Flat Lattices Research Articles

A relation algebraic model of robust correctness

We propose a new and uniform abstract relational approach to demonic nondeterminism and robust correctness similar to Hoare's chaos semantics. It is based on a specific set of relations on flat lattices. This set forms a complete lattice. Furthermore, we deal with the refinement of programs. Among other things, we show the correctness of the unfold/fold method for demonic nondeterminism and robust correctness as refinement relation and investigate relationships to Dijkstra's wp-calculus and Morgan's specification statement.

Informational Models for the Configurational Entropy of Regular Solid Solutions: Flat Lattices

The present study attempts to improve the results of the well-known quasi-chemical (pair) approximation for systems with short-range interactions. Using an analogy between expressions arising in information theory and in statistical thermodynamics, the procedure of derivation of configurational factors for 2D Ising models was found which guarantees nonnegative sign of the configurational entropy at any set of point and pair probabilities. The present approximation predicts values of critical parameters for 2D Ising models of a ferromagnetic with the accuracy of 1−3% in comparison to the exact results for square, honeycomb, and triangular lattices.

Reduction of a Z(3) gauge theory on the flat lattices to the spin-1 BEG model

The Z(3) gauge model with double plaquette representation of the action on the flat triangular and square lattices is constructed. It is reduced to the spin-1 Blume-Emery-Griffiths (BEG) model. An Ising-type critical line of a second-order phase transition is found.

Character expansion methods for matrix models of dually weighted graphs

We consider generalized one-matrix models in which external fields allow control over the coordination numbers on both the original and dual lattices. We rederive in a simple fashion a character expansion formula for these models originally due to Itzykson and Di Francesco, and then demonstrate how to take the large N limit of this expansion. The relationship to the usual matrix model resolvent is elucidated. Our methods give as a by-product an extremely simple derivation of the Migdal integral equation describing the large $N$ limit of the Itzykson-Zuber formula. We illustrate and check our methods by analyzing a number of models solvable by traditional means. We then proceed to solve a new model: a sum over planar graphs possessing even coordination numbers on both the original and the dual lattice. We conclude by formulating equations for the case of arbitrary sets of even, self-dual coupling constants. This opens the way for studying the deep problem of phase transitions from random to flat lattices.

Quenching 2D quantum gravity

We simulate the Ising model on a set of fixed random ø 3 graphs, which corresponds to a quenched coupling to 2D gravity rather than the annealed coupling that is usually considered. We investigate the critical exponents in such a quenched ensemble and compare them with measurements on dynamical ø 3 graphs, flat lattices and a single fixed ø 3 graph.

Clathrin and HA2 adaptors: effects of potassium depletion, hypertonic medium, and cytosol acidification.

The effects of methods known to perturb endocytosis from clathrin-coated pits on the localization of clathrin and HA2 adaptors in HEp-2 carcinoma cells have been studied by immunofluorescence and ultrastructural immunogold microscopy, using internalization of transferrin as a functional assay. Potassium depletion, as well as incubation in hypertonic medium, remove membrane-associated clathrin lattices: flat clathrin lattices and coated pits from the plasma membrane, and clathrin-coated vesicles from the cytoplasm, as well as those budding from the TGN. In contrast, immunofluorescence microscopy using antibodies specific for the alpha- and beta-adaptins, respectively, and immunogold labeling of cryosections with anti-alpha-adaptin antibodies shows that under these conditions HA2 adaptors are aggregated at the plasma membrane to the same extent as in control cells. After reconstitution with isotonic K(+)-containing medium, adaptor aggregates and clathrin lattices colocalize at the plasma membrane as normally and internalization of transferrin resumes. Acidification of the cytosol affects neither clathrin nor HA2 adaptors as studied by immunofluorescence microscopy. However, quantitative ultrastructural observations reveal that acidification of the cytosol results in formation of heterogeneously sized and in average smaller clathrin-coated pits at the plasma membrane and buds on the TGN. Collectively, our observations indicate that the methods to perturb formation of clathrin-coated vesicles act by different mechanisms: acidification of the cytosol by affecting clathrin-coated membrane domains in a way that interferes with budding of clathrin-coated vesicles from the plasma membrane as well as from the TGN; potassium depletion and incubation in hypertonic medium by preventing clathrin and adaptors from interacting. Furthermore our observations show that adaptor aggregates can exist at the plasma membrane independent of clathrin lattices and raise the possibility that adaptor aggregates can form nucleation sites for clathrin lattices.

The appendage domain of the AP-2 subunit is not required for assembly or invagination of clathrin-coated pits.

Coated pits contain a resident membrane molecule(s) that binds clathrin AP-2 with high affinity. AP-2 binding to this site is likely to be the first step in coated pit assembly because this subunit functions as a template for the polymerization of clathrin into flat polygonal lattices. Integral membrane proteins involved in receptor mediated endocytosis cluster in the newly assembled pits as they invaginate and bud from the membrane. The AP-2 subunit is a multi-domain, molecular complex that can be separated by proteolysis into a brick-shaped core and ear-like appendage domains. We have used this property to identify the domain involved in the various stages of coated pit assembly and budding. We found that the core of AP-2 is the domain that binds both to membranes and to triskelions during assembly. Triskelions are perfectly capable of forming lattices on the membrane bound cores. Clathrin lattices bound only to core domains were also able to invaginate normally. Limited proteolysis was also useful for further characterizing the AP-2 binding site. Elastase treatment of the inside membrane surface released a peptide fraction that is able to bind AP-2 in solution and prevent it from interacting with membranes. Affinity purification of binding activity yielded a collection of peptides that was dominated by a 45-kD species. This is the candidate peptide for containing the AP-2-binding site. Therefore, the appendage domain does not directly participate in any of the assembly or invagination events required for coated pit function.

Multicritical phases of the O( n) model on a random lattice

We exhibit the multicritical phase structure of the loop gas model on a random surface. The dense phase is reconsidered, with special attention paid to the topological points g = 1/ p. This phase is complementary to the dilute and higher multicritical phases in the sense that dense models contain the same spectrum of bulk operators (found in the continuum by Lian and Zuckerman) but a different set of boundary operators. This difference illuminates the well-known ( p, q) asymmetry of the matrix chain models. Higher multicritical phases are constructed, generalizing both Kazakov's multicritical models as well as the known dilute phase models. They are quite likely related to multicritical polymer theories recently considered independently by Saleur and Zamolodchikov. Our results may be of help in defining such models on flat honeycomb lattices; an unsolved problem in polymer theory. The phase boundaries correspond again to “topological” points with g = p/1 integer, which we study in some detail. Two qualitatively different types of critical points are discovered for each such g. For the special point g = 2 we demonstrate that the dilute phase O(−2) model does not correspond to the Parisi-Sourlas model, a result likely to hold as well for the flat case. Instead it is proven that the first multicritical O(−2) point possesses the Parisi-Sourlas supersymmetry.

Translocation of the glucose transporter (GLUT4) to the cell surface in permeabilized 3T3-L1 adipocytes: effects of ATP insulin, and GTP gamma S and localization of GLUT4 to clathrin lattices.

Insulin stimulates the movement of two glucose transporter isoforms (GLUT1 and GLUT4) to the plasma membrane (PM) in adipocytes. To study this process we have prepared highly purified PM fragments by gently sonicating 3T3-L1 adipocytes grown on glass coverslips. Using confocal laser immunofluorescence microscopy we observed increased PM labeling for GLUT1 (2.3-fold) and GLUT4 (eightfold) after insulin treatment in intact cells. EM immunolabeling of PM fragments indicated that in the nonstimulated state GLUT4 was mainly localized to flat clathrin lattices. Whereas GLUT4 labeling of clathrin lattices was only slightly increased after insulin treatment, labeling of uncoated PM regions was markedly increased with insulin. These data suggest that GLUT4 recycles from the cell surface both in the presence and absence of insulin. In streptolysin-O permeabilized adipocytes, insulin, and GTP gamma S increased PM levels of GLUT4 to a similar extent as observed with insulin in intact cells. In the absence of an exogenous ATP source the magnitude of these effects was considerably reduced. Removal of ATP per se caused a significant increase in cell surface levels of GLUT4 suggesting that ATP may be required for intracellular sequestration of these transporters. When insulin and GTP gamma S were added together, in the presence of ATP, PM GLUT4 levels were similar to levels observed when either insulin or GTP gamma S was added individually. Addition of GTP gamma S was able to overcome this ATP dependence of insulin-stimulated GLUT4 movement. GTP gamma S had no effect on constitutive secretion of adipsin in permeabilized cells. In addition, there was no effect of insulin or GTP gamma S on GLUT4 movement to the PM in noninsulin sensitive streptolysin-O-permeabilized 3T3-L1 fibroblasts overexpressing GLUT4. We conclude that the insulin-stimulated movement of GLUT4 to the cell surface in adipocytes may require ATP early in the insulin signaling pathway and a GTP-binding protein(s) at a later step(s). We propose that the association of GLUT4 with clathrin lattices may be important in maintaining the exclusive intracellular location of this transporter in the absence of insulin.

Transferrin receptors promote the formation of clathrin lattices

Gold conjugates have been used to quantitate human transferrin receptors (hTfnRs) on transfected chick embryo fibroblasts. No relationship could be found between the number of hTfnRs and the number of clathrincoated pits. However, hTfnRs are also associated with flat clathrin lattices that lie outside invaginated pits. With increasing levels of receptor expression, the density of hTfnRs within flat lattices increases, and at the highest levels of expression the total area of flat lattice increases up to 3-fold. These results show that increased receptor numbers can promote clathrin lattice growth and suggest that the recruitment of receptors like hTfnRs is an essential step in lattice construction. We conclude that the process of invagination, which gives rise to coated pits, is regulated separately.

Can LCF be topped? Flat lattice models of typed λ-calculus

Plotkin ((1977) Theoret. Comput. Sci. 5 : 223–256) examines the denotational semantics of PCF (essentially typed λ-calculus with arithmetic and looping). The standard Scott semantics V is computationally adequate but not fully abstract; with the addition of some parallel facilities, it becomes fully abstract, and with the addition of an existential operator, denotationally universal. We consider carrying out the same program for ⊙, the Scott models built from flat lattices rather than flat cpo's. Surprisingly, no computable extension of PCF can be denotationally universal; perfectly reasonable semantic values such as supremum and Plotkin's “parallel or” cannot be definable. There is an unenlightening fully abstract extension L A ( approx ), based on Gödel numbering and syntactic analysis. Unfortunately, this is the best we can do; operators defined by PCF-style rules cannot give a fully abstract language. (There is a natural and desirable property, operational extensionality , which prevents full abstraction with respect to ⊙.) However, we show that Plotkin's program can be carried out for a nonconfluent evaluator.

Effects of cytoplasmic acidification on clathrin lattice morphology.

Reducing the internal pH of cultured cells by several different protocols that block endocytosis is found to alter the structure of clathrin lattices on the inside of the plasma membrane. Lattices curve inward until they become almost spherical yet remain stubbornly attached to the membrane. Also, the lattices bloom empty "microcages" of clathrin around their edges. Correspondingly, broken-open cells bathed in acidified media demonstrate similar changes in clathrin lattices. Acidification accentuates the normal tendency of lattices to round up in vitro and also stimulates them to nucleate microcage formation from pure solutions of clathrin. On the other hand, several conditions that also inhibit endocytosis have been found to create, instead of unusually curved clathrin lattices with extraneous microcages, a preponderance of unusually flat lattices. These treatments include pH-"clamping" cells at neutrality with nigericin, swelling cells with hypotonic media, and sticking cells to the surface of a culture dish with soluble polylysine. Again, the unusually flat lattices in such cells display a tendency to round up and to nucleate clathrin microcage formation during subsequent in vitro acidification. This indicates that regardless of the initial curvature of clathrin lattices, they all display an ability to grow and increase their curvature in vitro, and this is enhanced by lowering ambient pH. Possibly, clathrin lattice growth and curvature in vivo may also be stimulated by a local drop in pH around clusters of membrane receptors.

Clathrin assembly proteins: affinity purification and a model for coat assembly.

Assembly protein (AP) preparations from bovine brain coated vesicles have been fractionated by clathrin-Sepharose affinity chromatography. Two distinct fractions that possess coat assembly activity were obtained and are termed AP-1 and AP-2. The AP-1, not retained on the resin, has principal components with molecular weights of 108,000, 100,000, 47,000, and 19,000. The AP-2, bound to the resin and eluted by Tris-HCl at a concentration that parallels the latter's effect on coat disassembly, corresponds to the active complex described previously (Zaremba, S., and J. H. Keen, 1983, J. Cell Biol., 97:1339-1347). Its composition is similar to that of the AP-1 in that it contains 100,000-, 50,000-, and 16,000-mol-wt polypeptides in equimolar amounts; minor amounts of 112,000- and 115,000-mol-wt polypeptides are also present. Both are distinct from a recently described assembly protein of larger subunit molecular weight that we term AP-3. These results indicate the existence of a family of assembly proteins within cells. On incubation with clathrin both AP-1 and AP-2 induce the formation of coat structures, those containing AP-1 slightly smaller (mean diameter = 72 nm) than those formed in the presence of AP-2 (mean diameter = 79 nm); both structures have been detected previously in coated vesicle preparations from brain. Coats formed in the presence of AP-2 consistently contain approximately one molecule each of the 100,000-, 50,000-, and 16,000-mol-wt polypeptides per clathrin trimer. By low angle laser light scattering the molecular weight of native AP-2 was determined to be approximately 343,000, indicating that it is a dimer of each of the three subunits, and implying that it is functionally bivalent in clathrin binding. A model for AP-mediated coat assembly is proposed in which a bivalent AP-2 molecule bridges the distal legs or terminal domains of two clathrin trimers that are destined to occupy adjacent vertices in the assembled coat. Binding of a second AP-2 molecule locks these two trimers in register for assembly and further addition of AP-2 to free trimer legs promotes completion of the clathrin lattice. Effects of AP binding on the angle and flexibility of the legs at the hub of the trimer (the "pucker") are suggested to account for the characteristic size distributions of coats formed under varied conditions and, more speculatively, to contribute to the transformation of flat clathrin lattices to curved coated vesicles that are thought to occur during endocytosis.

Mathematical modelling of the formation of coated vesicles. A theoretical geometrical approach.

The mechanism by which flat hexagonal lattices of clathrin trimers transform into pentagonal/hexagonal spheres remains a mystery. In light of the geometrical nature of this process we have pursued a mathematical approach to the question. Through the geometrical analysis of flat hexagonal lattices we have discovered three possible forms of transformation to introduce curvature into the centre of the lattice: hub-centre transformation; hub-edge transformation; fringe transformation. Hub-edge and fringe transformations are used first to close the lattice while introducing localized curvature at the edges of the lattice. Hub-centre transformation is used after closure to relax the severely localized curvature generated during closure. This scheme not only maximizes the size of the coated vesicle generated, but also minimizes the number of transformations, thus minimizing the energy expended.

Clathrin lattice reorganization: theoretical considerations

We wish to postulate a mechanism by which flat hexagonal lattices of clathrin trimers transform into coated pits. Using an established model for packing trimers into lattices, we explored the assembly process by single addition of trimers to form polygons. Subject to favorable conditions, removal of a single trimer from a hexagon could lead to the formation of a pentagon. Elimination of trimers from polygonal sheets can occur either at the center of the network or at the edges. Removal of a trimer from the center of these adjacent polygons, "hub transformation," is possible in very few instances, whereas removal from the edges of a polygonal sheet, "fringe transformation," is possible in a host of cases. These hypothetical constructs can be used effectively to explain intermediate structures actually observed in flat hexagonal lattices. The geometry of a purely hexagonal lattice seems to dictate that the first step in transformation must be a "fringe transformation," which then will allow subsequent "hub transformation" to take place leading to the introduction of pentagons into the center of the lattice and ultimately to the curvature of the clathrin lattice.

Bearing capacity of a spiral heat exchanger with a rigid core and flat tube lattices

Bearing capacity of a spiral heat exchanger with a rigid core and flat tube lattices