Three-dimensional Pattern Formation Research Articles

A collagen-rich arch in the urochordate notochord coordinates cell shaping and multi-tissue elongation

Cell and tissue reshaping is crucial for coordinating three-dimensional pattern formation, in which the size and shape of the cells must be accurately regulated via signal transport and communication among tissues. However, the identity of signaling and transportation mechanisms in this process remains elusive. In our study, we identified an extracellular matrix (ECM) structure with a vertebra-like shape surrounding the central notochord tissue in the larval tail of the urochordate Ciona. Additionally, we verified that the ECM structure was formed de novo, mainly from collagens secreted by notochord cells. Fluorescence recovery after photobleaching and simulation results revealed that this structure was formed via diffusional collagen flow from a notochord that was restricted and molded in the spaces among tail tissues. We revealed that the collagen structure was essential for notochord cell arrangement and elongation. Furthermore, we observed that the central notochord connects with the epidermis through this ECM structure. The disruption of this structure by collagen knockdown and loss-of-collagen function caused the failure of notochord elongation. More importantly, the epidermis could not elongate proportionally with notochord, indicating that the collagen-rich structure serves as a scaffold to coordinate the concurrent elongation of the tail tissues. These findings provide insights into how the central tissue forms and molds its surrounding ECM structure, by not only regulating its own morphogenesis but also functioning as a scaffold for signal transmission to orchestrate the coordinated morphologic reshaping of the surrounding tissues.

Progenitors oppositely polarize WNT activators and inhibitors to orchestrate tissue development.

To spatially co-exist and differentially specify fates within developing tissues, morphogenetic cues must be correctly positioned and interpreted. Here, we investigate mouse hair follicle development to understand how morphogens operate within closely spaced, fate-diverging progenitors. Coupling transcriptomics with genetics, we show that emerging hair progenitors produce both WNTs and WNT inhibitors. Surprisingly, however, instead of generating a negative feedback loop, the signals oppositely polarize, establishing sharp boundaries and consequently a short-range morphogen gradient that we show is essential for three-dimensional pattern formation. By establishing a morphogen gradient at the cellular level, signals become constrained. The progenitor preserves its WNT signaling identity and maintains WNT signaling with underlying mesenchymal neighbors, while its overlying epithelial cells become WNT-restricted. The outcome guarantees emergence of adjacent distinct cell types to pattern the tissue.

Mathematical modeling and computer simulation of the three-dimensional pattern formation of honeycombs

We present a mathematical model, a numerical scheme, and computer simulations of the three-dimensional pattern formation of a honeycomb structure by using the immersed boundary method. In our model, we assume that initially the honeycomb cells have a hollow hemisphere mounted by a hollow circular cylinder shape at their birth and there is force acting upon the entire side of the cell. The net force from the individual cells is a key factor in their transformation from a hollow hemisphere mounted by a hollow circular cylinder shape to a rounded rhombohedral surfaces mounted by a hexagonal cylinder shape. Numerical simulations of the proposed mathematical model equation produce the rounded rhombohedral surfaces mounted by a hexagonal cylinder patterns observed in honeybee colonies.

Interplay between Spinodal Decomposition and Gelation and Their Role in Two- and Three-Dimensional Pattern Formation at the Gelatin Gel Surface

Understanding the mechanisms underlying the spontaneous formation of Liesegang patterns based on the prenucleation and postnucleation models is of critical importance for understanding pattern formation in nature. In contrast to the rapid experimental and theoretical advances in understanding within the framework of the prenucleation model, discussion of the postnucleation model is mostly limited to numerical analysis. To construct a standard model for a chemical experiment discussed in terms of the postnucleation model, we have investigated the pattern formation mechanism in a mixed system containing gelatin, starch, and sugar. Fluorescence and differential interference contrast microscopy revealed a process of two-dimensional spinodal decomposition into gelatin-rich and gelatin-poor phases. Since the gelation temperature of the gelatin-rich phase was higher, spatially periodic gelation proceeded selectively in the gelatin-rich phase. As both two-dimensional spinodal decomposition and three-dimensional g...

Phase-field simulations of spiral growth during directional ternary eutectic solidification

A wide variety of growth patterns has been observed during the directional solidification of three-phase ternary eutectics in multi-component alloys. One hypothesized pattern is the spiral growth of two rod-like phases rotating around each other, which are embedded in a matrix phase. Possible evidence of spiral growth is found experimentally in longitudinal micrographs of the ternary eutectic system Ag–Al–Cu. The phase-field method allows the study of such spatially complex microstructures in order to gain deeper insights into the three-dimensional pattern formation of ternary eutectics. Based on the simulation parameters, which produced a high tilt angle in 2D lamellar growth, determined via systematic parameter studies, multiple spirals are detected in large-scale 3D simulations. Through the presented phase-field simulations, the possibility of the existence of spiral growth in ternary eutectics, reported from experiments, is confirmed.

Pattern selection near the onset of convection in binary mixtures in cylindrical cells

We report numerical investigations of three-dimensional pattern formation of binary mixtures in a vertical cylindrical container heated from below. Negative separation ratio mixtures, for which the onset of convection occurs via a subcritical Hopf bifurcation, are considered. We focus on the dynamics in the neighbourhood of the initial oscillatory instability and analyze the spatio-temporal properties of the patterns for different values of the aspect ratio of the cell, (, where R is the radius of the cell and d its height). Despite the oscillatory nature of the primary instability, for highly constrained geometries, , only pure thermal stationary modes are selected after long transients. As the aspect ratio of the cell increases, for intermediate aspect ratio cells such as , multistability and coexistence of stationary and time-dependent patterns is observed. In highly extended cylinders, , the dynamics near the onset is completely different from the pure fluid case, and a startling diversity of confined patterns is observed. Many of these patterns are consistent with experimental observations. Remarkably, though, we have obtained persistent large amplitude highly localized states not reported previously.

Compositionally modulated ripples during composite film growth: Three-dimensional pattern formation at the nanoscale

Three-dimensional, ion-induced nanoscale pattern formation in the growth mode is studied for a bicomponent thin film. C:Ni films were grown by dual ion beam cosputtering applying an assisting oblique-incidence low-energy ${\text{Ar}}^{+}$ ion beam. Their microstructure was determined by scanning electron, atomic force, and transmission electron microscopy, as well as by grazing-incidence small-angle x-ray scattering. The role of ion-induced collisional effects was investigated by binary collision computer simulations. The formation of compositionally modulated ripples on the C:Ni film surface is demonstrated. They consist of metal-enriched topographic crests and carbon-enriched valleys. Since the surface is constantly covered by incoming species, this pattern is transferred into the bulk as a periodic array of ${\text{Ni}}_{3}\text{C}$ nanoparticles or of Ni-enriched regions in a carbon matrix. Lateral ripple propagation is shown to be one of the crucial phenomena for the film morphology. The essential experimental features are reproduced by the computer simulations. The results reveal the importance of ion-induced preferential displacements as the driving factor for a surface instability, which gives rise to the observed pattern formation.

Multi-component hierarchically structured polymer brushes

Polymer brushes offer patternable, robust, two-dimensional coatings that can be used for, among other applications, the capture and release of medicine and nanoparticles. In this contribution, we demonstrate a technique for extending the patterning ability into the third dimension. This technique, which combines photolithography and ATRP, uses a photoresist to mask and expose chosen areas of a polymer brush. The polymer brush grows only in the exposed areas and subsequent masking and growth processes can be repeated owing to the living nature of ATRP, and thus allowing complex three-dimensional pattern formation. The concept is demonstrated by creating multi-component block-copolymer brush patterns of poly(2-hydroxyethyl methacrylate) (PHEMA), poly(N-isopropylacrylamide) (PNIPAM) and poly(glycidyl methacrylate) (PGMA). Colorimetric visualization of the swell–deswell transition of the partially hydrated and fully hydrated patterns confirm that PGMA patterned on top of PNIPAM alters swelling and the kinetics of the transition.

Three-dimensional pattern formation of magnetically labeled microgel beads for biological tissue engineering

We commenced basic research on the three-dimensional (3D) pattern formation of microgel beads for applications in biological tissue engineering. In this new technique, microgel beads are premagnetized by doping them with magnetic nanoparticles. Living cells will be included in the beads for actual use. If a nonuniform magnetic field is applied to a solution containing these magnetized beads, the beads will align, contact, and form a 3D structure. The structure is controlled by the seed pattern of the magnetic particles plugged in a substrate and the profile of the magnetic field distribution. We constructed tubes, which imitate blood vessels, for demonstration using gel beads whose diameters are of the order of several tens of micrometers. The diameter of the demonstrated tube was less than 0.5 mm and its length was 6.6 mm, although living cells were not included in the beads. Numerical calculations by using the discrete element method were conducted to confirm the formation of the tube and to predict the effect of centrifugal force, which will be applied to fill cells in the space between magnetically patterned beads. Although this unique technology is in the nascent stage, this 3D pattern formation technique by the control of the magnetic field has potential to be one of the effective engineering technologies for manufacturing 3D patterned biological tissues in the future.

Laser Direct-Write Processing

AbstractDirect-write techniques enable computer-controlled two- and three-dimensional pattern formation in a serial fashion. Among these techniques, the versatility offered by laser-based direct-write methods is unique, given their ability to add, remove, and modify different types of materials without physical contact between a tool or nozzle and the material of interest. Laser pulses used to generate the patterns can be manipulated to control the composition, structure, and even properties of individual three-dimensional volumes of materials across length scales spanning six orders of magnitude, from nanometers to millimeters. Such resolution, combined with the ability to process complex or delicate material systems, enables laser direct-write tools to fabricate structures that are not possible to generate using other serial or parallel fabrication techniques. The goal of the articles in this issue of MRS Bulletin is to illustrate the range of materials processing capabilities, fundamental research opportunities, and commercially viable applications that can be achieved using recently developed laser direct-write techniques. We hope that the articles provide the reader with a fresh perspective on the challenges and opportunities that these powerful techniques offer for the fabrication of novel devices and structures.

DIMENSIONALITY EFFECTS IN TURING PATTERN FORMATION

The problem of morphogenesis and Turing instability are revisited from the point of view of dimensionality effects. First the linear analysis of a generic Turing model is elaborated to the case of multiple stationary states, which may lead the system to bistability. The difference between two- and three-dimensional pattern formation with respect to pattern selection and robustness is discussed. Preliminary results concerning the transition between quasi-two-dimensional and three-dimensional structures are presented and their relation to experimental results are addressed.

SELF-SUSTAINED OSCILLATIONS IN FLOWS AROUND LONG BLUNT PLATES

The presence of flow separation from both leading and trailing edges of elongated bluff bodies leads to vortex interactions and resonances not observed in shorter bodies such as circular and square cylinders. Stepwise behaviour in the Strouhal number with increasing plate chord-to-thickness ratio has been observed for long bodies in a number of different situations: natural shedding, under transverse forcing, and with excited duct modes. In the present study, an investigation is made of the predicted unforced laminar flow around long plates (up to chord, c, to thickness, t, ratioc /t=16). The two main types of plate geometry considered are rectangular plates and plates with an aerodynamic leading edge. The rectangular plate represents a geometrical extension of the normal flat and square plates. The aerodynamic leading-edge plate is a natural precursor to the rectangular plate because the vortex shedding is only from the trailing edge. The natural flow around rectangular plates is of greater complexity due to the interaction between the leading- and trailing-edge shedding. The previously neglected influence of the trailing-edge vortex shedding is found to play an important role in the stepwise progression of the Strouhal number with chord-to-thickness ratio. In addition, the formation of three-dimensional patterns in the boundary layer along the plate and in the trailing-edge wake is predicted. The predicted boundary layer hairpin vortices are compared with previous observations and the predicted streamwise modes in the wake are compared with those found in the case of circular cylinders.

Three-dimensional pattern formation, multiple homogeneous soft modes, and nonlinear dielectric electroconvection

Patterns forming spontaneously in extended, three-dimensional, dissipative systems are likely to excite several homogeneous soft modes ( approximately hydrodynamic modes) of the underlying physical system, much more than quasi-one- (1D) and two-dimensional (2D) patterns are. The reason is the lack of damping boundaries. This paper compares two analytic techniques to derive the pattern dynamics from hydrodynamics, which are usually equivalent but lead to different results when applied to multiple homogeneous soft modes. Dielectric electroconvection in nematic liquid crystals is introduced as a model for 3D pattern formation. The 3D pattern dynamics including soft modes are derived. For slabs of large but finite thickness the description is reduced further to a 2D one. It is argued that the range of validity of 2D descriptions is limited to a very small region above threshold. The transition from 2D to 3D pattern dynamics is discussed. Experimentally testable predictions for the stable range of ideal patterns and the electric Nusselt numbers are made. For most results analytic approximations in terms of material parameters are given.

Human SPRY2 inhibits FGF2 signalling by a secreted factor

Growth factor signalling pathways and their inhibitors coordinate the formation of three-dimensional patterns of vertebrates and invertebrates. Temporal and spatial restriction of the response to a few well-defined cells is crucial and needs the integration of positive and negative signals. Recently, Spry has been identified as an inhibitor of fibroblast growth factor (FGF) signalling during Drosophila trachea development. Spry has been described as an intracellular protein that can exert its function in a cell autonomous or a paracrine manner. Here we describe the role of SPRY2, a human homologue of Spry, in human FGF2 signalling. We show that in primary human dermal endothelial cells (MVEC) SPRY2 mRNA is transiently upregulated in response to FGF2. Overexpression of SPRY2 in A375 cells leads to the secretion of a soluble factor that inhibits FGF2- but not VEGF-stimulated proliferation of MVEC. Direct administration of recombinant SPRY2 protein has no effect on MVEC proliferation. However, SPRY2 protein binds the intracellular adaptor protein GRB2, indicating an intracellular localization. A SPRY2/GFP fusion protein remains in the cell, further supporting the intracellular localization of SPRY2. So the intracellular protein SPRY2 is involved in the non-cell autonomous inhibitory effect indirectly, via regulating the secretion of an inhibitor of FGF2 signalling in vertebrates, the evidence of which is presented here for the first time.

Pattern formation in three-dimensional reaction–diffusion systems

Existing group theoretic analysis of pattern formation in three dimensions [T.K. Callahan, E. Knobloch, Symmetry-breaking bifurcations on cubic lattices, Nonlinearity 10 (1997) 1179–1216] is used to make specific predictions about the formation of three-dimensional patterns in two models of the Turing instability, the Brusselator model and the Lengyel–Epstein model. Spatially periodic patterns having the periodicity of the simple cubic (SC), face-centered cubic (FCC) or body-centered cubic (BCC) lattices are considered. An efficient center manifold reduction is described and used to identify parameter regimes permitting stable lamellæ, SC, FCC, double-diamond, hexagonal prism, BCC and BCCI states. Both models possess a special wavenumber k * at which the normal form coefficients take on fixed model-independent ratios and both are described by identical bifurcation diagrams. This property is generic for two-species chemical reaction–diffusion models with a single activator and inhibitor.

Three-dimensional pattern formation during growth of ice dendrites — its relation to universal law of dendritic growth

In-situ observations of ice crystals grown in a sample of supercooled water were conducted, using a new growth apparatus combined with a Mach-Zehnder interferometer. Three-dimensional patterns of ice dendrites were first analyzed from the distributions of interference fringes observed on the ice crystals. Two distinct tip radii in addition to the tip growth velocity were independently measured from the pictures of interference fringes as a function of supercooling. Characteristics of three-dimensional pattern formations of ice dendrites are emphasized on the basis of the Langer and Müller-Krumbhaar theory for dendritic growth.