Hartmut Schwetlick

Existence of Dynamic Phase Transitions in a One-Dimensional Lattice Model with Piecewise Quadratic Interaction Potential

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

Calculation of long time classical trajectories: algorithmic treatment and applications for molecular systems

L^2(\mathbb{R})

On the geometric flow of kirchhoff elastic rods

. It is demonstrated that the kinetic relation can be easily inferred from this framework.

Subsonic Phase Transition Waves in Bistable Lattice Models with Small Spinodal Region

We study the problem of finding a path that joins a given initial state with a final one, where the evolution is governed by classical (Hamiltonian) dynamics. A new algorithm for the computation of long time transition trajectories connecting two configurations is presented. In particular, a strategy for finding transition paths between two stable basins is established. The starting point is the formulation of the equation of motion of classical mechanics in the framework of Jacobis principle; a shortening procedure inspired by Birkhoffs method is then applied to find geodesic solutions. Numerical examples are given for Mullers potential and the collinear reaction H(2) + H --> H + H(2).

On selection criteria for problems with moving inhomogeneities

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 Ki...

The Computation of Long Time Hamiltonian Trajectories for Molecular Systems via Global Geodesics

Although phase transition waves in atomic chains with double-well potential play a fundamental role in materials science, very little is known about their mathematical properties. In particular, the only available results about waves with large amplitudes concern chains with piecewise-quadratic pair potential. In this paper we consider perturbations of a bi-quadratic potential and prove that the corresponding three-parameter family of waves persists as long as the perturbation is small and localized with respect to the strain variable. As a standard Lyapunov--Schmidt reduction cannot be used due to the presence of an essential spectrum, we characterize the perturbation of the wave as a fixed point of a nonlinear and nonlocal operator and show that this operator is contractive on a small ball in a suitable function space. Moreover, we derive a uniqueness result for phase transition waves with certain properties and discuss the kinetic relations.

A systems biology approach uncovers the core gene regulatory network governing iridophore fate choice from the neural crest

We study mechanical problems with multiple solutions and introduce a thermodynamic framework to formulate two different selection criteria in terms of macroscopic energy productions and fluxes. Simple examples for lattice motion are then studied to compare the implications for both resting and moving inhomogeneities.

Finding the elasticae by means of geometric gradient flows

A string method for the computation of Hamiltonian trajectories linking two given points is presented, based on the Maupertuis principle; trajectories then correspond to geodesics. For local geodesics, convergence of an algorithm based on Birkhoff’s method has been shown recently in Schwetlick and Zimmer (Submitted). We demonstrate how to extend this approach to global geodesics and thus arbitrary boundary values of the corresponding Hamiltonian problem. Numerical illustrations of the algorithm are given, as well as situations are shown in which the method converges to a degenerate solution.

Existence of travelling fronts for Nonlinear transport equations

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.

Convergence of the Yamabe flow for ``large'' energies

This article is an announcement of our recent result on finding the critical points of a Kirchhoff elastic energy, or the so called elasticae, by means of geometric gradient flows. The reader can find out the details of the proof in [8]. In order to keep the model problem simple, we only consider a special isotropic Kirchhoff elastic energy in the following. Suppose f : I = R/Z —> M is the centerline of a closed rod. Let 7 = \dxf\, ds = 7 dx the arclength element, and ds = j~ dx the arclength differentiation. Denote by T = dsf the unit tangent vector, and K = d^f the curvature vector of /. A rod configuration F is a framed curve described by {/ (s); T ( s ) , MI (s), MZ (s)}, where the material frame {T, Mi,M2} forms an orthonormal frame field along /. Thus, we can write the skew-symmetric system