Reaction-diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal-mesenchymal coupling in development, and symmetry-breaking in cell polarization. We develop a modeling framework for bilayer reaction-diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction-diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

Cardiff School of Mathematics Cardiff University Senghennydd Road Cardiff CF24 4AG UK

Department of Mathematics FNSPE Czech Technical University Prague Trojanova 13 120 00 Prague Czech Republic

Microsoft Research 21 Station Rd Cambridge CB1 2FB UK

Wolfson Centre for Mathematical Biology Mathematical Institute University of Oxford Andrew Wiles Building Radcliffe Observatory Quarter Woodstock Road Oxford OX2 6GG UK

Zobrazit více v PubMed

Anguige K, Röger M. Global existence for a bulk/surface model for active-transport-induced polarisation in biological cells. J Math Anal Appl. 2017;448(1):213–244.

Asllani M, Busiello DM, Carletti T, Fanelli D, Planchon G. Turing patterns in multiplex networks. Phys Rev E. 2014;90(4):042814. PubMed

Asllani M, Challenger JD, Pavone FS, Sacconi L, Fanelli D. The theory of pattern formation on directed networks. Nat Commun. 2014;5:4517. PubMed

Balagaddé FK, Song H, Ozaki J, Collins CH, Barnet M, Arnold FH, Quake SR, You L. A synthetic Escherichia coli predator-prey ecosystem. Mol Syst Biol. 2008;4(1):187. PubMed PMC

Basu S, Gerchman Y, Collins CH, Arnold FH, Weiss R. A synthetic multicellular system for programmed pattern formation. Nature. 2005;434(7037):1130–1134. PubMed

Benson DL, Maini PK, Sherratt JA. Unravelling the turing bifurcation using spatially varying diffusion coefficients. J Math Biol. 1998;37(5):381–417.

Berding C. On the heterogeneity of reaction-diffusion generated pattern. Bull Math Biol. 1987;49(2):233–252. PubMed

Boehm CR, Grant PK, Haseloff J. Programmed hierarchical patterning of bacterial populations. Nat Commun. 2018;9:776. PubMed PMC

Brauns F, Pawlik G, Halatek J, Kerssemakers J, Frey E, Dekker C (2020) Bulk-surface coupling reconciles Min-protein pattern formation in vitro and in vivo. bioRxiv, page 2020.03.01.971952

Budrene EO, Berg HC. Complex patterns formed by motile cells of Escherichia coli. Nature. 1991;349(6310):630. PubMed

Budrene EO, Berg HC. Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature. 1995;376(6535):49. PubMed

Cao Y, Feng Y, Ryser MD, Zhu K, Herschlag G, Cao C, Marusak K, Zauscher S, You L. Programmable assembly of pressure sensors using pattern-forming bacteria. Nat Biotechnol. 2017;35(11):1087–1093. PubMed PMC

Cao Y, Ryser MD, Payne S, Li B, Rao CV, You L. Collective space-sensing coordinates pattern scaling in engineered bacteria. Cell. 2016;165(3):620–630. PubMed PMC

Catllá AJ, McNamara A, Topaz CM. Instabilities and patterns in coupled reaction-diffusion layers. Phys Rev E. 2012;85(2):026215. PubMed

Chaplain MAJ, Ganesh M, Graham IG. Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. J Math Biol. 2001;42(5):387–423. PubMed

Chapman SJ, Erban R, Isaacson SA. Reactive boundary conditions as limits of interaction potentials for Brownian and Langevin dynamics. SIAM J Appl Math. 2016;76(1):368–390.

COMSOL Multiphysics

Conte SD, De Boor C. Elementary numerical analysis: an algorithmic approach. Philadelphia: SIAM; 2017.

Crampin EJ, Gaffney EA, Maini PK. Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull Math Biol. 1999;61(6):1093–1120. PubMed

Cross MC, Hohenberg PC. Pattern formation outside of equilibrium. Rev Mod Phys. 1993;65(3):851.

Cruywagen GC, Murray JD. On a tissue interaction model for skin pattern formation. J Nonlinear Sci. 1992;2(2):217–240. PubMed

Cusseddu D, Edelstein-Keshet L, Mackenzie JA, Portet S, Madzvamuse A. A coupled bulk-surface model for cell polarisation. J Theor Biol. 2018;481:119–135. PubMed

Dalchau N, Smith MJ, Martin S, Brown JR, Emmott S, Phillips A. Towards the rational design of synthetic cells with prescribed population dynamics. J R Soc Interface. 2012;9(76):2883–2898. PubMed PMC

De Kepper P, Castets V, Dulos E, Boissonade J. Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction. Physica D. 1991;49(1–2):161–169.

Denk J, Kretschmer S, Halatek J, Hartl C, Schwille P, Frey E. MinE conformational switching confers robustness on self-organized Min protein patterns. Proc Nat Acad Sci. 2018;115(18):4553–4558. PubMed PMC

Dewel G, Borckmans P, De Wit A, Rudovics B, Perraud J-J, Dulos E, Boissonade J, De Kepper P. Pattern selection and localized structures in reaction-diffusion systems. Physica A. 1995;213(1–2):181–198.

Epstein IR, Berenstein IB, Dolnik M, Vanag VK, Yang L, Zhabotinsky AM. Coupled and forced patterns in reaction-diffusion systems. Philos Trans R Soc A Math Phys Eng Sci. 2007;366(1864):397–408. PubMed

Frey E, Halatek J, Kretschmer S, Schwille P. Protein pattern formation. In: Bassereau P, Sens P, editors. Physics of biological membranes. Cham: Springer; 2018. pp. 229–260.

Fujita H, Kawaguchi M. Pattern formation by two-layer turing system with complementarysynthesis. J Theor Biol. 2013;322:33–45. PubMed

Fussell EF, Krause AL, Van Gorder RA. Hybrid approach to modeling spatial dynamics of systems with generalist predators. J Theor Biol. 2019;462:26–47. PubMed

Geßele R, Halatek J, Würthner L, Frey E. Geometric cues stabilise long-axis polarisation of PAR protein patterns in C. elegans. Nat Commun. 2020;11(1):1–12. PubMed PMC

Glock P, Brauns F, Halatek J, Frey E, Schwille P. Design of biochemical pattern forming systems from minimal motifs. eLife. 2019;8:e48646. PubMed PMC

Glover JD, Wells KL, Matthäus F, Painter KJ, Ho W, Riddell J, Johansson JA, Ford MJ, Jahoda CAB, Klika V, et al. Hierarchical patterning modes orchestrate hair follicle morphogenesis. PLoS Biol. 2017;15(7):e2002117. PubMed PMC

D. Gomez, M. J. Ward, and J. Wei. The linear stability of symmetric spike patterns for a bulk-membrane coupled Gierer-Meinhardt model. arXiv:1810.09588 (2018)

Gomez S, Diaz-Guilera A, Gomez-Gardenes J, Perez-Vicente CJ, Moreno Y, Arenas A. Diffusion dynamics on multiplex networks. Phys Rev Lett. 2013;110(2):028701. PubMed

Grant PK, Dalchau N, Brown JR, Federici F, Rudge TJ, Yordanov B, Patange O, Phillips A, Haseloff J. Orthogonal intercellular signaling for programmed spatial behavior. Mol Syst Biol. 2016;12(1):849. PubMed PMC

Green JBA, Sharpe J. Positional information and reaction-diffusion: two big ideas in developmental biology combine. Development. 2015;142(7):1203–1211. PubMed

Gurtin ME, Fried E, Anand L. The mechanics and thermodynamics of continua. Cambridge: Cambridge University Press; 2013.

Haim L, Hagberg A, Meron E. Non-monotonic resonance in a spatially forced Lengyel-Epstein model. Chaos Interdiscip J Nonlinear Sci. 2015;25(6):064307. PubMed

Halatek J, Brauns F, Frey E. Self-organization principles of intracellular pattern formation. Philos Trans R Soc B Biol Sci. 2018;373(1747):20170107. PubMed PMC

Halatek J, Frey E. Highly canalized MinD transfer and MinE sequestration explain the origin of robust MinCDE-protein dynamics. Cell Rep. 2012;1(6):741–752. PubMed

Halatek J, Frey E. Rethinking pattern formation in reaction-diffusion systems. Nat Phys. 2018;14(5):507–514.

Hausberg S, Röger M. Well-posedness and fast-diffusion limit for a bulk-surface reaction-diffusion system. Nonlinear Differ Equ Appl. 2018;25:1–32.

Higham NJ. Functions of matrices: theory and computation. Philadelphia: SIAM; 2008.

Ide Y, Izuhara H, Machida T. Turing instability in reaction-diffusion models on complex networks. Physica A. 2016;457:331–347.

Karig D, Martini KM, Lu T, DeLateur NA, Goldenfeld N, Weiss R. Stochastic turing patterns in a synthetic bacterial population. Proc Nat Acad Sci. 2018;115(26):6572–6577. PubMed PMC

Klika V, Baker RE, Headon D, Gaffney EA. The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organisation. Bull Math Biol. 2012;74(4):935–957. PubMed

Klika V, Kozák M, Gaffney EA. Domain size driven instability: self-organization in systems with advection. SIAM J Appl Math. 2018;78(5):2298–2322.

Klünder B, Freisinger T, Wedlich-Söldner R, Frey E. GDI-mediated cell polarization in yeast provides precise spatial and temporal control of Cdc42 signaling. PLoS Comput Biol. 2013;9(12):e1003396. PubMed PMC

Kolokolnikov T, Wei J. Pattern formation in a reaction-diffusion system with space-dependent feed rate. SIAM Rev. 2018;60(3):626–645.

Kondo S, Miura T. Reaction-diffusion model as a framework for understanding biological pattern formation. Science. 2010;329(5999):1616–1620. PubMed

Korvasova K, Gaffney EA, Maini PK, Ferreira MA, Klika V. Investigating the Turing conditions for diffusion-driven instability in the presence of a binding immobile substrate. J Theor Biol. 2015;367:286–295. PubMed

Kouvaris NE, Hata S, Díaz-Guilera A. Pattern formation in multiplex networks. Sci Rep. 2015;5(1):1–9. PubMed PMC

Kozák M, Gaffney EA, Klika V. Pattern formation in reaction-diffusion systems with piece-wise kinetic modulation: an example study of heterogeneous kinetics. Phys Rev E. 2019;100(4):042220. PubMed

Krause AL, Ellis MA, Van Gorder RA. Influence of curvature, growth, and anisotropy on the evolution of turing patterns on growing manifolds. Bull Math Biol. 2019;81(3):759–799. PubMed PMC

Krause AL, Klika V, Woolley TE, Gaffney EA. From one pattern into another: analysis of turing patterns in heterogeneous domains via WKBJ. J R Soc Interface. 2020;17:20190621. PubMed PMC

Kretschmer S, Schwille P. Pattern formation on membranes and its role in bacterial cell division. Curr Opin Cell Biol. 2016;38:52–59. PubMed

Levin PA, Angert ER. Small but mighty: cell size and bacteria. Cold Spring Harbour Perspect Biol. 2015;7(7):a019216. PubMed PMC

Levine H, Rappel W-J. Membrane-bound turing patterns. Phys Rev E. 2005;72(6):061912. PubMed

Loose M, Fischer-Friedrich E, Ries J, Kruse K, Schwille P. Spatial regulators for bacterial cell division self-organize into surface waves in vitro. Science. 2008;320(5877):789–792. PubMed

Macfarlane FR, Chaplain MA, Lorenzi T (2020) A hybrid discrete-continuum approach to model turing pattern formation. arXiv:2007.04195 PubMed

Madzvamuse A, Chung AH, Venkataraman C. Stability analysis and simulations of coupled bulk-surface reaction-diffusion systems. Proc R Soc A. 2015;471(2175):20140546. PubMed PMC

Madzvamuse A, Gaffney EA, Maini PK. Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains. J Math Biol. 2010;61(1):133–164. PubMed

Maini PK, Woolley TE, Baker RE, Gaffney EA, Lee SS. Turing’s model for biological pattern formation and the robustness problem. Interface focus. 2012;2(4):487–496. PubMed PMC

Mou C, Jackson B, Schneider P, Overbeek PA, Headon DJ. Generation of the primary hair follicle pattern. Proc Nat Acad Sci. 2006;103(24):9075–9080. PubMed PMC

Muller DE. A method for solving algebraic equations using an automatic computer. Math Tables Other Aids Comput. 1956;10(56):208–215.

Murray JD. Mathematical biology II: spatial models and biomedical applications. 3. Berlin: Springer; 2003.

Nakao H, Mikhailov AS. Turing patterns in network-organized activator-inhibitor systems. Nat Phys. 2010;6(7):544.

Nauman J, Campbell P, Lanni F, Anderson J. Diffusion of insulin-like growth factor-i and ribonuclease through fibrin gels. Biophys J . 2007;92(12):4444–50. PubMed PMC

Othmer HG, Scriven L. Instability and dynamic pattern in cellular networks. J Theor Biol. 1971;32(3):507–537. PubMed

Page K, Maini PK, Monk NA. Pattern formation in spatially heterogeneous turing reaction-diffusion models. Physica D. 2003;181(1–2):80–101.

Page KM, Maini PK, Monk NAM. Complex pattern formation in reaction-diffusion systems with spatially varying parameters. Physica D. 2005;202(1–2):95–115.

Paquin-Lefebvre F, Nagata W, Ward MJ (2018) Pattern formation and oscillatory dynamics in a 2-d coupled bulk-surface reaction-diffusion system. arXiv:1810.00251

Payne S, Li B, Cao Y, Schaeffer D, Ryser MD, You L. Temporal control of self-organized pattern formation without morphogen gradients in bacteria. Mol Syst Biol. 2013;9(1):697. PubMed PMC

Plaza RG, Sanchez-Garduno F, Padilla P, Barrio RA, Maini PK. The effect of growth and curvature on pattern formation. J Dyn Diff Equat. 2004;16(4):1093–1121.

Rätz A. Turing-type instabilities in bulk-surface reaction-diffusion systems. J Comput Appl Math. 2015;289:142–152.

Rätz A, Röger M. Symmetry breaking in a bulk-surface reaction-diffusion model for signalling networks. Nonlinearity. 2014;27(8):1805.

Sánchez-Garduño F, Krause AL, Castillo JA, Padilla P. Turing-Hopf patterns on growing domains: the torus and the sphere. J Theor Biol. 2019;481:136–150. PubMed

Sekine R, Shibata T, Ebisuya M. Synthetic mammalian pattern formation driven by differential diffusivity of nodal and lefty. Nat Commun. 2018;9(1):1–11. PubMed PMC

Shaw LJ, Murray JD. Analysis of a model for complex skin patterns. SIAM J Appl Math. 1990;50(2):628–648.

Spill F, Andasari V, Mak M, Kamm RD, Zaman MH. Effects of 3d geometries on cellular gradient sensing and polarization. Phys Biol. 2016;13(3):036008. PubMed PMC

Tabor JJ, Salis HM, Simpson ZB, Chevalier AA, Levskaya A, Marcotte EM, Voigt CA, Ellington AD. A synthetic genetic edge detection program. Cell. 2009;137(7):1272–1281. PubMed PMC

Thalmeier D, Halatek J, Frey E. Geometry-induced protein pattern formation. Proc Nat Acad Sci. 2016;113(3):548–553. PubMed PMC

Turing AM. The chemical basis of morphogenesis. Philos Trans R Soc Lond Ser B Biol Sci. 1952;237(641):37–72.

Tyson R, Lubkin S, Murray JD. A minimal mechanism for bacterial pattern formation. Proc R Soc Lond Ser B Biol Sci. 1999;266(1416):299–304. PubMed PMC

Van Gorder RA, Klika V, Krause AL (2019) Turing conditions for pattern forming systems on evolving manifolds. arXiv:1904.09683 [nlin.PS] PubMed

Varea C, Aragon JL, Barrio RA. Turing patterns on a sphere. Phys Rev E. 1999;60(4):4588. PubMed

Verschelde J. Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans Math Softw (TOMS) 1999;25(2):251–276.

Vilaca LM, Milinkovitch MC, Ruiz-Baier R. Numerical approximation of a 3d mechanochemical interface model for skin patterning. J Comput Phys. 2019;384:383–404.

Woolley T. Visions of mathematics, chapter 48: mighty morphogenesis. Oxford: Oxford University Press; 2014.

Wu F, Halatek J, Reiter M, Kingma E, Frey E, Dekker C. Multistability and dynamic transitions of intracellular Min protein patterns. Mol Syst Biol. 2016;12(6):873. PubMed PMC

Yang L, Dolnik M, Zhabotinsky AM, Epstein IR. Spatial resonances and superposition patterns in a reaction-diffusion model with interacting turing modes. Phys Rev Lett. 2002;88(20):208303. PubMed

Yang L, Epstein IR. Oscillatory turing patterns in reaction-diffusion systems with two coupled layers. Phys Rev Lett. 2003;90(17):178303. PubMed

Modern perspectives on near-equilibrium analysis of Turing systems

Isolating Patterns in Open Reaction-Diffusion Systems