The Turing mechanism in vertebrate limb patterning

Multiscale modeling of vertebrate limb development

We review the current state of mathematical modeling of cartilage pattern formation in vertebrate limbs. We place emphasis on several reaction-diffusion type models that have been proposed in the last few years. These models are grounded in more detailed knowledge of the relevant regulatory processes than previous ones but generally refer to different molecular aspects of these processes. Considering these models in light of comparative phylogenomics permits framing of hypotheses on the evolutionary order of appearance of the respective mechanisms and their roles in the fin-to-limb transition. This article is categorized under: Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Mechanistic Models Developmental Biology > Developmental Processes in Health and Disease Analytical and Computational Methods > Analytical Methods.

Importance of the Voronoi domain partition for position-jump reaction-diffusion processes on nonuniform rectilinear lattices

Position-jump processes are used for the mathematical modeling of spatially extended chemical and biological systems with increasing frequency. A large subset of the literature concerning such processes is concerned with modeling the effect of stochasticity on reaction-diffusion systems. Traditionally, computational domains have been divided into regular voxels. Molecules are assumed well mixed within each of these voxels and are allowed to react with other molecules within the same voxel or to jump to neighboring voxels with predefined transition rates. For a variety of reasons implementing position-jump processes on irregular grids is becoming increasingly important. However, it is not immediately clear what form an appropriate irregular partition of the domain should take if it is to allow the derivation of mean molecular concentrations that agree with a given partial differential equation for molecular concentrations. It has been demonstrated, in one dimension, that the Voronoi domain partition is the appropriate method with which to divide the computational domain. In this Brief Report, we investigate theoretically the propriety of the Voronoi domain partition as an appropriate method to partition domains for position-jump models in higher dimensions. We also provide simulations of diffusion processes in two dimensions in order to corroborate our results.

Dynamical patterning modules: physico-genetic determinants of morphological development and evolution

The shapes and forms of multicellular organisms arise by the generation of new cell states and types and changes in the numbers and rearrangements of the various kinds of cells. While morphogenesis and pattern formation in all animal species are widely recognized to be mediated by the gene products of an evolutionarily conserved 'developmental-genetic toolkit', the link between these molecular players and the physics underlying these processes has been generally ignored. This paper introduces the concept of 'dynamical patterning modules' (DPMs), units consisting of one or more products of the 'toolkit' genes that mobilize physical processes characteristic of chemically and mechanically excitable meso- to macroscopic systems such as cell aggregates: cohesion, viscoelasticity, diffusion, spatiotemporal heterogeneity based on lateral inhibition and multistable and oscillatory dynamics. We suggest that ancient toolkit gene products, most predating the emergence of multicellularity, assumed novel morphogenetic functions due to change in the scale and context inherent to multicellularity. We show that DPMs, acting individually and in concert with each other, constitute a 'pattern language' capable of generating all metazoan body plans and organ forms. The physical dimension of developmental causation implies that multicellular forms during the explosive radiation of animal body plans in the middle Cambrian, approximately 530 million years ago, could have explored an extensive morphospace without concomitant genotypic change or selection for adaptation. The morphologically plastic body plans and organ forms generated by DPMs, and their ontogenetic trajectories, would subsequently have been stabilized and consolidated by natural selection and genetic drift. This perspective also solves the apparent 'molecular homology-analogy paradox', whereby widely divergent modern animal types utilize the same molecular toolkit during development by proposing, in contrast to the Neo-Darwinian principle, that phenotypic disparity early in evolution occurred in advance of, rather than closely tracked, genotypic change.