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Multipotent neural crest (NC) progenitors generate an astonishing array of derivatives, including neuronal, skeletal components and pigment cells (chromatophores), but the molecular mechanisms allowing balanced selection of each fate remain unknown. In zebrafish, melanocytes, iridophores and xanthophores, the three chromatophore lineages, are thought to share progenitors and so lend themselves to investigating the complex gene regulatory networks (GRNs) underlying fate segregation of NC progenitors. Although the core GRN governing melanocyte specification has been previously established, those guiding iridophore and xanthophore development remain elusive. Here we focus on the iridophore GRN, where mutant phenotypes identify the transcription factors Sox10, Tfec and Mitfa and the receptor tyrosine kinase, Ltk, as key players. Here we present expression data, as well as loss and gain of function results, guiding the derivation of an initial iridophore specification GRN. Moreover, we use an iterative process of mathematical modelling, supplemented with a Monte Carlo screening algorithm suited to the qualitative nature of the experimental data, to allow for rigorous predictive exploration of the GRN dynamics. Predictions were experimentally evaluated and testable hypotheses were derived to construct an improved version of the GRN, which we showed produced outputs consistent with experimentally observed gene expression dynamics. Our study reveals multiple important regulatory features, notably a sox10-dependent positive feedback loop between tfec and ltk driving iridophore specification; the molecular basis of sox10 maintenance throughout iridophore development; and the cooperation between sox10 and tfec in driving expression of pnp4a, a key differentiation gene. We also assess a candidate repressor of mitfa, a melanocyte-specific target of sox10. Surprisingly, our data challenge the reported role of Foxd3, an established mitfa repressor, in iridophore regulation. Our study builds upon our previous systems biology approach, by incorporating physiologically-relevant parameter values and rigorous evaluation of parameter values within a qualitative data framework, to establish for the first time the core GRN guiding specification of the iridophore lineage.

Neural crest cells are highly multipotent stem cells, but it remains unclear how their fate restriction to specific fates occurs. The direct fate restriction model hypothesises that migrating cells maintain full multipotency, whilst progressive fate restriction envisages fully multipotent cells transitioning to partially-restricted intermediates before committing to individual fates. Using zebrafish pigment cell development as a model, we show applying NanoString hybridization single cell transcriptional profiling and RNAscope in situ hybridization that neural crest cells retain broad multipotency throughout migration and even in post-migratory cells in vivo, with no evidence for partially-restricted intermediates. We find that leukocyte tyrosine kinase early expression marks a multipotent stage, with signalling driving iridophore differentiation through repression of fate-specific transcription factors for other fates. We reconcile the direct and progressive fate restriction models by proposing that pigment cell development occurs directly, but dynamically, from a highly multipotent state, consistent with our recently-proposed Cyclical Fate Restriction model. Neural crest cells are highly multipotent stem cells, but it remains unclear how their fate restriction to specific fates occurs. Here, the authors show in zebrafish that broad multipotency is retained even after migration, suggesting that fate restriction occurs directly, but dynamically.

In this article, we study the evolution of open inextensible planar curves with hinged ends. We obtain long time existence of C ∞ -smooth solutions during the evolution, given the initial curves that are only C 2 -smooth with vanishing curvature at the boundary. Moreover, the asymptotic limits of this flow are inextensible elasticae. Our method and result extend the work by Wen (Duke Math. J. 70(3):683–698, 1993 ).

In this paper we present a method to untangle smooth knots by a gradient flow for a suitable energy. We show that the flow of smooth initial knots remains smooth for all time and approaches asymptotically an “optimal embedding” in its isotopy type. The method is to set up a gradient flow for the total energy of knots, which consists of bending energy and the Möbius energy of knots.

The aim of this article is to analyse travelling waves for a lattice model of phase transitions, specifically the Fermi–Pasta–Ulam chain with piecewise quadratic interaction potential. First, for fixed, sufficiently large subsonic wave speeds, we rigorously prove the existence of a family of travelling wave solutions. Second, it is shown that this family of solutions gives rise to a kinetic relation which depends on the jump in the oscillatory energy in the solution tails. Third, our constructive approach provides a very good approximate travelling wave solution.

Recently, rod theory has been applied to the mathematical modeling of bacterial fibers and biopolymers (e.g., DNA) to study their mechanical properties and shapes (e.g., supercoiling). In static rod theory, an elastic rod in equilibrium is the critical point of an elastic energy. This induces a natural question of how to find elasticae. In this paper, we focus on how to find the critical points by means of gradient flows. We relate a geometric function of curves to the isotropic Kirchhoff elastic energy of rods so that the generalized elastic curves are the centerlines of elastic rods in equilibrium. Thus, the variational problem for rods is formulated in curve geometry. This problem turns out to be a generalization of curve-straightening flows, which induce nonlinear fourth-order evolution equations. We establish the long time existence of length-preserving gradient flow for the geometric energy. Furthermore, by studying the asymptotic behavior, we show that the limit curves are the centerlines of the Kirchhoff elastic rods in equilibrium.

In this paper we study the problem on how to find an equilibrium state of a Kirchhoff elastic rod by evolving it in a certain way, called a geometric flow. The elastic energy of rods would decrease during the geometric flow. We show that rods remain smooth during the geometric flow as long as they stay embedded, e.g., self-penetrations do no occur. Furthermore, rods would approach an equilibrium configuration asymptotically if self-penetrations are avoided during the flow.

On a Riemann surface with smooth boundary, we consider Riemannian metrics conformal to a given background metric. Let κ be a smooth, positive function on . If K denotes the Gauss curvature, then the L ∞ -norm of K / κ gives rise to a functional on the space of all admissible metrics. We study minimizers subject to an area constraint. Under suitable conditions, we construct a minimizer with the property that | K |/ κ is constant. The sign of K can change, but this happens only on the nodal set of the solution of a linear partial differential equation.

We consider a sixth and an eighth order conformal flow on Riemannian surfaces, which arise as gradient flows for the Calabi energy with respect to a higher order metric. Motivated by a work of Struwe which unified the approach to the Hamilton-Ricci and Calabi flow, we extend the method to these higher order cases. Our results contain global existence and exponentially fast convergence to a constant scalar curvature metric. Uniform bounds on the conformal factor are obtained via the concentration-compactness result for conformal metrics. In the case of the sphere we use the idea of DeTurck's gauge flow to derive bounds up to conformal transformation. We prove exponential convergence by showing that the Calabi energy decreases exponentially fast. The problem of the non-trivial kernel in the evolution of the Calabi energy on the sphere is resolved by using Kazdan-Warner's identity. [PUBLICATION ABSTRACT]

The existence of travelling waves in an atomistic model for martensitic phase transitions is the focus of this study. The elastic energy is assumed to be piecewise quadratic, with two wells representing two stable phases. We develop a framework such that the existence of subsonic heteroclinic waves in a bi-infinite chain of atoms can be proved rigorously. The key is to represent the solution as a sum of a (here explicitly given) profile and a corrector in$L^2(\\mathbb{R})$ . It is demonstrated that the kinetic relation can be easily inferred from this framework.