Acoustic modes of a spherical thin-walled tank for liquid propellant mass gauging: Theory and experiment.

Acoustic response of a thin-walled spherical flight tank filled with water is explored theoretically and experimentally as a testbed for an application of Weyl's law to the problem of mass-gauging propellants in zero-gravity in space. Weyl's law relates the mode counting function of a resonator to its volume and can be used to infer the volume of liquid in a tank from the tank's acoustic response. One of the challenges of applying Weyl's law to real tanks is to account for the boundary conditions which are neither Neumann nor Dirichlet.

A micromagnetic theory of skyrmion lifetime in ultrathin ferromagnetic films.

We use the continuum micromagnetic framework to derive the formulas for compact skyrmion lifetime due to thermal noise in ultrathin ferromagnetic films with relatively weak interfacial Dzyaloshinskii-Moriya interaction. In the absence of a saddle point connecting the skyrmion solution to the ferromagnetic state, we interpret the skyrmion collapse event as "capture by an absorber" at microscale. This yields an explicit Arrhenius collapse rate with both the barrier height and the prefactor as functions of all the material parameters, as well as the dynamical paths to collapse.

Efficient Approximations for Stationary Single-Channel Ca Nanodomains across Length Scales.

We consider the stationary solution for the Ca concentration near a point Ca source describing a single-channel Ca nanodomain in the presence of a single mobile Ca buffer with 1:1 Ca binding. We present computationally efficient approximants that estimate stationary single-channel Ca nanodomains with great accuracy in broad regions of parameter space. The presented approximants have a functional form that combines rational and exponential functions, which is similar to that of the well-known excess buffer approximation and the linear approximation but with parameters estimated using two novel, to our knowledge, methods.

Spherical Caps in Cell Polarization.

Intracellular symmetry breaking plays a key role in wide range of biological processes, both in single cells and in multicellular organisms. An important class of symmetry-breaking mechanisms relies on the cytoplasm/membrane redistribution of proteins that can autocatalytically promote their own recruitment to the plasma membrane. We present an analytical construction and a comprehensive parametric analysis of stable localized patterns in a reaction-diffusion model of such a mechanism in a spherical cell.

Domain structure of ultrathin ferromagnetic elements in the presence of Dzyaloshinskii-Moriya interaction.

Recent advances in nanofabrication make it possible to produce multilayer nanostructures composed of ultrathin film materials with thickness down to a few monolayers of atoms and lateral extent of several tens of nanometers. At these scales, ferromagnetic materials begin to exhibit unusual properties, such as perpendicular magnetocrystalline anisotropy and antisymmetric exchange, also referred to as Dzyaloshinskii-Moriya interaction (DMI), because of the increased importance of interfacial effects. The presence of surface DMI has been demonstrated to fundamentally alter the structure of domain walls.

On well-posedness of variational models of charged drops.

Electrified liquids are well known to be prone to a variety of interfacial instabilities that result in the onset of apparent interfacial singularities and liquid fragmentation. In the case of electrically conducting liquids, one of the basic models describing the equilibrium interfacial configurations and the onset of instability assumes the liquid to be equipotential and interprets those configurations as local minimizers of the energy consisting of the sum of the surface energy and the electrostatic energy. Here we show that, surprisingly, this classical geometric variational model is mathematically ill-posed irrespective of the degree to which the liquid is electrified.

Uniqueness of one-dimensional Néel wall profiles.

We study the domain wall structure in thin uniaxial ferromagnetic films in the presence of an in-plane applied external field in the direction normal to the easy axis. Using the reduced one-dimensional thin-film micromagnetic model, we analyse the critical points of the obtained non-local variational problem. We prove that the minimizer of the one-dimensional energy functional in the form of the Néel wall is the unique (up to translations) critical point of the energy among all monotone profiles with the same limiting behaviour at infinity.

Local accumulation times for source, diffusion, and degradation models in two and three dimensions.

We analyze the transient dynamics leading to the establishment of a steady state in reaction-diffusion problems that model several important processes in cell and developmental biology and account for the diffusion and degradation of locally produced chemical species. We derive expressions for the local accumulation time, a convenient characterization of the time scale of the transient at a given location, in two- and three-dimensional systems with first-order degradation kinetics, and analyze their dependence on the model parameters. We also study the relevance of the local accumulation time as a single measure of timing for the transient and demonstrate that, while it may be sufficient for describing the local concentration dynamics far from the source, a more delicate multi-scale description of the transient is needed near a tightly localized source in two and three dimensions.

Self-similar dynamics of morphogen gradients.

Morphogen gradients are concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues and play a fundamental role in various aspects of embryonic development. We discovered a family of self-similar solutions in a canonical class of nonlinear reaction-diffusion models describing the formation of morphogen gradients. These solutions are realized in the limit of infinitely high production rate at the tissue boundary and are given by the product of the steady state concentration profile and a function of the diffusion similarity variable.

Local kinetics of morphogen gradients.

Some aspects of pattern formation in developing embryos can be described by nonlinear reaction-diffusion equations. An important class of these models accounts for diffusion and degradation of a locally produced single chemical species. At long times, solutions of such models approach a steady state in which the concentration decays with distance from the source of production.

Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle.

A detailed asymptotic study of the effect of small Gaussian white noise on a relaxation oscillator undergoing a supercritical Hopf bifurcation is presented. The analysis reveals an intricate stochastic bifurcation leading to several kinds of noise-driven mixed-mode oscillations at different levels of amplitude of the noise. In the limit of strong time-scale separation, five different scaling regimes for the noise amplitude are identified.

Noise can play an organizing role for the recurrent dynamics in excitable media.

We analyze patterns of recurrent activity in a prototypical model of an excitable medium in the presence of noise. Without noise, this model robustly predicts the existence of spiral waves as the only recurrent patterns in two dimensions. With small noise, however, we found that this model is also capable of generating coherent target patterns, another type of recurrent activity that is widely observed experimentally.

Quantitative models of developmental pattern formation.

Pattern formation in developing organisms can be regulated at a variety of levels, from gene sequence to anatomy. At this level of complexity, mechanistic models of development become essential for integrating data, guiding future experiments, and predicting the effects of genetic and physical perturbations. However, the formulation and analysis of quantitative models of development are limited by high levels of uncertainty in experimental measurements, a large number of both known and unknown system components, and the multiscale nature of development.

Non-meanfield deterministic limits in chemical reaction kinetics.

A general mechanism is proposed by which small intrinsic fluctuations in a system far from equilibrium can result in nearly deterministic dynamical behaviors which are markedly distinct from those realized in the meanfield limit. The mechanism is demonstrated for the kinetic Monte Carlo version of the Schnakenberg reaction where we identified a scaling limit in which the global deterministic bifurcation picture is fundamentally altered by fluctuations. Numerical simulations of the model are found to be in quantitative agreement with theoretical predictions.

Two distinct mechanisms of coherence in randomly perturbed dynamical systems.

We carefully examine two mechanisms--coherence resonance and self-induced stochastic resonance--by which small random perturbations of excitable systems with large time scale separation may lead to the emergence of new coherent behaviors in the form of limit cycles. We analyze what controls the degree of coherence in these two mechanisms and classify their very different properties. In particular we show that coherence resonance arises only at the onset of bifurcation and is rather insensitive against variations in the noise amplitude and the time scale separation ratio.

Signal propagation and failure in discrete autocrine relays.

A mechanistic model of discrete one-dimensional arrays of autocrine cells interacting via diffusible signals is investigated. Under physiologically relevant assumptions, the model is reduced to a system of ordinary differential equations for the intracellular variables, with a particular, biophysically derived type of long-range coupling between cells. Exact discrete traveling wave and static kink solutions are obtained in the model with sharp threshold nonlinearity.

Discrete models of autocrine cell communication in epithelial layers.

Pattern formation in epithelial layers heavily relies on cell communication by secreted ligands. Whereas the experimentally observed signaling patterns can be visualized at single-cell resolution, a biophysical framework for their interpretation is currently lacking. To this end, we develop a family of discrete models of cell communication in epithelial layers.

Long-range signal transmission in autocrine relays.

Intracellular signaling induced by peptide growth factors can stimulate secretion of these molecules into the extracellular medium. In autocrine and paracrine networks, this can establish a positive feedback loop between ligand binding and ligand release. When coupled to intercellular communication by autocrine ligands, this positive feedback can generate constant-speed traveling waves.

Transitions in the model of epithelial patterning.

We analyze pattern formation in the model of cell communication in Drosophila egg development. The model describes the regulatory network formed by the epidermal growth factor receptor (EGFR) and its ligands. The network is activated by the oocyte-derived input that is modulated by feedback loops within the follicular epithelium.

Modeling and computational analysis of EGF receptor-mediated cell communication in Drosophila oogenesis.

Autocrine signaling through the Epidermal Growth Factor Receptor (EGFR) operates at various stages of development across species. A recent hypothesis suggested that a distributed network of EGFR autocrine loops was capable of spatially modulating a simple single-peaked input into a more complex two-peaked signaling pattern, specifying the formation of a pair organ in Drosophila oogenesis (two respiratory appendages on the eggshell). To test this hypothesis, we have integrated genetic and biochemical information about the EGFR network into a mechanistic model of transport and signaling.