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Twist grain boundaries in three-dimensional lamellar Turing structures - PubMed

  • ️Wed Jan 01 1997

Twist grain boundaries in three-dimensional lamellar Turing structures

De Wit A et al. Proc Natl Acad Sci U S A. 1997.

Abstract

Steady spatial self-organization of three-dimensional chemical reaction-diffusion systems is discussed with the emphasis put on the possible defects that may alter the Turing patterns. It is shown that one of the stable defects of a three-dimensional lamellar Turing structure is a twist grain boundary embedding a Scherk minimal surface.

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Figures

Figure 1
Figure 1

3D Turing bccπ pattern obtained by numerically integrating the reaction-diffusion Brusselator model: ∂X/∂t = A − (B + 1)X + X2Y + DX▿2X, ∂Y/∂t = BX − X2Y + DY▿2Y in a cube of size lx = ly = lz = 40 with periodic boundary conditions along the three axes. The Turing instability occurs when B = BcT = [1 + A(DX/DY)1/2]2. The parameters are A = 4.5, DX = 2, DY = 16, B = 6.9 (BcT = 6.71). Isoconcentration surfaces of the variable X are looked at along the [111] axis of the cube in which the equations are discretized. (A) The spheres of lower isoconcentrations (as an example here X = 2.737) are organized with the bcc symmetry. (B) The higher isoconcentrations (here X = 5.153) fill in the interspace between the lower isoconcentrations bcc.

Figure 2
Figure 2

Numerical bifurcation diagram for the amplitude of the concentration X of the Brusselator model with A = 4.5, DX = 2, DY = 16 in the system with size lx = ly = lz = 40. The amplitude is here defined as the difference between the maximum and minimum value of the variable X in the system.

Figure 3
Figure 3

Scherk surface obtained by numerically integrating the reaction-diffusion Brusselator model with A = 4.5, DX = 2, DY = 16, and B = 7.2 in a box of sizes lx = ly = 60, lz = 30. Periodic boundary conditions are applied along x and y while no-flux conditions are imposed in the z direction. To have a better visualization of the saddles region, typical of the Scherk surface, we zoom in on half the system. The representation is then lx = ly = lz = 30 with lz being on the vertical axis. The drawn isoconcentration surface corresponds to X = 4.5, the concentration of the uniform unstable reference state. The Scherk surface joins two orthogonal sets of three planes, one set being the x = constant planes while the other set is the y = constant planes.

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