Cyrill B. Muratov

Theory of domain patterns in systems with long-range interactions of Coulomb type

We develop a theory of the domain patterns in systems with competing short-range attractive interactions and long-range repulsive Coulomb interactions. We take an energetic approach, in which patterns are considered as critical points of a mean-field free energy functional. Close to the microphase separation transition, this functional takes on a universal form, allowing us to treat a number of diverse physical situations within a unified framework. We use asymptotic analysis to study domain patterns with sharp interfaces. We derive an interfacial representation of the patterns free energy which remains valid in the fluctuating system, with a suitable renormalization of the Coulomb interactions coupling constant. We also derive integro-differential equations describing stationary domain patterns of arbitrary shapes and their thermodynamic stability, coming from the first and second variations of the interfacial free energy. We show that the length scale of a stable domain pattern must obey a certain scaling law with the strength of the Coulomb interaction. We analyzed the existence and stability of localized (spots, stripes, annuli) and periodic (lamellar, hexagonal) patterns in two dimensions. We show that these patterns are metastable in certain ranges of the parameters and that they can undergo morphological instabilities leading to the formation of more complex patterns. We discuss nucleation of the domain patterns by thermal fluctuations and pattern formation scenarios for various thermal quenches. We argue that self-induced disorder is an intrinsic property of the domain patterns in the systems under consideration.

On an Isoperimetric Problem with a Competing Nonlocal Term II: The General Case

This paper is the continuation of a previous paper (H. Knupfer and C. B. Muratov, Comm. Pure Appl. Math. 66 (2013), 1129‐1162). We investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance. In this work, we treat the case of a general space dimension. We obtain basic existence results for minimizers with sufficiently small masses. For certain ranges of the exponent in the kernel, we also obtain nonexistence results for sufficiently large masses, as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality. The physically important special case of three space dimensions and Coulombic repulsion is included in all the results mentioned above. In particular, our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter. In all cases the minimal energy scales linearly with mass for large masses, even if the infimum of energy cannot be attained.

Static spike autosolitons in the Gray-Scott model

We construct asymptotically the solutions to a classical reaction-diffusion system (the Gray-Scott model of an autocatalytic reaction) in the form of static spike autosolitons (self-sustained solitary pulses, spots and clots). We show that solutions in the form of static spike autosolitons exist over a wide range of system parameters in one dimension, and in a narrower range of parameters in two and three dimensions. We study the properties of these solutions.

STABILITY OF THE STATIC SPIKE AUTOSOLITONS IN THE GRAY--SCOTT MODEL

We performed an asymptotic linear stability analysis of the static spike autosolitons (ASs)---self-sustained solitary inhomogeneous states---in the Gray--Scott model of an autocatalytic chemical reaction. We found that in one dimension these ASs destabilize with respect to pulsations or the onset of traveling motion when the inhibitor is slow enough. In higher dimensions, the one-dimensional static spike ASs are always unstable with respect to corrugation and wriggling. The higher-dimensional radially symmetric static spike ASs may destabilize with respect to the radially nonsymmetric fluctuations leading to their splitting when the inhibitor is fast or with respect to pulsations when the inhibitor is slow.

Droplet Phases in Non-local Ginzburg-Landau Models with Coulomb Repulsion in Two Dimensions

We establish the behavior of the energy of minimizers of non-local Ginzburg-Landau energies with Coulomb repulsion in two space dimensions near the onset of multi-droplet patterns. Under suitable scaling of the background charge density with vanishing surface tension the non-local Ginzburg-Landau energy becomes asymptotically equivalent to a sharp interface energy with screened Coulomb interaction. Near the onset the minimizers of the sharp interface energy consist of nearly identical circular droplets of small size separated by large distances. In the limit the droplets become uniformly distributed throughout the domain. The precise asymptotic limits of the bifurcation threshold, the minimal energy, the droplet radii, and the droplet density are obtained.

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.

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.

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.

Homogenization of boundary conditions for surfaces with regular arrays of traps

The problem of trapping of diffusing particles by nonoverlapping absorbing patches randomly or regularly located on a surface arises in numerous settings. Examples include diffusion current to ensembles of microelectrodes, ligand binding to cells, mass transfer to heterogeneous surfaces, ligand accumulation in cell culture assays, etc. see Refs. 1–15 and references therein . The problem is extremely complicated because the boundary conditions on the surface are nonuniform: absorbing on the patches and reflecting otherwise. There is, however, an approximation that greatly simplifies the analysis when the layer of medium above the surface is sufficiently thick. The approximation is based on the fact that, far from the surface, fluxes and concentrations become uniform in the lateral direction and, therefore, indistinguishable from those in the case of uniformly absorbing surface. Keeping this in mind, one can replace the nonuniform boundary conditions on the surface by a uniform radiationtype boundary condition with a properly chosen trapping rate see e.g., Ref. 16 and references therein . We have demonstrated how this procedure works in the case of randomly distributed traps in Refs. 17 and 18. Here we consider the problem with traps regularly distributed over the surface. Our aim is to predict the dependence of on the trap concentration and parameters of the traps and diffusing particles over the entire concentration range. At low concentrations is equal to the product of the concentration and the trapping rate constant of an isolated trap. It turns out that the low-concentration linear dependence of on the trap concentration fails very early, and grows with the concentration much faster even when only a small fraction of the surface is covered by the traps. This happens because “interaction” between traps decays very slowly, as 1 /L, where L is the intertrap distance. As a consequence, collective effects due to this interaction, which lead to the enhancement of the trapping rate, manifest themselves already at low concentrations. In Ref. 17 we reported a boundary homogenization approach for surfaces randomly covered by nonoverlapping circular traps. To describe the enhancement of the trapping rate compared to the linear regime, we introduced the function F of the trap surface fraction and suggested an approximate formula for this function. In Ref. 18 we found that the enhancement due to the collective effects was insensitive to whether the traps were identical or polydisperse and their radii are allowed to fluctuate. This suggests that the enhancement depends only on and is weakly sensitive to the details of the trap arrangement on the surface. To check this hypothesis here we study homogenization of boundaries with regular arrangements of identical traps. Our results support this hypothesis. We find that the values of for the three different arrangements and for random distribution of traps are close to each other the difference is within 20% . In addition, here we study homogenization of periodic nonuniform boundaries formed by alternating absorbing and reflecting stripes. Such boundaries are special because the conventional ideology based on the existence of a trivial limiting behavior of when →0 fails in this case since a stationary flux to an isolated absorbing strip does not exist. Nevertheless, we are able to overcome this difficulty and find the effective trapping rate for such a boundary. In our analysis we use the computer-assisted boundary homogenization approach suggested in Ref. 17: First, based on the limiting behavior and dimensional arguments we express in terms of an unknown dimensionless function of the dimensionless trap surface fraction ,F . We then determine this function using the dependence , which is found by Brownian dynamics simulations as described in Ref. 17 or by solving THE JOURNAL OF CHEMICAL PHYSICS 124, 1 2006

Theory of 360° domain walls in thin ferromagnetic films

An analytical and computational study of 360° domain walls in thin uniaxial ferromagnetic films is presented. The existence of stable one-dimensional 360° domain wall solutions both with and without the applied field is demonstrated in a reduced thin film micromagnetic model. The wall energy is found to depend rather strongly on the orientation of the wall and the wall width significantly grows when the strength of the magnetostatic forces increases. It is also shown that a critical reverse field is required to break up a 360° domain wall into a pair of 180° walls. The stability of the 360° walls in two-dimensional films of finite extent is demonstrated numerically and the stability with respect to slow modulations in extended films is demonstrated analytically. These domain wall solutions are shown to play an important role in magnetization reversal. In particular, it is found that the presence of 360° domain walls may result in nonuniqueness of the observed magnetization patterns during repeated cycles ...