Plasma equilibrium in a double-dipole magnetic confinement system with a separatrix

Results are presented from theoretical studies of plasma equilibrium consistent with the convective stability of ideal interchange modes in axisymmetric configurations with an outward-decreasing field that may have a separatrix limiting the plasma volume. A two-dimensional numerical code is developed to solve the Grad-Shafranov equation with a convectively stable pressure distribution at an arbitrary value of β. The problem is solved for an actual geometry of the magnetic field produced by thin current rings. Configurations of a double-dipole confinement system are calculated for the parameters measured in experiments carried out in the Magnetor device, as well as for higher β values. A configuration of a model mirror system with a divertor is also calculated. The code allows one to optimize confinement systems operating at high β values at which equilibrium still can exist.     

Physical constraints on early blastomere packings

At very early embryonic stages, when embryos are composed of just a few cells, establishing the correct packing arrangements (contacts) between cells is essential for the proper development of the organism. As early as the 4-cell stage, the observed cellular packings in different species are distinct and, in many cases, differ from the equilibrium packings expected for simple adherent and deformable particles. It is unclear what are the specific roles that different physical parameters, such as the forces between blastomeres, their division times, orientation of cell division and embryonic confinement, play in the control of these packing configurations. Here we simulate the non-equilibrium dynamics of cells in early embryos and systematically study how these different parameters affect embryonic packings at the 4-cell stage. In the absence of embryo confinement, we find that cellular packings are not robust, with multiple packing configurations simultaneously possible and very sensitive to parameter changes. Our results indicate that the geometry of the embryo confinement determines the packing configurations at the 4-cell stage, removing degeneracy in the possible packing configurations and overriding division rules in most cases. Overall, these results indicate that physical confinement of the embryo is essential to robustly specify proper cellular arrangements at very early developmental stages.     

Some problems of axisymmetric toroidal equilibria in orthogonal flux coordinates

Formulations of axisymmetric equilibrium problems are presented in orthogonal flux coordinates with boundary conditions that do not assume fixation of the boundary shape for both levitron-type traps with a plasma-embedded conductor and conductor-free configurations, including a tokamak. Illustrative examples of numerical solutions of these problems are presented. For traps with a conductor, it is demonstrated how the geometry of the equilibrium with an isodynamic magnetic surface can be found using flux coordinates.     

Thermodynamic equilibrium of pure electron plasmas in a Malmberg-Penning trap

A new class of annular confinement configurations of a single-charged plasma corresponding to global thermodynamic equilibria in a cylindrical Malmberg-Penning trap with an axial conductor is investigated both numerically and analytically. In the case of an infinite plasma length, the density turns out to be essentially constant inside a surface of revolution and to fall off abruptly outside of it. Analytical limiting cases are calculated explicitly in the limit of small Debye lengths. In the case of a finite plasma length, the self-consistent solution to Poisson's equation describing thermodynamic equilibrium is obtained numerically and the dependence of the plasma density distribution on the various parameters of the system is investigated.     

Electron states of semiconductor quantum ring with geometry and size variations

Heterostructured semiconductor quantum rings (QR) are studied by a three-dimensional model in a single sub-band approach with energy dependent electron effective mass. Nonlinear confinement energy problem is numerically solved by the finite element method. Effects of boundary conditions corresponding to various QR/substrate configurations are analyzed. General relation for the size dependence of the QR confined energy is derived. The contribution to confinement energy due to the presence of external magnetic field is compared with the effect of variation of QR geometrical parameters. Applicability of the model is tested by comparison with experimental data.     

Simulations of knotting in confined circular DNA

The packing of DNA inside bacteriophages arguably yields the simplest example of genome organization in living organisms. As an assay of packing geometry, the DNA knot spectrum produced upon release of viral DNA from the P4 phage capsid has been analyzed, and compared to results of simulation of knots in confined volumes. We present new results from extensive stochastic sampling of confined self-avoiding and semiflexible circular chains with volume exclusion. The physical parameters of the chains (contour length, cross section, and bending rigidity) have been set to match those of P4 bacteriophage DNA. By using advanced sampling techniques, involving multiple Markov chain pressure-driven confinement combined with a thermodynamic reweighting technique, we establish the knot spectrum of the circular chains for increasing confinement up to the highest densities for which available algorithms can exactly classify the knots. Compactified configurations have an enclosing hull diameter ∼2.5 times larger than the P4 caliper size. The results are discussed in relation to the recent experiments on DNA knotting inside the capsid of a P4 tailless mutant. Our investigation indicates that confinement favors chiral knots over achiral ones, as found in the experiments. However, no significant bias of torus over twist knots is found, contrary to the P4 results. The result poses a crucial question for future studies of DNA packaging in P4: is the discrepancy due to the insufficient confinement of the equilibrium simulation or does it indicate that out-of-equilibrium mechanisms (such as rotation by packaging motors) affect the genome organization, hence its knot spectrum in P4?     

Analytic examples of force-free toroidal MHD equilibria

Analytically described toroidal (axisymmetric and three-dimensional) equilibrium magnetic field configurations with a “flat” current density, j=λB (λ = const), are proposed. Such configurations are superpositions of several force-free two-dimensional configurations with plane, axial, or helical coordinate symmetry. Each of them is generated by an exact partial solution to the corresponding Grad-Shafranov equation. A variety of toroidal configurations thus obtained allows one to model topological changes of magnetic surfaces, such as magnetic axis splitting (doublets) in axisymmetric equilibrium configurations and the appearance and interaction of magnetic islands and ergodic lines in three-dimensional configurations.     

Longitudinal structure of MHD perturbations at the boundary of convective stability in the Kruskal-Oberman model

It is shown that, in contrast to the MHD model, a perturbation at the boundary of convective stability of a finite-pressure plasma in confinement systems without an averaged minB in the Kruskal-Oberman model is not generally a purely flute one. The reasons for this discrepancy are clarified. The analysis is carried out for axisymmetric configurations formed by a poloidal magnetic field.     

Self-consistent quasineutral dust structures

Distributions of the dusty plasma parameters (electron, ion, and dust densities; dust grain charge; and ion drift velocity) in quasineutral dust structures whose dimensions are much greater than the mean free path of the ions in their interactions with neutral particles are calculated numerically under conditions such that ionization sources are located outside the structures. Planar, cylindrical, and spherical structures are investigated. It is shown that static equilibrium structures are governed by a single (basic) parameter: the electrostatic potential drop between the center of the structure and its boundary. It is found that the maximum value of the basic parameter (in energy units) does not exceed the electron temperature. The basic parameter also determines the total number of dust grains in the structure and the power of external ionization sources that are necessary to sustain this structure. The fact that the basic parameter varies within a limited range allows one to consider all the possible structures with a given dimensionality (planar, cylindrical, and spherical).     

Islands and ergodicity of the plasma current in toroidal magnetic confinement systems

Magnetic and current structures arising due to resonant perturbations of an equilibrium current-carrying magnetic configuration are analyzed using the Hamiltonian formalism. Special attention is paid to axisymmetric tokamak and pinch configurations. It is shown that, due to the very different dependences of the magnetic and current rotational transforms on the plasma pressure, the resonances (islands) of the magnetic field may not coincide with those of the current. The perturbed force-free equilibrium of a cylindrical pinch in which the field and current islands overlap is analyzed. The long-lived ribbon structures observed in the JET tokamak are explained as a manifestation of a force-free magneto-current island.     

Adhesion and hemifusion of cytoplasmic myelin lipid membranes are highly dependent on the lipid composition

We report the effects of calcium ions on the adhesion and hemifusion mechanisms of model supported myelin lipid bilayer membranes of differing lipid composition. As in our previous studies Min et al. [1,2], the lipid compositions used mimic "healthy" and "diseased-like" (experimental autoimmune encephalomyelitis, EAE) membranes. Our results show that the interaction forces as a function of membrane separation distance are well described by a generic model that also (and in particular) includes the hydrophobic interaction arising from the hydrophobically exposed (interior) parts of the bilayers. The model is able to capture the mechanical instability that triggers the onset of the hemifusion event, and highlights the primary role of the hydrophobic interaction in membrane fusion. The effects of lipid composition on the fusion mechanism, and the adhesion forces between myelin lipid bilayers, can be summarized as follows: in calcium-free buffer, healthy membranes do not present any signs of adhesion or hemifusion, while diseased membranes hemifuse easily. Addition of 2mM calcium favors adhesion and hemifusion of the membranes independently of their composition, but the mechanisms involved in the two processes were different: healthy bilayers systematically presented stronger adhesion forces and lower energy barriers to fusion compared to diseased bilayers. These results are of particular relevance for understanding lesion development (demyelination, swelling, vacuolization and/or vesiculation) in myelin associated diseases such as multiple sclerosis and its relationship to lipid domain formation in myelin membranes.     

Three-dimensional discrete element method for the prediction of protoplasmic seepage through membrane in a biological cell

This paper presents a three-dimensional and compressible biological cell model based on discrete element method using multiple interacting agent that represent cellular structures within a simulated environment. The cytoplasm and nucleoplasm fluid behavior in the cell is time dependent. When taking this approach, it is important to calibrate protoplasmic flow behaviors through simulation techniques such as compressing the cell and examining the agents representing the cell cytoplasm seeping between the ones representing the confining cell membrane. This type of modelling may motivate future work on simulating simultaneous operations and interactions of multiple cellular agents in an attempt to re-create and predict the appearance of complex phenomena such as protoplasmic seepage that is caused by the force actuations of neighboring cells. Seepage occurs when a cytoplasm agent passes between three membrane particles connected in a triangular network. Based on the force–deformation response of spheres having variable size and stiffness, semi-analytic expressions are developed for the force required to cause seepage and solved numerically to find the maximum resistance offered by the membrane against cytoplasm seepage. The equations are based on force equilibrium and the constitutive relations for particle contact and membrane stiffness. In multi-particle representations of an individual cell undergoing deformation, different modes of cytoplasm seepage through confining cell membranes can occur. This can be avoided if simple criteria are satisfied. These findings can lead to certain fundamental laws for the improvement of novel cell-to-organ simulation techniques based on discrete element method.     

Bifurcation of the equilibrium of a current-carrying plasma column

The classical (for the Grad-Shafranov equation) formulation of the equilibrium problem for a cylindrical current-carrying plasma column is shown to admit multiple solutions. The multiple solutions are bifurcational in character and appear due to the nonlinearity of the equilibrium equation. This was demonstrated analytically using a stepped current profile as an example. Bifurcational solutions found for the cylindrical case survive in toroidal geometry too.     

Analysis of adhesion of large vesicles to surfaces.

An experimental procedure that can be used to measure the interfacial free energy density for the adhesion of membranes of large vesicles to other surfaces is outlined and analyzed. The approach can be used for both large phospholipid bilayer vesicles and red blood cells when the membrane force resultants are dominated by isotropic tension. The large vesicle or red cell is aspirated by a micropipet with sufficient suction pressure to form a spherical segment outside the pipet. The vesicle is then brought into close proximity of the surface to be tested and, the suction pressure reduced to permit adhesion, and the new equilibrium configuration is established. The mechanical analysis of the equilibrium shape provides the interfacial free energy density for the surface affinity. With this approach, the measurable range of membrane surface affinity is 10(-4)-3 erg/cm2 for large phospholipid bilayer vesicles and 10(-2)-10 erg/cm2 for red blood cells.     

The Tethered Infinitesimal Tori and Spheres Algorithm: A Versatile Calculator for Axisymmetric Problems in Equilibrium Membrane Mechanics

Constrained minimization of energy functionals is a central part, and usually the difficult part, of solving problems in the equilibrium mechanics of biological and biomimetic membranes. The inherent difficulties of the conventional variational-calculus approach prevents the numerical calculation involved from being made routine in the analyses of experimental results. We have developed a simulated annealing-based computational technique for routinizing the task of constrained minimization of energy functionals governing whole, or small patches of whole, fluid membranes with axisymmetry, spherical topology, and no domains of inhomogeneity. In this article, we describe the essential principles of the technique and apply it to five examples to demonstrate its versatility. It gives membrane shapes that are automatically stable to axisymmetric perturbations. Presently, it can account for constraints on 1), the membrane area or the effective membrane tension; 2), the enclosed volume or the effective pressure difference across the membrane thickness; and 3), the axial end-to-end distance or the applied axial point force.     

Theoretical and experimental intravascular gas embolism absorption dynamics.

Multifocal cerebrovascular gas embolism occurs frequently during cardiopulmonary bypass and is thought to cause postoperative neurological dysfunction in large numbers of patients. We developed a mathematical model to predict the absorption time of intravascular gas embolism, accounting for the bubble geometry observed in vivo. We modeled bubbles as cylinders with hemispherical end caps and solved the resulting governing gas transport equations numerically. We validated the model using data obtained from video-microscopy measurements of bubbles in the intact cremaster microcirculation of anesthetized male Wistar rats. The theoretical model with the use of in vivo geometry closely predicted actual absorption times for experimental intravascular gas embolisms and was more accurate than a model based on spherical shape. We computed absorption times for cerebrovascular gas embolism assuming a range of bubble geometries, initial volumes, and parameters relevant to brain blood flow. Results of the simulations demonstrated absorption time maxima and minima based on initial geometry, with several configurations taking as much as 50% longer to be absorbed than would a comparable spherical bubble.     

Organization and dynamics of cross-linked actin filaments in confined environments

The organization of the actin cytoskeleton is impacted by the interplay between physical confinement, features of cross-linking proteins, and deformations of semiflexible actin filaments. Some cross-linking proteins preferentially bind filaments in parallel, although others bind more indiscriminately. However, a quantitative understanding of how the mode of binding influences the assembly of actin networks in confined environments is lacking. Here we employ coarse-grained computer simulations to study the dynamics and organization of semiflexible actin filaments in confined regions upon the addition of cross-linkers. We characterize how the emergent behavior is influenced by the system shape, the number and type of cross-linking proteins, and the length of filaments. Structures include isolated clusters of filaments, highly connected filament bundles, and networks of interconnected bundles and loops. Elongation of one dimension of the system promotes the formation of long bundles that align with the elongated axis. Dynamics are governed by rapid cross-linking into aggregates, followed by a slower change in their shape and connectivity. Cross-linking decreases the average bending energy of short or sparsely connected filaments by suppressing shape fluctuations. However, it increases the average bending energy in highly connected networks because filament bundles become deformed, and small numbers of filaments exhibit long-lived, highly unfavorable configurations. Indiscriminate cross-linking promotes the formation of high-energy configurations due to the increased likelihood of unfavorable, difficult-to-relax configurations at early times. Taken together, this work demonstrates physical mechanisms by which cross-linker binding and physical confinement impact the emergent behavior of actin networks, which is relevant both in cells and in synthetic environments.     

Isodynamic axisymmetric equilibrium near the magnetic axis

Plasma equilibrium near the magnetic axis of an axisymmetric toroidal magnetic confinement system is described in orthogonal flux coordinates. For the case of a constant current density in the vicinity of the axis and magnetic surfaces with nearly circular cross sections, expressions for the poloidal and toroidal magnetic field components are obtained in these coordinates by using expansion in the reciprocal of the aspect ratio. These expressions allow one to easily derive relationships between quantities in an isodynamic equilibrium, in which the absolute value of the magnetic field is constant along the magnetic surface (Palumbo’s configuration).     

Atomistic simulations of liquid–liquid coexistence in confinement: comparison of thermodynamics and kinetics with bulk

We review a few simulation methods and results related to the structure and non-equilibrium dynamics in the coexistence region of immiscible symmetric binary fluids, in bulk as well as under confinement, with special emphasis on the latter. Monte Carlo methods to estimate interfacial tensions for flat and curved interfaces have been discussed. The latter, combined with a thermodynamic integration technique, provides contact angles for coexisting fluids attached to the wall. For such three-phase coexistence, results for the line tension are also presented. For the kinetics of phase separation, various mechanisms and corresponding theoretical expectations have been discussed. A comparative picture between the domain growth in bulk and confinement (including thin-film and semi-infinite geometry) has been presented from molecular dynamics simulations. Applications of finite-size scaling technique have been discussed in both equilibrium and non-equilibrium contexts.