The term morphogen is used rigorously to describe a particular type of signal- ing molecule that acts on cells directly to induce distinct cellular responses in a concentration dependent manner [109]. In nature, various pattern formations occur due to the variation of concentration of morphogens within each cell, which, both react and diffuse within the system owing to instabilities. A va- riety of species exhibits remarkable types of patterns in their skins, shapes or sizes. Patterns imitating the spots in leopards, spikes in cactus, stripes in ze- bras etc. has created tremendous interest amongst researchers and biologists.
These pattern formation mechanisms are actually governed by simple mathe- matical models of reaction and diffusion. The signaling molecules involved in the chemical processes spread away from their source to form concentration gradients.
132Higher-order compact simulation of pattern formation in Mathematical Biology
Reaction and diffusion are the two fundamental phenomena which govern the mechanism of morphogenesis. Within a system, reaction and diffusion occur between morphogens resulting in various patterns. Due to instabilities within the system, morphogens react and diffuse to change their cell concentrations which eventually give rise to different pattern formations.
Study on pattern formation dates to 1952, when British mathematician, Alan Turing, in his pioneering work [113] described a reaction-diffusion model for morphogen concentrations. This model hypothesizes the existence of two dif- ferent molecules, an activator and an inhibitor. Turing’s reaction-diffusion model is of the form
∂U
∂t = Du∇2U+F(U, V) (6.1)
∂V
∂t = Dv∇2V +G(U, V).
where U (activator) and V (inhibitor) are the concentrations of the reacting chemicals, Du and Dv are the diffusion coefficients and F and G are some functions of U and V (generally non-linear).
In his work, Turing showed that the activator and the inhibitor can give rise to spatial patterns starting from a nearly homogenous state. Application of the Turing mechanism to the formation of a given pattern is mainly identified by the specific dynamics of the reaction-diffusion system. This work was followed by numerous researchers which include Newman and Frisch (1979) [81], Bard (1981) [12], Jung et al. (1998) [54] etc. In 1993, an alternative mechanism for the spatial patterning was proposed by Murray et al. [79]. This mecha- nism follows a mechanochemical approach where it is assumed that patterns arise due to the physical interaction of cells with their external environment resulting in cell aggregations. With the advent of these models, there has been an immense growth of research in the field of pattern formation. Researchers have continued to carry out their studies on both of these model problems and it has been revealed that these models can simulate many exciting patterns that are observed in nature.
6.1. Introduction 133
Pattern formation, apart from being studied by researchers to propose numer- ical methods for simulating those structures and predicting new ones, is also of great importance to science and industry. It is useful in studying pattern formation and regulation in Hydra and other biological patterning processes which include phyllotaxis, veins, spatial ecology, segmentation and colour- ful pigmentation in sea-shells [75]. Apart from biology, pattern formation is also applicable to microstructure engineering, sputtering and lithography [85].
Spontaneous pattern formation from reaction-diffusion systems reduces certain complications like diffraction and splashing from masks and therefore can be used to obtain higher accuracy [39]. Moreover, sensitiveness of pattern forma- tions to initial conditions and various environmental factors can be utilized in sensing application [39] and in cellular engineering [102]. These features can further be utilized to devices like cameras and in detection of antioxidant [39].
Following the works of Turing, several other reaction-diffusion models have been proposed, notable ones include the Gray-Scott model [35, 36] and the Gierer-Meinhardt model [64]. In recent years, there has been a rapid rise in the field of Mathematical Biology and quite prominent ones include the study of pattern formation. Amongst the recent works worth mentioning are: Turk [114], Pearson [87], Aragon et al. [8], Asai et al. [9], Barrio et al. [13], Fleury [28], Liu et al. [72], Wu et al. [121] and Zhang et al. [128]. Because of non-linearity, these models pose considerable difficulty in direct computation and as such researchers have been driven to adopt novel numerical strategies from time to time. In this chapter, we use the second order time accurate HOC scheme developed in chapter 5 by modifying it so as to fit the equations of the form (6.1). In the process, we carry out the simulation of various pat- terns, viz. spikes, spots, stripes and labyrinths for varying range of parameters.
The governing unsteady reaction-diffusion model in its two-dimensional (2D) form may be written as
∂φ
∂t −ε∇2φ+r(x, y, t)φ(x, y, t) =f(x, y, t) (6.2) where φ(x, y, t) is a transport variable in some domain Ω ⊆ R2 with bound-
134Higher-order compact simulation of pattern formation in Mathematical Biology
ary ∂Ω, ε > 0 is the diffusion coefficient, r(x, y, t) is the reaction coefficient, f(x, y, t) is the forcing function and ∇2 is the Laplacian operator.
Considering the HOC formulation (5.8) described in chapter 5, we can approx- imate Equation (6.2) as follows
1 +
z }| { h2
12εδx2+ k2 12εδy2
δt+φnij −Aijδx2φnij −Bijδy2φnij (6.3)
− (h2+k2)
12 εδx2δy2φnij +Gijφnij =Fijn
where the coefficients Aij, Bij, Gij and Fij are now as follows:
Aij = ε− h2 12εrijk
Bij = ε− k2 12εrijk Fij =
1 + h2
12εδx2+ k2 12εδy2
fij
Gij =
1 + h2
12εδx2+ k2 12εδy2
rij (6.4)
δ+t is the forward difference operator and the superscriptn stands for then-th time level.
The process of basic discretization and matrix computation follows a similar approach as was adopted in chapter 5 for the computation of 3D flows.