Resonances and circuit theory for the interaction of metallic disks and annuli with an electromagnetic field.

To understand the nature of the electromagnetic resonances of finite metallic surfaces, we formulate a rigorous and rapidly convergent circuit theory for the interaction of a metallic disk and a metallic annulus with an electromagnetic field. Expressions for the current induced and the resonance condition are derived. A new understanding of the nature of the resonances is obtained. For half of the resonances we find a divergent electric field at the edge of the disk, even though it is smooth in shape. For the disk, we compare with previous results using vector spheroidal wave functions and found good agreement for the resonance condition. Our approach can be generalized to other finite surfaces.

Crystallization mechanisms of hemoglobin C in the R state.

Crystallization of the mutated hemoglobin, HbC, which occurs inside red blood cells of patients expressing betaC-globin and exhibiting the homozygous CC and the heterozygous SC (in which two mutant beta-globins, S and C, are expressed) diseases, is a convenient model for processes underlying numerous condensation diseases. As a first step, we investigated the molecular-level mechanisms of crystallization of this protein from high-concentration phosphate buffer in its stable carbomonoxy form using high-resolution atomic force microscopy. We found that in conditions of equilibrium with the solution, the crystals' surface reconstructs into four-molecule-wide strands along the crystallographic a (or b) axis. However, the crystals do not grow by the alignment of such preformed strands. We found that the crystals grow by the attachment of single molecules to suitable sites on the surface. These sites are located along the edges of new layers generated by two-dimensional nucleation or by screw dislocations. During growth, the steps propagate with random velocities, with the mean being an increasing function of the crystallization driving force. These results show that the crystallization mechanisms of HbC are similar to those found for other proteins. Therefore, strategies developed to control protein crystallization in vitro may be applicable to pathology-related crystallization systems.

Dissipating step bunches during crystallization under transport control.

In studies of crystal formation by the generation and spreading of layers, equidistant step trains are considered unstable-bunches and other spatiotemporal patterns of the growth steps are viewed as ubiquitous. We provide an example to the opposite. We monitor the spatiotemporal dynamics of steps and the resulting step patterns during crystallization of the proteins ferritin and apoferritin using the atomic force microscope. The variations in step velocity and density are not correlated, indicating the lack of a long-range attraction between the steps. We show that (i) because of its coupling to bulk transport, nucleation of new layers is chaotic and occurs at the facet edges, where the interfacial supersaturation is higher; (ii) step bunches self-organize via the competition for supply from the solution; and, (iii) bunches of weakly interacting steps decay as they move along the face. Tests by numerical modeling support the conclusions about the mechanisms underlying our observations. The results from these systems suggest that during crystallization controlled by transport, with weakly or noninteracting growth steps, the stable kinetic state of the surface is an equidistant step train, and step bunches only arise during nucleation of new layers. Since nucleation only occurs at a few sites on the surface, the surface morphology may be controllably patterned or smoothened by locally controlling nucleation.

Direct observation of nucleus structure and nucleation pathways in apoferritin crystallization.

Using atomic force microscopy (AFM) in situ during the crystallization of the protein apoferritin from its solution, we imaged the arrangement of the molecules in near-critical clusters, larger or smaller than the crystal nucleus, that are representative of the nucleus structure. At supersaturations Delta mu/k(B)T of 1.1 -- 1.6 -- 2.3, the nuclei contain about 50 -- 20 -- 10 molecules. The molecular arrangement within the nuclei is similar to that in the crystal bulk. Contrary to the general belief, the observed nuclei are not compact molecular clusters, but are planar arrays of several rods of 4--7 molecules set in one or two monomolecular layers. Similarly unexpected nuclei structures might be common, especially for anisotropic molecules. Hence, the nucleus structure should be considered as a variable by advanced theoretical treatments.

Molecular mechanisms of microheterogeneity-induced defect formation in ferritin crystallization.

We apply in situ atomic force microscopy to the crystallization of ferritins from solutions containing approximately 5% (w/w) of their inherent molecular dimers. Molecular resolution imaging shows that the dimers consist of two bound monomers. The constituent monomers are likely partially denatured, resulting in increased hydrophobicity of the dimer surface. Correspondingly, the dimers strongly adsorb on the crystal surface. The adsorbed dimers hinder step growth and on incorporation by the crystal initiate stacks of up to 10 triple and single vacancies in the subsequent crystal layers. The molecules around the vacancies are shifted by approximately 0.1 molecular dimensions from their crystallographic positions. The shifts strain the lattice and, as a consequence, at crystal sizes > 200 microm, the accumulated strain is resolved by a plastic deformation whereupon the crystal breaks into mosaic blocks 20-50 microm in size. The critical size for the onset of mosaicity is similar for ferritin and apoferritin and close to the value for a third protein, lysozyme; it also agrees with theoretical predictions. Trapped microcrystals in ferritin and apoferritin induce strain with a characteristic length scale equal to that of a single point defect, and, as a consequence, trapping does not contribute to the mosaicity. The sequence of undesired phenomena that include heterogeneity generation, adsorption, incorporation, and the resulting lattice strain and mosaicity in this and other proteins systems, could be avoided by improved methods to separate similar proteins species (microheterogeneity) or by increasing the biochemical stability of the macromolecules against oligomerization.

Molecular-level thermodynamic and kinetic parameters for the self-assembly of apoferritin molecules into crystals.

The self-assembly of apoferritin molecules into crystals is a suitable model for protein crystallization and aggregation; these processes underlie several biological and biomedical phenomena, as well as for protein and virus self-assembly. We use the atomic force microscope in situ, during the crystallization of apoferritin to visualize and quantify at the molecular level the processes responsible for crystal growth. To evaluate the governing thermodynamic parameters, we image the configuration of the incorporation sites, "kinks", on the surface of a growing crystal. We show that the kinks are due to thermal fluctuations of the molecules at the crystal-solution interface. This allows evaluation of the free energy of the intermolecular bond phi=3.0 k(B)T=7.3 kJ/mol. The crystallization free energy, extracted from the protein solubility, is -42 kJ/mol. Published determinations of the second virial coefficient and the protein solubility between 0 and 40 degrees C revealed that the enthalpy of crystallization is close to zero. Analyses based on these three values suggest that the main component in the crystallization driving force is the entropy gain of the water molecules bound to the protein molecules in solution and released upon crystallization. Furthermore, monitoring the incorporation of individual molecules in to the kinks, we determine the characteristic frequency of attachment of individual molecules at one set of conditions. This allows a correlation between the mesoscopic kinetic coefficient for growth and the molecular-level thermodynamic and kinetic parameters determined here. We found that step growth velocity, scaled by the molecular size, equals the product of the kink density and attachment frequency, i.e. the latter pair are the molecular-level parameters for self-assembly of the molecules into crystals.

Quasi-planar nucleus structure in apoferritin crystallization.

First-order phase transitions of matter, such as condensation and crystallization, proceed through the formation and subsequent growth of 'critical nuclei' of the new phase. The thermodynamics and kinetics of the formation of these critical nuclei depend on their structure, which is often assumed to be a compact, three-dimensional arrangement of the constituent molecules or atoms. Recent molecular dynamics simulations have predicted compact nucleus structures for matter made up of building blocks with a spherical interaction field, whereas strongly anisotropic, dipolar molecules may form nuclei consisting of single chains of molecules. Here we show, using direct atomic force microscopy observations, that the near-critical-size clusters formed during the crystallization of apoferritin, a quasi-spherical protein, and which are representative of the critical nucleus of this system, consist of planar arrays of one or two monomolecular layers that contain 5-10 rods of up to 7 molecules each. We find that these clusters contain between 20 and 50 molecules each, and that the arrangement of the constituent molecules is identical to that found in apoferritin crystals. We anticipate that similarly unexpected critical nucleus structures may be quite common, particularly with anisotropic molecules, suggesting that advanced nucleation theories should treat the critical nucleus structure as a variable.

Holomorphic curves in surfaces of general type.

This note answers some questions on holomorphic curves and their distribution in an algebraic surface of positive index. More specifically, we exploit the existence of natural negatively curved "pseudo-Finsler" metrics on a surface S of general type whose Chern numbers satisfy c(2)1>2c2 to show that a holomorphic map of a Riemann surface to S whose image is not in any rational or elliptic curve must satisfy a distance decreasing property with respect to these metrics. We show as a consequence that such a map extends over isolated punctures. So assuming that the Riemann surface is obtained from a compact one of genus q by removing a finite number of points, then the map is actually algebraic and defines a compact holomorphic curve in S. Furthermore, the degree of the curve with respect to a fixed polarization is shown to be bounded above by a multiple of q - 1 irrespective of the map.

Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity.

We find some integrability conditions for low-dimensional manifolds to admit metrics with nonnegative scalar curvature. In particular, we solve the positive action conjecture in general relativity in the affirmative.

Incompressible minimal surfaces, three-dimensional manifolds with nonnegative scalar curvature, and the positive mass conjecture in general relativity.

We study three-dimensional Riemannian manifolds with nonnegative scalar curvature. We find new topological obstruction for such manifolds. Our method turns out to be useful in studying the positive mass conjecture in general relativity.

Calabi's conjecture and some new results in algebraic geometry.

We announce a proof of Calabi's conjectures on the Ricci curvature of a compact Kähler manifold and then apply it to prove some new results in algebraic geometry and differential geometry. For example, we prove that the only Kähler structure on a complex projective space is the standard one.

On the structure of complete simply-connected Kähler manifolds with nonpositive curvature.

We prove that a complete simply-connected Kähler manifold with nonpositive sectional curvature is biholomorphic to the complex Euclidean space if the curvature is suitably small at infinity.

On the Fundamental Group of Manifolds of Non-positive Curvature.

We prove theorems on the structure of the fundamental group of a compact riemannian manifold of non-positive curvature. In particular, a conjecture of J. Wolf [J. Differential Geometry, 2, 421-446 (1968)] is proved.