During gastrulation, the pluripotent epiblast self-organizes into the 3 germ layers—endoderm, mesoderm and ectoderm, which eventually form the entire embryo. Decades of research in the mouse embryo have revealed that a signaling cascade involving the Bone Morphogenic Protein (BMP), WNT, and NODAL pathways is necessary for gastrulation. In vivo, WNT and NODAL ligands are expressed near the site of gastrulation in the posterior of the embryo, and knockout of these ligands leads to a failure to gastrulate. These data have led to the prevailing view that a signaling gradient in WNT and NODAL underlies patterning during gastrulation; however, the activities of these pathways in space and time have never been directly observed. In this study, we quantify BMP, WNT, and NODAL signaling dynamics in an in vitro model of human gastrulation. Our data suggest that BMP signaling initiates waves of WNT and NODAL signaling activity that move toward the colony center at a constant rate. Using a simple mathematical model, we show that this wave-like behavior is inconsistent with a reaction-diffusion–based Turing system, indicating that there is no stable signaling gradient of WNT/NODAL. Instead, the final signaling state is homogeneous, and spatial differences arise only from boundary effects. We further show that the durations of WNT and NODAL signaling control mesoderm differentiation, while the duration of BMP signaling controls differentiation of CDX2-positive extra-embryonic cells. The identity of these extra-embryonic cells has been controversial, and we use RNA sequencing (RNA-seq) to obtain their transcriptomes and show that they closely resemble human trophoblast cells in vivo. The domain of BMP signaling is identical to the domain of differentiation of these trophoblast-like cells; however, neither WNT nor NODAL forms a spatial pattern that maps directly to the mesodermal region, suggesting that mesoderm differentiation is controlled dynamically by the combinatorial effect of multiple signals. We synthesize our data into a mathematical model that accurately recapitulates signaling dynamics and predicts cell fate patterning upon chemical and physical perturbations. Taken together, our study shows that the dynamics of signaling events in the BMP, WNT, and NODAL cascade in the absence of a stable signaling gradient control fate patterning of human gastruloids.

We study patterns that arise in the wake of an externally triggered, spatially propagating instability in the complex Ginzburg-Landau equation. We model the trigger by a spatial inhomogeneity moving with constant speed. In the comoving frame, the trivial state is unstable to the left of the trigger and stable to the right. At the trigger location, spatio-temporally periodic wavetrains nucleate. Our results show existence of coherent, "heteroclinic" profiles when the speed of the trigger is slightly below the speed of a free front in the unstable medium. Our results also give expansions for the wavenumber of wavetrains selected by these coherent fronts. A numerical comparison yields very good agreement with observations, even for moderate trigger speeds. Technically, our results provide a heteroclinic bifurcation study involving an equilibrium with an algebraically double pair of complex eigenvalues. We use geometric desingularization and invariant foliations to describe the unfolding. Leading order terms are determined by a condition of oscillations in a projectivized flow, which can be found by intersecting absolute spectra with the imaginary axis

We study invasion fronts in a class of simple, two-species reaction-diffusion systems that occur as models for recurrent precipitation and undercooled liquids. We exhibit several different modes of front propagation: the invasion of an unstable homogeneous equilibrium can create persistent periodic patterns, transient patterns, or simply a homogeneous state. We give criteria that distinguish between these different modes of invasion, corroborate our predictions with numerical simulations, and point to a rich variety of more subtle phenomena and bifurcations.Running head: Wavenumber selection in precipitation Corresponding author: Arnd Scheel

During gastrulation, the pluripotent epiblast is patterned into the three germ layers, which form the embryo proper. This patterning requires a signaling cascade involving the BMP, WNT and NODAL pathways; however, how these pathways regulate one another in space and time to generate cell-fate patterns remains unknown. Using a human gastruloid model, we show that BMP signaling initiates a wave of WNT signaling, which, in turn, initiates a wave of NODAL signaling. While WNT propagation depends on continuous BMP activity, NODAL propagates independently of upstream signals. We further show that the duration of BMP signaling determines the position of mesodermal differentiation while WNT and NODAL synergize to achieve maximal differentiation. The waves of both WNT and NODAL signaling activity extend farther into the colony than mesodermal differentiation. Combining dynamic measurements of signaling activity with mathematical modeling revealed that the formation of signaling waves is inconsistent with WNT and NODAL forming a stable spatial pattern in signaling activities, and the final signaling state is spatially homogeneous. Thus, dynamic events in the BMP, WNT, and NODAL signaling cascade, in the absence of a signaling gradient, have the potential to mediate epiblast patterning.

We study the effect of domain growth on the orientation of striped phases in a Swift–Hohenberg equation. Domain growth is encoded in a step‐like parameter dependence that allows stripe formation in a half plane, and suppresses patterns in the complement, while the boundary of the pattern‐forming region is propagating with fixed normal velocity. We construct front solutions that leave behind stripes in the pattern‐forming region that are parallel to or at a small oblique angle to the boundary. Technically, the construction of stripe formation parallel to the boundary relies on ill‐posed, infinite‐dimensional spatial dynamics. Stripes forming at a small oblique angle are constructed using a functional‐analytic, perturbative approach. Here, the main difficulties are the presence of continuous spectrum and the fact that small oblique angles appear as a singular perturbation in a traveling‐wave problem. We resolve the former difficulty using a farfield‐core decomposition and Fredholm theory in weighted spaces. The singular perturbation problem is resolved using preconditioners and boot‐strapping.

We study pattern-forming dissipative systems in growing domains. We characterize classes of boundary conditions that allow for defect-free growth and derive universal scaling laws for the wavenumber in the bulk of the domain. Scalings are based on a description of striped patterns in semi-bounded domains via strain-displacement relations. We compare predictions with direct simulations in the Swift-Hohenberg, the Complex Ginzburg-Landau, the Cahn-Hilliard, and reactiondiffusion equations.

We present results on stripe formation in the Swift-Hohenberg equation with a directional quenching term. Stripes are "grown" in the wake of a moving parameter step line, and we analyze how the orientation of stripes changes depending on the speed of the quenching line and on a lateral aspect ratio. We observe stripes perpendicular to the quenching line, but also stripes created at oblique angles, as well as periodic wrinkles created in an otherwise oblique stripe pattern. Technically, we study stripe formation as traveling-wave solutions in the Swift-Hohenberg equation and in reduced Cahn-Hilliard and Newell-Whitehead-Segel models, analytically, through numerical continuation, and in direct simulations.

Pattern-forming fronts are often controlled by an external stimulus which progresses through a stable medium at a fixed speed, rendering it unstable in its wake. By controlling the speed of excitation, such stimuli, or "triggers," can mediate pattern forming fronts which freely invade an unstable equilibrium and control which pattern is selected. In this work, we analytically and numerically study when the trigger perturbs an oscillatory pushed free front. In such a situation, the resulting patterned front, which we call a pushed trigger front, exhibits a variety of interesting phenomenon, including snaking, non-monotonic wavenumber selection, and hysteresis. Assuming the existence of a generic oscillatory pushed free front, we use heteroclinic bifurcation techniques to prove the existence of trigger fronts in an abstract setting motivated by the spatial dynamics approach. We then derive a leading order expansion for the selected wavenumber in terms of the trigger speed. Furthermore, we show that such a bifurcation curve is governed by the difference of certain strong-stable and weakly-stable spatial eigenvalues associated with the decay of the free pushed front. We also study prototypical examples of these phenomena in the cubic-quintic complex Ginzburg Landau equation and a modified Cahn-Hilliard equation.Running head: Triggered pushed fronts