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. We show that the liquid modes in a thin-walled spherical tank correspond to the spectrum of a slightly larger spherical tank with infinitely compliant wall (Dirichlet boundary condition), where Weyl's law can be applied directly. The mass of the liquid enclosed by this "effective" tank's wall is found to equal the actual mass of the liquid plus the mass of the wall. This finding is generalized to thin-walled tanks and liquid configurations of arbitrary shapes and thus provides a calculable correction factor for the propellant mass inferred using Weyl's law with Dirichlet boundary conditions.

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 Ca2+ Nanodomains across Length Scales.

We consider the stationary solution for the Ca2+ concentration near a point Ca2+ source describing a single-channel Ca2+ nanodomain in the presence of a single mobile Ca2+ buffer with 1:1 Ca2+ binding. We present computationally efficient approximants that estimate stationary single-channel Ca2+ 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. One of the methods involves interpolation between the short-range Taylor series of the free buffer concentration and its long-range asymptotic series in inverse powers of distance from the channel. Although this method has already been used to find Padé (rational-function) approximants to single-channel Ca2+ and buffer concentrations, extending this method to interpolants combining exponential and rational functions improves accuracy in a significant fraction of the relevant parameter space. A second method is based on the variational approach and involves a global minimization of an appropriate functional with respect to parameters of the chosen approximations. An extensive parameter-sensitivity analysis is presented, comparing these two methods with previously developed approximants. Apart from increased accuracy, the strength of these approximants is that they can be extended to more realistic buffers with multiple binding sites characterized by cooperative Ca2+ binding, such as calmodulin and calretinin.

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. The constructed patterns take the form of high-concentration patches localized into spherical caps, similar to the patterns observed in the studies of symmetry breaking in single cells and early embryos.

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. Here we use the micromagnetic modelling framework to analyse the existence and structure of chiral domain walls, viewed as minimizers of a suitable micromagnetic energy functional. We explicitly construct the minimizers in the one-dimensional setting, both for the interior and edge walls, for a broad range of parameters. We then use the methods of Γ-convergence to analyse the asymptotics of the two-dimensional magnetization patterns in samples of large spatial extent in the presence of weak applied magnetic fields.

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. Thus, we establish uniqueness of the one-dimensional monotone Néel wall profile in the considered setting. We also obtain some uniform estimates for general one-dimensional domain wall profiles.

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. Specifically, we demonstrate that an isolated spherical droplet is never a local minimizer, no matter how small is the total charge on the droplet, as the energy can always be lowered by a smooth, arbitrarily small distortion of the droplet's surface. This is in sharp contrast to the experimental observations that a critical amount of charge is needed in order to destabilize a spherical droplet. We discuss several possible regularization mechanisms for the considered free boundary problem and argue that well-posedness can be restored by the inclusion of the entropic effects resulting in finite screening of free charges.

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. We solved the boundary value problem for the similarity profile numerically and analyzed the implications of the discovered self-similarity on the dynamics of morphogenetic patterning.

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. We present analytical results that characterize the dynamics of this process and are in quantitative agreement with numerical solutions of the underlying nonlinear equations. The derived results provide an explicit connection between the parameters of the problem and the time needed to reach a steady state value at a given position. Our approach can be used for the quantitative analysis of tissue patterning by morphogen gradients, a subject of active research in biophysics and developmental biology.

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. As the noise amplitude is decreased, the dynamics of the system goes from the limit cycle due to self-induced stochastic resonance to the coherence resonance limit cycle, then to bursting relaxation oscillations, followed by rare clusters of several relaxation cycles (spikes), and finally to small-amplitude oscillations (or stable fixed point) with sporadic single spikes. These scenarios are corroborated by numerical simulations.

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. These patterns remain essentially deterministic despite the presence of the noise, yet their existence is impossible without it. Their degree of coherence can also be made arbitrarily high for wide ranges of the parameters, which does not require fine-tuning. Our findings demonstrate the need to reexamine current modeling approaches to active biological media.

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. At the same time, an expanding arsenal of experimental tools can constrain models and directly test their predictions, making the modeling efforts not only necessary, but feasible. Using a number of problems in fruit fly development, we discuss how models can be used to test the feasibility of proposed patterning mechanisms and characterize their systems-level properties.

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. In contrast, self-induced stochastic resonance may arise away from bifurcations and the properties of the limit cycle it induces are controlled by both 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. It is argued that the considered mechanism may be used extensively for transmission of information in tissues during homeostasis and development.

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. The models are based on the introduction of cell-to-cell coupling coefficients that characterize the spatial range of intercellular signaling by diffusing ligands. We derive the coupling coefficients as functions of geometric, cellular, and molecular parameters of the ligand transport problem. Using these coupling coefficients, we analyze a nonlinear model of positive feedback between ligand release and binding. In particular, we study criteria of existence of the patterns consisting of clusters of a few signaling cells, as well as the onset of signal propagation. We use our model to interpret recent experimental studies of the EGFR/Rhomboid/Spitz module in Drosophila development.

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. To demonstrate that, we propose a mechanistic model of autocrine relay systems. The model is relevant to the physiology of epithelial layers and to a number of in vitro experimental formats. Using asymptotic and numerical tools, we find that traveling waves in autocrine relays exist and have a number of unusual properties, such as an optimal ligand binding strength necessary for the maximal speed of propagation. We compare our results to recent observations of autocrine and paracrine systems and discuss the steps toward experimental tests of our predictions.

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. We analyze these dynamics within the framework of a recently proposed mathematical model of EGFR signaling (Shvartsman et al. [2002] Development 129:2577-2589). The emphasis is on the large-amplitude solutions of the model that can be correlated with the experimentally observed patterns of protein and gene expression. Our analysis of transitions between the major classes of patterns in the model can be used to interpret the experimentally observed phenotypic transitions in eggshell morphology in Drosophila melanogaster. The existence of complex patterns in the model can be used to account for complex eggshell morphologies in related fly species.