A “Numerical Evo-Devo” Synthesis for the Identification of Pattern-Forming Factors

Abstract

Animals display extensive diversity in motifs adorning their coat, yet these patterns have reproducible orientation and periodicity within species or groups. Morphological variation has been traditionally used to dissect the genetic basis of evolutionary change, while pattern conservation and stability in both mathematical and organismal models has served to identify core developmental events. Two patterning theories, namely instruction and self-organisation, emerged from this work. Combined, they provide an appealing explanation for how natural patterns form and evolve, but in vivo factors underlying these mechanisms remain elusive. By bridging developmental biology and mathematics, novel frameworks recently allowed breakthroughs in our understanding of pattern establishment, unveiling how patterning strategies combine in space and time, or the importance of tissue morphogenesis in generating positional information. Adding results from surveys of natural variation to these empirical-modelling dialogues improves model inference, analysis, and in vivo testing. In this evo-devo-numerical synthesis, mathematical models have to reproduce not only given stable patterns but also the dynamics of their emergence, and the extent of inter-species variation in these dynamics through minimal parameter change. This integrative approach can help in disentangling molecular, cellular and mechanical interaction during pattern establishment.

Fixed and Distributed Gene Expression Time Delays in Reaction-Diffusion Systems

“…Hence, the continuously distributed time delay model is a generalisation of the fixed time delay model. 1 The error function is given by erf(x) = 2 √ π…”

“…We remark that computing λ k for τ > 1.6 is numerically difficult; thus full numerical solutions were used to plot 'true' time-to-pattern against τ on a longer timescale, for both the LI model and GM models, to verify the linear relationship between the time delay and time to pattern. In Figure 7 we consider the time delay τ ∈ [1,16] at unit intervals for the LI model, with two different parameter sets of (a, b) and plot T, the time taken for a perturbation to grow up to a threshold value u T * = 0.1 from an initial perturbation of size σ IC = 10 −5 , which once more reveals a linearly increasing relationship between T and τ.…”

“…For example, the distribution of cells during the developmental process of an embryo, the growth of cancerous tissue [2], vertebrate limb development [30,18,29], and pattern formation on animal coats (e.g. spots on a jaguar [33], feathers on birds [1]). Wolpert [49] presented the idea that, underpinning the development of shape and form (morphogenesis) is a cell's ability to differentiate according to its position in space and time.…”

Symmetry Breaking as an Interdisciplinary Concept Unifying Cell and Developmental Biology

“…Connecting unicellular and multicellular scales, Yap and colleagues review epithelial cell extrusion, a form of collective behavior of cells within epithelial sheets that plays an important role in normal morphogenesis and development of cancer [ 29 ]. A review by Manceau, Bailleul, and Touboul provides an extensive overview of mechanisms and models of multicellular pattern formation [ 30 ].…”

A conserved paint box underlies color pattern diversity in Estrildid finches

“…Conversely, spontaneous spatial arrangement through tissue “selforganization” may reach stable states corresponding to complex morphologies (e.g. periodic coloration, teeth circumvolutions) that vary importantly through minimal parameter change (6,7). This inherent malleability provides an appealing explanation to morphological diversity, but does not ensure reproducibility.…”