Stability analysis and simulations of coupled bulk-surface reaction–diffusion systems

Abstract

In this article, we formulate new models for coupled systems of bulk-surface reaction–diffusion equations on stationary volumes. The bulk reaction–diffusion equations are coupled to the surface reaction–diffusion equations through linear Robin-type boundary conditions. We then state and prove the necessary conditions for diffusion-driven instability for the coupled system. Owing to the nature of the coupling between bulk and surface dynamics, we are able to decouple the stability analysis of the bulk and surface dynamics. Under a suitable choice of model parameter values, the bulk reaction–diffusion system can induce patterning on the surface independent of whether the surface reaction–diffusion system produces or not, patterning. On the other hand, the surface reaction–diffusion system cannot generate patterns everywhere in the bulk in the absence of patterning from the bulk reaction–diffusion system. For this case, patterns can be induced only in regions close to the surface membrane. Various numerical experiments are presented to support our theoretical findings. Our most revealing numerical result is that, Robin-type boundary conditions seem to introduce a boundary layer coupling the bulk and surface dynamics.

Analysis and Simulations of Coupled Bulk-surface Reaction-Diffusion Systems on Exponentially Evolving Volumes

“…and take m = 2 and m = 3 ( Figure 1) and vary the exponential growth rate k. With these parameter values, when there is no growth, k = 0, we recover the uniform steady state on stationary volumes given by [24] (u * , v * , r * , s…”

“…different diffusion coefficients in the bulk and on the surface to generate a wide range of patterns either concentrated in the bulk or on the surface (results not shown). We refer the interested reader to see [24] for such pattern diversity albeit on stationary volumes. Our second example shows how domain growth enhances patterning for the case where no patterning will occur in the absence of domain growth.…”

“…Example meshes for the bulk (top) and surface system (bottom). Part of the domain has been cut away and shown on the right to reveal some internal mesh structure (see [24] for details).…”

Dermal denticle patterning in the Cretaceous hybodont sharkTribodus limae(Euselachii, Hybodontiformes), and its implications for the evolution of patterning in the chondrichthyan dermal skeleton

“…However, since the mid-20th century, the fit of increasingly elaborate mathematical models of reaction-diffusion (R-D) chemical dynamics (e.g., Turing, 1952;Yang et al, 2002;Barrio et al, 2009;Badugu et al, 2012;Maini et al, 2012;Madzvamuse et al, 2015) to different biological case studies of cutaneous tissues (e.g., Kondo and Asai, 1995;Yamaguchi et al, 2007;Barrio et al, 2009;Kondo and Miura, 2010) suggests that such conceptual models may potentially approximate the actual dynamics of spatial patterning in cutaneous tissues generated by combinations of activator/inhibitor transcription factors (e.g., the WNT/DKK interactions in feather bud spatial patterning; Sick et al, 2006). As applied to dermal denticles, for example, such models are intuitively appealing because they operationalize the inhibitory field and odontode regulation theories of Reif (1980Reif ( , 1982, who-with remarkable foresight-postulated that denticle development occurred as the outcome of diffusion gradients of an inhibiting signal through the epidermis.…”

“…Reaction-diffusion models that involve coupling between layers (Yang et al, 2002;Barrio et al, 2009;Madzvamuse et al, 2015) approximate the interaction of morphogens through the basal lamina between the epidermal and dermal layers. Such coupled models can recover patterns of two-size spatial arrangements, regardless of both the kind of interactions taking place across layers and the assumed kinetics of the R-D system (Barrio 2008).…”