. Biol. Systems, 11 (2003), 293-324. doi: 10.1142/S0218339003000889. Google Scholar

[13]

A. Marciniak-Czochra, Receptor-based models with hysteresis for pattern formation in Hydra, Math. Biosci., 199 (2006), 97-119. doi: 10.1016/j.mbs.2005.10.004. Google Scholar

[14]

A. Marciniak-Czochra, Strong two-scale convergence and corrector result for the receptor-based model of the intercellular communication, IMA J. Appl. Math., 77 (2012), 855-868. doi: 10.1093/imamat/hxs052. Google Scholar

[15]

A. Marciniak-CzochraG. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl., 99 (2013), 509-543. doi: 10.1016/j.matpur.2012.09.011. Google Scholar

[16]

A. Marciniak-CzochraG. Karch and K. Suzuki, Instability of turing patterns in reaction-diffusion-ODE systems, J. Math. Biol., (2016). doi: 10.1007/s00285-016-1035-z. Google Scholar

[17]

A. Marciniak-Czochra and M. Kimmel, Modelling of early lung cancer progression: Influence of growth factor production and cooperation between partially transformed cells, Math. Models Methods Appl. Sci., 17 (2007), 1693-1719. doi: 10.1142/S0218202507002443. Google Scholar

[18]

A. Marciniak-CzochraM. Nakayama and I. Takagi, Pattern formation in a diffusion-ODE model with hysteresis, Differential Integral Equations, 28 (2015), 655-694. Google Scholar

[19]

A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenisation techniques, SIAM J. Math. Anal., 40 (2008), 215-237. doi: 10.1137/050645269. Google Scholar

[20]

M. MimuraM. Tabata and Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal., 11 (1980), 613-631. doi: 10.1137/0511057. Google Scholar

[21]

J. D. Murray, Mathematical Biology. Ⅱ. Spatial Models and Biomedical Applications 3rd edition, Interdisciplinary Applied Mathematics, 18, 2003, Springer-Verlag, New York.Google Scholar

[22] W. A. Müller, Developmental Biology, Springer-Verlag, New York, 1997. Google Scholar
[23]

K. PhamA. ChauviereH. HatzikirouX. LiH.M. ByrneV. Cristini and J. Lowengrub, Density-dependent quiescence in glioma invasion: Instability in a simple reaction-diffusion model for the migration/proliferation dichotomy, J. Biol. Dyn., 6 (2012), 54-71. doi: 10.1080/17513758.2011.590610. Google Scholar

[24]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, 1984, Springer-Verlag, Berlin.Google Scholar

[25] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2 edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar
[26]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012. Google Scholar

[27]

D. M. UmulisM. SerpeM. B. O'Connor and H. G. Othmer, Robust, bistable patterning of the dorsal surface of the Drosophila embryo, PNAS, 103 (2006), 11613-11618. doi: 10.1073/pnas.0510398103. Google Scholar

[28]

H. F. Weinberger, A simple system with a continuum of stable inhomogeneous steady states, Nonlinear Partial Differential Equations in Applied Science; Proceedings of the U.S.-Japan Seminar, North-Holland Math. Stud., 81 (1983), 345-359. Google Scholar

[29] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Heidelberg/Dordrecht/London/New York, 2010. doi: 10.1007/978-3-642-04631-5. Google Scholar
Figure 2.1.  Illustration of the topology applied to problem (1.1) for scalar $u$. Model (1.1) exhibits steady states with jump discontinuity and global existence of classical solutions. $\tilde{u}$ represents a steady state while $u(t,x)$ represents a solution for some $t$
Figure 3.1.  Plot of the nullclines of $f_r(u,v)=-(1+v)u+m_1(u^2/(1+ku^2))$ for $u\neq 0$ and $g_r(u,v)=-(\mu_3+u)v+m_2 (u^2/(1+ku^2))$.
Figure 3.2.  Numerically obtained solution to model (3.3)-(3.5) for parameters $m_1 =1.44, m_2 = 2,\mu_3 \approx 4.1, k=0.01, D=1$. We observe convergence towards a steady state with jump discontinuity. Left: Non-diffusive component $u$. Right: Diffusive component $v$.
Figure 3.3.  Numerically obtained solution component $u$ of model (3.3)-(3.5) for parameters $m_1 =1.44, m_2 = 2,\mu_3 \approx 4.1, k=0.01$ with varying diffusion coefficient: upper left: $D=5$, upper right: $D=1$, lower left: $D=0.5$, lower right: $D=0.1$. We observe emergence of more jump-type discontinuities for smaller diffusion coefficient. Initial conditions are $u_0(x)=1.725-0.1\cos(2 \pi x^2), v_0(x)=2.48615$
Figure 3.4.  Numerically obtained solution component $u$ of model (3.3)-(3.5) for parameters $m_1 =1.44, m_2 = 2,\mu_3 \approx 4.1, k=0.01, D=5$. We observe emergence of jump-type discontinuities around local maxima of the initial conditions. Initial conditions are $u_0(x)=1.725-0.1x^4\cos(8 \pi x^2), v_0(x)=2.48615$
Figure 5.1.  Nullclines of $f_r$ and $g_r$ and $v_{f_r,g_r}$ for parameters $m_1 = 1.44, m_2 = 2, \mu_3 = 4.2, k=0.1$
Figure 5.2.  Illustration of the right-hand side of $-\partial^2 v/\partial x^2 = g_r(u,v)$ for different branches of the solution $u(v)$ of $u_t=f_r(u,v)=0$. The parameters for illustration are $D=1,m_1=1.44,m_2=2,\mu_3=4.2$. We can observe that all nontrivial homogeneous steady states are of type $u_-$
Figure 5.3.  Illustration of the construction of weak steady states in the proof of Lemma 3.6
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