Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

In this article we present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many different phenomena in areas such as developmental and cancer biology, cell motility and material science. Often one is interested in identifying parameters which will lead to a particular pattern. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present various examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally we see that if two or more eigenvalues are in a permissible range then the inhomogeneous steady state can be a linear combination of the respective eigenfunctions. Finally we show an example which suggests that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary.

Universality of quantum critical dynamics in a planar optical parametric oscillator

We analyze the critical quantum fluctuations in a coherently driven planar optical parametric oscillator. We show that the presence of transverse modes combined with quantum fluctuations changes the behavior of the quantum image critical point. This zero-temperature nonequilibrium quantum system has the same universality class as a finite-temperature magnetic Lifshitz transition.

Knockdown of zebrafish crim1 results in a bent tail phenotype with defects in somite and vascular development

The Crim1 gene encodes a transmembrane protein containing six cysteine-rich repeats similar to those found in the BMP antagonist, chordin (chd). To investigate its physiological role, zebrafish crim1 was cloned and shown to be both maternally and zygotically expressed during zebrafish development in sites including the vasculature, intermediate cell mass. notochord, and otic vesicle. Bent or hooked tails with U-shaped somites were observed in 85% of morphants from 12 hpf. This was accompanied by a loss of muscle pioneer cells. While morpholino knockdown of crim1 showed some evidence of ventralisation, including expansion of the intermediate cell mass (ICM), reduction in head size bent tails and disruption to the somites and notochord, this did not mimic the classically ventralised phenotype, as assessed by the pattern of expression of the dorsal markers chordin, otx2 and the ventral markers eve1, pax2.1, tall and gata1 between 75% epiboly and six-somites. From 24 hpf, morphants displayed an expansion of the ventral mesoderm-derived ICM, as evidenced by expansion of tall. Imo2 and crim1 itself. Analysis of the crim1 morphant phenotype in Tg(fli:EGFP) fish showed a clear reduction in the endothelial cells forming the intersegmental vessels and a loss of the dorsal longitudinal anastomotic vessel (DLAV). Hence, the primary role of zebrafish crim1 is likely to be the regulation of somitic and vascular development. (c) 2006 Elsevier Ireland Ltd. All rights reserved.

Generation of primary hepatocyte microarrays by piezoelectric printing

We demonstrate a single-step method for the generation of collagen and poly-l-Lysine (PLL) micropatterns on a poly(ethylene glycol) (PEG) functionalized glass surface for cell based assays. The method involves establishing a reliable silanization method to create an effective non-adhesive PEG layer on glass that inhibits cell attachment, followed by the spotting of collagen or PLL solutions using non-contact piezoelectric printing. We show for the first time that the spotted protein micropatterns remain stable on the PEG surface even after extensive washing, thus significantly simplifying protein pattern formation. We found that adherence and spreading of NIH-3T3 fibroblasts was confined to PLL and collagen areas of the micropatterns. In contrast, primary rat hepatocytes adhered and spread only on collagen micropatterns, where they formed uniform, well defined functionally active cell arrays. The differing affinity of hepatocytes and NIH-3T3 fibroblasts for collagen and PLL patterns was used to develop a simple technique for creating a co-culture of the two cell types. This has the potential to form structured arrays that mimic the in vivo hepatic environment and is easily integrated within a miniaturized analytical platform for developing high throughput toxicity analysis in vitro.

A preliminary study into emergent behaviours in a lattice of interacting nonlinear resonators and oscillators

Future sensor arrays will be composed of interacting nonlinear components with complex behaviours with no known analytic solutions. This paper provides a preliminary insight into the expected behaviour through numerical and analytical analysis. Specically, the complex behaviour of a periodically driven nonlinear Duffing resonator coupled elastically to a van der Pol oscillator is investigated as a building block in a 2D lattice of such units with local connectivity. An analytic treatment of the 2-device unit is provided through a two-time-scales approach and the stability of the complex dynamic motion is analysed. The pattern formation characteristics of a 2D lattice composed of these units coupled together through nearest neighbour interactions is analysed numerically for parameters appropriate to a physical realisation through MEMS devices. The emergent patterns of global and cluster synchronisation are investigated with respect to system parameters and lattice size.

Travelling waves in a complex Ginzburg-Landau equation with time-delay feedback

One of the simplest ways to create nonlinear oscillations is the Hopf bifurcation. The spatiotemporal dynamics observed in an extended medium with diffusion (e.g., a chemical reaction) undergoing this bifurcation is governed by the complex Ginzburg-Landau equation, one of the best-studied generic models for pattern formation, where besides uniform oscillations, spiral waves, coherent structures and turbulence are found. The presence of time delay terms in this equation changes the pattern formation scenario, and different kind of travelling waves have been reported. In particular, we study the complex Ginzburg-Landau equation that contains local and global time-delay feedback terms. We focus our attention on plane wave solutions in this model. The first novel result is the derivation of the plane wave solution in the presence of time-delay feedback with global and local contributions. The second and more important result of this study consists of a linear stability analysis of plane waves in that model. Evaluation of the eigenvalue equation does not show stabilisation of plane waves for the parameters studied. We discuss these results and compare to results of other models.

In-cavity pulse shaping and dissipative parametric instability mode-locking in fibre lasers

At the level of fundamental research, fibre lasers provide convenient and reproducible experimental settings for the study of a variety of nonlinear dynamical processes, while at the applied research level, pulses with different and optimised features – e.g., in terms of pulse duration, temporal and/or spectral shape, energy, repetition rate and emission bandwidth – are sought with the general constraint of developing efficient cavity architectures. In this work, we review our recent progress on the realisation of pulse shaping in passively- mode-locked fibre lasers by inclusion of an amplitude and phase spectral filter into the laser cavity. We present a fibre laser design in which pulse shaping occurs through filtering of a spectrally nonlinearly broadened pulse in the cavity. This strategy of pulse shaping is illustrated through the numerical demonstration of the laser operation in different pulse-generation regimes, including parabolic, flattop and triangular waveform generations, depending on the amplitude profile of the in-cavity spectral filter [1]. As an application of this general approach, we show that the use of an in-cavity flat-top spectral filter makes it possible to directly generate sinc-shaped Nyquist pulses of high quality and of a widely tunable bandwidth from the laser [2]. We also report on a recently-developed versatile erbium-doped fibre laser, in which conventional soliton, dispersion-managed soliton (stretched-pulse) and dissipative soliton mode-locking regimes can be selectively and reliably targeted by programming different group-velocity dispersion profiles and bandwidths on an in-cavity programmable filter [3]. Further, we report on our recent results on the passive mode locking of a Raman fibre laser by a recently predicted new type of parametric instability – the dissipative Faraday instability [4], where spatially periodic zig-zag modulation of spectrally dependent losses can lead to pattern formation in the temporal domain. High-order harmonic mode locking is achieved in a very simple experimental configuration, with the laser cavity including an optical fibre and two chirped fibre Bragg gratings, and no additional mode-locking elements. The results not only open up new possibilities for the design of mode-locked lasers, but extend beyond fibre optics to other fields of physics and engineering. References [1] S. Boscolo, C. Finot, H. Karakuzu, P. Petropoulos, “Pulse shaping in mode-locked fiber laser by in-cavity spectral filter,” Opt. Lett., vol. 39, pp. 438–441, 2014. [2] S. Boscolo, C. Finot, S. K. Turitsyn, “Bandwidth programmable optical Nyquist pulse generation in passively mode-locked fiber laser,” IEEE Photon. J., vol. 7, 7802008(8), 2015. [3] J. Peng, S. Boscolo, “Filter-based dispersion-managed versatile ultrafast fibre laser,” Sci. Rep., 2016, In press. [4] A. M. Perego, N. Tarasov, D. V. Churkin, S. K. Turitsyn, K. Staliunas, “Pattern generation by dissipative parametric instability,” Phys. Rev. Lett., vol. 116, 028701, 2016.

Interactions between endothelin receptor B and transcription factors Sox10 and Pax3 in the melanocyte lineage

Genetic interactions that underlie developmental processes such as cell differentiation and pattern formation are complex and difficult to elucidate. Neural Crest (NC) cells and their derivatives offer an optimal system in which to probe for these complex interactions as they acquire different cell fates and constitute a variety of structures. The transcription factors Sox10 and Pax3 as well as the transmembrane receptor Endothelin receptor b (Ednrb) are temporally and spatially co-expressed early in NC cells and mutations in these genes lead to similar hypopigmentation phenotypes due to a reduced number of NC-derived melanocyte precursors, the melanoblasts. The goal of this study was to establish whether Sox10 and Ednrb or Pax3 and Ednrb interact to promote normal murine melanocyte development. Crosses of Sox10 or Pax3 with Ednrb heterozygous mutants showed that the double heterozygous hypopigmentation phenotype was significantly more pronounced than phenotypes of single heterozygotes, implying that a synergistic interaction exists between Sox10 and Ednrb and Pax3 and Ednrb. This interaction was further explored by the attempt to rescue the Sox10 and Pax3 hypopigmentation phenotypes by the transgenic addition of Ednrb to melanoblasts. Pigmentation was completely restored in the Sox10 and partially restored in the Pax3 mutant mice. The comparison of the number of melanoblasts in transgenic and non-transgenic Sox10 mutant embryos showed that the transgenic rescue occurred as early as E11.5, a critical time for melanoblast population expansion. Cell survival assays indicated that the rescue was not due to an effect of the transgene on melanoblast survival. A novel phenotype arose when studying the interaction between Ednrb and Pax3. Newborns appeared normal but by 3.5 weeks of age, the affected pups were smaller than normal littermates and developed a dome-shaped head; some also developed thoracic kyphosis. Affected pups were dead by 4 weeks of age: 80% were Pax3Sp/+ and 75% were female. When compared to normal littermates, affected mice had brains with enlarged 4th ventricles and more glia while skeletal staining showed kyphosis, wider rib cages and pelvic differences. An epistatic interaction resulting from the mixing of genetic backgrounds that is exacerbated in the presence of Pax3 heterozygosity is suspected.

Interactions between Endothelin Receptor B and Transcription Factors Sox10 and Pax3 in the Melanocyte Lineage

Genetic interactions that underlie developmental processes such as cell differentiation and pattern formation are complex and difficult to elucidate. Neural Crest (NC) cells and their derivatives offer an optimal system in which to probe for these complex interactions as they acquire different cell fates and constitute a variety of structures. The transcription factors Sox10 and Pax3 as well as the transmembrane receptor Endothelin receptor b (Ednrb) are temporally and spatially co-expressed early in NC cells and mutations in these genes lead to similar hypopigmentation phenotypes due to a reduced number of NC-derived melanocyte precursors, the melanoblasts. The goal of this study was to establish whether Sox10 and Ednrb or Pax3 and Ednrb interact to promote normal murine melanocyte development. Crosses of Sox10 or Pax3 with Ednrb heterozygous mutants showed that the double heterozygous hypopigmentation phenotype was significantly more pronounced than phenotypes of single heterozygotes, implying that a synergistic interaction exists between Sox10 and Ednrb and Pax3 and Ednrb. This interaction was further explored by the attempt to rescue the Sox10 and Pax3 hypopigmentation phenotypes by the transgenic addition of Ednrb to melanoblasts. Pigmentation was completely restored in the Sox10 and partially restored in the Pax3 mutant mice. The comparison of the number of melanoblasts in transgenic and non-transgenic Sox10 mutant embryos showed that the transgenic rescue occurred as early as E11.5, a critical time for melanoblast population expansion. Cell survival assays indicated that the rescue was not due to an effect of the transgene on melanoblast survival. A novel phenotype arose when studying the interaction between Ednrb and Pax3. Newborns appeared normal but by 3.5 weeks of age, the affected pups were smaller than normal littermates and developed a dome-shaped head; some also developed thoracic kyphosis. Affected pups were dead by 4 weeks of age: 80% were Pax3Sp/+ and 75% were female. When compared to normal littermates, affected mice had brains with enlarged 4th ventricles and more glia while skeletal staining showed kyphosis, wider rib cages and pelvic differences. An epistatic interaction resulting from the mixing of genetic backgrounds that is exacerbated in the presence of Pax3 heterozygosity is suspected.

Self-pulsating nonlinear systems via dissipative parametric instability

The recently discovered dissipative parametric instability is presented in the framework of the universal complex Ginzburg-Landau equation. The pattern formation associated with the instability is discussed in connection to the relevant applications in nonlinear photonics especially as a new tool for pulsed lasers design.

Nonlinear Dynamics Near the Cell Membrane of Chara Corallina

The phenomenon of patterned distribution of pH near the cell membrane of the algae Chara corallina upon illumination is well-known. In this paper, we develop a mathematical model, based on the detailed kinetic analysis of proton fluxes across the cell membrane, to explain this phenomenon. The model yields two coupled nonlinear partial differential equations which describe the spatial dynamics of proton concentration changes and transmembrane potential generation. The experimental observation of pH pattern formation, its period and amplitude of oscillation, and also its hysteresis in response to changing illumination, are all reproduced by our model. A comparison of experimental results and predictions of our theory is made. Finally, a mechanism for pattern formation in Chara corallina is proposed.

Hexagonal patterns in finite domains

In many mathematical models for pattern formation, a regular hexagonal pattern is stable in an infinite region. However, laboratory and numerical experiments are carried out in finite domains, and this imposes certain constraints on the possible patterns. In finite rectangular domains, it is shown that a regular hexagonal pattern cannot occur if the aspect ratio is rational. In practice, it is found experimentally that in a rectangular region, patterns of irregular hexagons are often observed. This work analyses the geometry and dynamics of irregular hexagonal patterns. These patterns occur in two different symmetry types, either with a reflection symmetry, involving two wavenumbers, or without symmetry, involving three different wavenumbers. The relevant amplitude equations are studied to investigate the detailed bifurcation structure in each case. It is shown that hexagonal patterns can bifurcate subcritically either from the trivial solution or from a pattern of rolls. Numerical simulations of a model partial differential equation are also presented to illustrate the behaviour.

Three-dimensional kinetic simulations of active suspensions: effect of chemotaxis and steric interaction

In this work, the effects of chemotaxis and steric interactions in active suspensions are analyzed by extending the kinetic model proposed by Saintillan and Shelley [1, 2]. In this model, a conservation equation for the active particle configuration is coupled to the Stokes equation for the flow arising from the force dipole exerted by the particles on the fluid. The fluid flow equations are solved spectrally and the conservation equation is solved by second-order finite differencing in space and second-order Adams-Bashforth time marching. First, the dynamics in suspensions of oxytactic run-and-tumble bacteria confined in thin liquid films surrounded by air is investigated. These bacteria modify their tumbling behavior by making temporal comparisons of the oxygen concentration, and, on average, swim towards high concentrations of oxygen. The kinetic model proposed by Saintillan and Shelley [1, 2] is modified to include run-and-tumble effects and oxygentaxis. The spatio-temporal dynamics of the oxygen and bacterial concentration are analyzed. For small film thicknesses, there is a weak migration of bacteria to the boundaries, and the oxygen concentration is high inside the film as a result of diffusion; both bacterial and oxygen concentrations quickly reach steady states. Above a critical film thickness (approximately 200 micron), a transition to chaotic dynamics is observed and is characterized by turbulent-like 3D motion, the formation of bacterial plumes, enhanced oxygen mixing and transport into the film, and hydrodynamic velocities of magnitudes up to 7 times the single bacterial swimming speed. The simulations demonstrate that the combined effects of hydrodynamic interactions and oxygentaxis create collective three-dimensional instabilities which enhances oxygen availability for the bacteria. Our simulation results are consistent with the experimental findings of Sokolov et al. [3], who also observed a similar transition with increasing film thickness. Next, the dynamics in concentrated suspensions of active self-propelled particles in a 3D periodic domain are analyzed. We modify the kinetic model of Saintillan and Shelley [1, 2] by including an additional nematic alignment torque proportional to the local concentration in the equation for the rotational velocity of the particles, causing them to align locally with their neighbors (Doi and Edwards [4]). Large-scale three- dimensional simulations show that, in the presence of such a torque both pusher and puller suspensions are unstable to random fluctuations and are characterized by highly nematic structures. Detailed measures are defined to quantify the degree and direction of alignment, and the effects of steric interactions on pattern formation will be presented. Our analysis shows that steric interactions have a destabilizing effect in active suspensions.

Coerced Mechanical Coarsening of Nanoparticle Assemblies

Coarsening is a ubiquitous phenomenon [1-3] that underpins countless processes in nature, including epitaxial growth [1,3,4], the phase separation of alloys, polymers and binary fluids [2], the growth of bubbles in foams5, and pattern formation in biomembranes6. Here we show, in the first real-time experimental study of the evolution of an adsorbed colloidal nanoparticle array, that tapping-mode atomic force microscopy (TM-AFM) can drive the coarsening of Au nanoparticle assemblies on silicon surfaces. Although the growth exponent has a strong dependence on the initial sample morphology, our observations are largely consistent with modified Ostwald ripening processes [7-9]. To date, ripening processes have been exclusively considered to be thermally activated, but we show that nanoparticle assemblies can be mechanically coerced towards equilibrium, representing a new approach to directed coarsening. This strategy enables precise control over the evolution of micro- and nanostructures.

Stable spike clusters for the one-dimensional Gierer-Meinhardt system

We consider the Gierer-Meinhardt system with precursor inhomogeneity in a one-dimensional interval. A spike cluster is the combination of several spikes which all approach the same point in the singular limit of small activator diffusivity. We rigorously prove the existence of a steady-state spike cluster consisting of N spikes near a non-degenerate local minimum point of the smooth inhomogeneity, where N is an arbitrary positive integer. Further, we show that this solution is linearly stable. We explicitly compute all eigenvalues, both large (of order O(1)) and small (of order o(1)). The main features of studying the Gierer-Meinhardt system in this setting are as follows: (i) it is biologically relevant since it models a hierarchical process (pattern formation of small-scale structures induced by a pre-existing large-scale inhomogeneity), (ii) it contains three different spatial scales two of which are small. (iii) all the expressions can be made explicit and often have a particularly simple form.