Unique Extension Research Articles

Extensions of 2-homogeneous polynomials on c 0

We prove that every 2-homogeneous polynomial on the complex c 0 has a unique normpreserving extension to its bidual l∞.

A Characterization of Box-Mengerian Matroid Ports

Let M be a matroid on E∪{l}, where l ∉ E is a distinguished element of M. The l-port of M is the set 𝒫= {P: P ⊆ E with P∪{l} a circuit of M}. Let A be the 𝒫-E incidence matrix. Let U2, 4 be the uniform matroid on four elements of rank two, let F7 be the Fano matroid, let F7* be the dual of F7, and let F7+ be the unique series extension of F7. In this paper, we prove that the system Ax≥1, x≥0 is box-totally dual integral (box-TDI) if and only if M has no U2, 4-minor using l, no F7*-minor using l, and no F7+-minor using l as a series element. Our characterization yields a number of interesting results in combinatorial optimization.

Notions of Sameness by Default and their Application to Anaphora, Vagueness, and Uncertain Reasoning

We motivate and formalize the idea of sameness by default: two objects are considered the same if they cannot be proved to be different. This idea turns out to be useful for a number of widely different applications, including natural language processing, reasoning with incomplete information, and even philosophical paradoxes. We consider two formalizations of this notion, both of which are based on Reiter's Default Logic. The first formalization is a new relation of indistinguishability that is introduced by default. We prove that the corresponding default theory has a unique extension, in which every two objects are indistinguishable if and only if their non-equality cannot be proved from the known facts. We show that the indistinguishability relation has some desirable properties: it is reflexive, symmetric, and, while not transitive, it has a transitive "flavor." The second formalization is an extension (modification) of the ordinary language equality by a similar default: two objects are equal if and only if their non-equality cannot be proved from the known facts. It appears to be less elegant from a formal point of view. In particular, it gives rise to multiple extensions. However, this extended equality is better suited for most of the applications discussed in this paper.

On Convexity of Generalized Wigner–Yanase–Dyson Information

The convexity of the Wigner–Yanase–Dyson information, as first proved by Lieb, is a deep and fundamental result because it leads to the strong subadditivity of quantum entropy. The Wigner–Yanase–Dyson information is a particular kind of quantum Fisher information with important applications in quantum estimation theory. But unlike the quantum entropy, which is the unique natural quantum extension of the classical Shannon entropy, there are many different variants of quantum Fisher information, and it is desirable to investigate their convexity. This article is devoted to studying the convexity of a direct generalization of the Wigner–Yanase–Dyson information. Some sufficient conditions are obtained, and some necessary conditions are illustrated. In a particular case, a surprising necessary and sufficient condition is obtained. Our results reveal the intricacy and subtlety of the convexity issue for general quantum Fisher information.

StAR-like Activity and Molten Globule Behavior of StARD6, A Male Germ-Line Protein

The steroidogenic acute regulatory protein (StAR) belongs to a family of 15 StAR-related lipid transfer (START) domain proteins termed StARD1-StARD15. StAR (StARD1) induces adrenal and gonadal steroidogenesis by moving cholesterol from the outer mitochondrial membrane to the inner mitochondrial membrane by an unclear process that involves conformational changes that have been characterized as a molten globule transition. We expressed, purified, and assessed the activity and cholesterol-binding behavior of StARD1 and StARD3-D7, showing that StARD6 had activity equal to StARD1, whereas StARD4, D5, and D7 had little or no activity with adrenal mitochondria in vitro. Partial proteolysis examined by mass spectrometry suggests that StARD6 has a protease-sensitive C-terminus, similar to but smaller than that of StARD1. Experiments using urea denaturation, stopped-flow kinetics and measurements of mitochondrial membrane association suggests that StARD1 and StARD6 both unfold and refold slowly with similar kinetic patterns. Isothermal titration calorimetry suggests that StARD6 interacts with mitochondrial membranes as well as or better than StARD1. Computational modeling of StARD6 suggests that it has a similar fold to StARD1, with a hydrophobic sterol-binding pocket and a unique C-terminal extension. StARD6, which is expressed only in male germ-line cells, thus exhibits biological and biophysical properties that imply a role in steroidogenesis.

Chemically Induced and Light-Independent Cryptochrome Photoreceptor Activation

The cryptochrome photoreceptors of higher plants are dimeric proteins. Their N-terminal photosensory domain mediates dimerization, and the unique C-terminal extension (CCT) mediates signaling. We made use of the human FK506-binding protein (FKBP) that binds with high affinity to rapamycin or rapamycin analogs (rapalogs). The FKBP-rapamycin complex is recognized by another protein, FRB, thus allowing rapamycin-induced dimerization of two target proteins. Here we demonstrate by bioluminescence resonance energy transfer (BRET) assays the applicability of this regulated dimerization system to plants. Furthermore, we show that fusion proteins consisting of the C-terminal domain of Arabidopsis cryptochrome 2 fused to FKBP and FRB and coexpressed in Arabidopsis cells specifically induce the expression of cryptochrome-controlled reporter and endogenous genes in darkness upon incubation with the rapalog. These results demonstrate that the activation of cryptochrome signal transduction can be chemically induced in a dose-dependent fashion and uncoupled from the light signal, and provide the groundwork for gain-of-function experiments to study specifically the role of photoreceptors in darkness or in signaling cross-talk even under light conditions that activate members of all photoreceptor families.

A Flexible Peptide Tether Controls Accessibility of a Unique C-terminal RNA-binding Domain in Leucyl-tRNA Synthetases

A unique C-terminal domain extension is required by most leucyl-tRNA synthetases (LeuRS) for aminoacylation. In one exception, the enzymatic activity of yeast mitochondrial LeuRS is actually impeded by its own C-terminal domain. It was proposed that the yeast mitochondrial LeuRS has compromised its aminoacylation activity to some extent and adapted its C terminus for a second role in RNA splicing, which is also essential. X-ray crystal structures of the LeuRS–tRNA complex show that the 60 residue C-terminal domain is tethered to the main body of the enzyme via a flexible peptide linker and allows interactions with the tRNALeu elbow. We hypothesized that this short peptide linker would facilitate rigid body movement of the C-terminal domain as LeuRS transitions between an aminoacylation and editing complex or, in the case of yeast mitochondrial LeuRS, an RNA splicing complex. The roles of the C-terminal linker peptide for Escherichia coli and yeast mitochondrial LeuRS were investigated via deletion mutagenesis as well as by introducing chimeric swaps. Deletions within the C-terminal linker of E. coli LeuRS determined that its length, rather than its sequence, was critical to aminoacylation and editing activities. Although deletions in the yeast mitochondrial LeuRS peptide linker destabilized the protein in general, more stable chimeric enzymes that contained an E. coli LeuRS C-terminal domain showed that shortening its tether stimulated aminoacylation activity. This suggested that limiting C-terminal domain accessibility to tRNALeu facilitates its role in protein synthesis and may be a unique adaptation of yeast mitochondrial LeuRS that accommodates its second function in RNA splicing.

Unique Hydrophobic Extension of the RGS2 Amphipathic Helix Domain Imparts Increased Plasma Membrane Binding and Function Relative to Other RGS R4/B Subfamily Members

RGS2 and RGS5 are inhibitors of G-protein signaling belonging to the R4/B subfamily of RGS proteins. We here show that RGS2 is a much more potent attenuator of M1 muscarinic receptor signaling than RGS5. We hypothesize that this difference is mediated by variation in their ability to constitutively associate with the plasma membrane (PM). Compared with full-length RGS2, the RGS-box domains of RGS2 and RGS5 both show reduced PM association and activity. Prenylation of both RGS-box domains increases activity to RGS2 levels, demonstrating that lipid bilayer targeting increases RGS domain function. Amino-terminal domain swaps confirm that key determinants of localization and function are found within this important regulatory domain. An RGS2 amphipathic helix domain mutant deficient for phospholipid binding (L45D) shows reduced PM association and activity despite normal binding to the M1 muscarinic receptor third intracellular loop and activated Galpha(q). Replacement of a unique dileucine motif adjacent to the RGS2 helix with corresponding RGS5 residues disrupts both PM localization and function. These data suggest that RGS2 contains a hydrophobic extension of its helical domain that imparts high efficiency binding to the inner leaflet of the lipid bilayer. In support of this model, disruption of membrane phospholipid composition with N-ethylmaleimide reduces PM association of RGS2, without affecting localization of the M1 receptor or Galpha(q). Together, these data indicate that novel features within the RGS2 amphipathic alpha helix facilitate constitutive PM targeting and more efficient inhibition of M1 muscarinic receptor signaling than RGS5 and other members of the R4/B subfamily.

Gαq Directly Activates p63RhoGEF and Trio via a Conserved Extension of the Dbl Homology-associated Pleckstrin Homology Domain

The coordinated cross-talk from heterotrimeric G proteins to Rho GTPases is essential during a variety of physiological processes. Emerging data suggest that members of the Galpha(12/13) and Galpha(q/11) families of heterotrimeric G proteins signal downstream to RhoA via distinct pathways. Although studies have elucidated mechanisms governing Galpha(12/13)-mediated RhoA activation, proteins that functionally couple Galpha(q/11) to RhoA activation have remained elusive. Recently, the Dbl-family guanine nucleotide exchange factor (GEF) p63RhoGEF/GEFT has been described as a novel mediator of Galpha(q/11) signaling to RhoA based on its ability to synergize with Galpha(q/11) resulting in enhanced RhoA signaling in cells. We have used biochemical/biophysical approaches with purified protein components to better understand the mechanism by which activated Galpha(q) directly engages and stimulates p63RhoGEF. Basally, p63RhoGEF is autoinhibited by the Dbl homology (DH)-associated pleckstrin homology (PH) domain; activated Galpha(q) relieves this autoinhibition by interacting with a highly conserved C-terminal extension of the PH domain. This unique extension is conserved in the related Dbl-family members Trio and Kalirin and we show that the C-terminal Rho-specific DH-PH cassette of Trio is similarly activated by Galpha(q).

Mucinous carcinoma of Vater's ampulla with a unique extension along the main pancreatic duct

We report a case of mucinous carcinoma of Vater's ampulla with a unique extension along only the main pancreatic duct (MPD) and microinvasion to the pancreas. A 52-year-old man was referred to our hospital for the evaluation and treatment of acute pancreatitis. Abdominal computed tomography (CT) demonstrated swelling in the head of the pancreas with a mass in the duodenum. Hypotonic duodenography and endoscopic examination revealed a well-defined mass, measuring about 25 mm in size, in Vater's ampulla. A biopsy specimen of the tumor showed moderately differentiated adenocarcinoma. A pylorus-preserving pancreaticoduodenectomy with a regional lymphadenectomy was performed, under a preoperative diagnosis of adenocarcinoma of Vater's ampulla with direct invasion into the head of the pancreas. The resected specimen of the duodenum confirmed the presence of the mass, which measured 22 x 15 mm in size, in Vater's ampulla. Microscopically, the tumor consisted of two components: moderately differentiated adenocarcinoma in the peripheral region of the tumor Vater's papilla and mucinous carcinoma in the central region of the tumor. The mucinous carcinoma component uniquely extended along only the MPD with microinvasion to the pancreas. Immunohistochemically, both the moderately differentiated adenocarcinoma and the mucinous carcinoma were positive for cytokeratin 20 (CK20) and negative for cytokeratin 7 (CK7) which is the pattern of intestinal-type carcinoma of Vater's ampulla. We concluded that the original site of this tumor may have been the duodenal epithelium of Vater's ampulla originally moderately differentiated adenocarcinoma-which subsequently changed to mucinous carcinoma that extended along only the MPD with microinvasion to the pancreas.

ATP-driven MalK Dimer Closure and Reopening and Conformational Changes of the “EAA” Motifs Are Crucial for Function of the Maltose ATP-binding Cassette Transporter (MalFGK2)

We have investigated conformational changes of the purified maltose ATP-binding cassette transporter (MalFGK(2)) upon binding of ATP. The transport complex is composed of a heterodimer of the hydrophobic subunits MalF and MalG constituting the translocation pore and of a homodimer of MalK, representing the ATP-hydrolyzing subunit. Substrate is delivered to the transporter in complex with periplasmic maltose-binding protein (MalE). Cross-linking experiments with a variant containing an A85C mutation within the Q-loop of each MalK monomer indicated an ATP-induced shortening of the distance between both monomers. Cross-linking caused a substantial inhibition of MalE-maltose-stimulated ATPase activity. We further demonstrated that a mutation affecting the "catalytic carboxylate" (E159Q) locks the MalK dimer in the closed state, whereas a transporter containing the "ABC signature" mutation Q140K permanently resides in the resting state. Cross-linking experiments with variants containing the A85C mutation combined with cysteine substitutions in the conserved EAA motifs of MalF and MalG, respectively, revealed close proximity of these residues in the resting state. The formation of a MalK-MalG heterodimer remained unchanged upon the addition of ATP, indicating that MalG-EAA moves along with MalK during dimer closure. In contrast, the yield of MalK-MalF dimers was substantially reduced. This might be taken as further evidence for asymmetric functions of both EAA motifs. Cross-linking also caused inhibition of ATPase activity, suggesting that transporter function requires conformational changes of both EAA motifs. Together, our data support ATP-driven MalK dimer closure and reopening as crucial steps in the translocation cycle of the intact maltose transporter and are discussed with respect to a current model.

Functional conservation between structurally diverse ribosomal proteins from Drosophila melanogaster and Saccharomyces cerevisiae: fly L23a can substitute for yeast L25 in ribosome assembly and function.

The proposed Drosophila melanogaster L23a ribosomal protein features a conserved C-terminal amino acid signature characteristic of other L23a family members and a unique N-terminal extension [Koyama et al. (Poly(ADP-ribose) polymerase interacts with novel Drosophila ribosomal proteins, L22 and l23a, with unique histone-like amino-terminal extensions. Gene 1999; 226: 339–345)], absent from Saccharomyces cerevisiae L25 that nearly doubles the size of fly L23a. The ability of fly L23a to replace the role of yeast L25 in ribosome biogenesis was determined by creating a yeast strain carrying an L25 chromosomal gene disruption and a plasmid-encoded FLAG-tagged L23a gene. Though affected by a reduced growth rate, the strain is dependent on fly L23a-FLAG function for survival and growth, demonstrating functional compatibility between the fly and yeast proteins. Pulse-chase experiments reveal a delay in rRNA processing kinetics, most notably at a late cleavage step that converts precursor 27S rRNA into mature 25S rRNA, likely contributing to the strain's slower growth pattern. Yet, given the essential requirement for L23(a)/L25 in ribosome biogenesis, there is a remarkable tolerance for accommodating the fly L23a N-terminal extension within the structure of the yeast ribosome. A search of available databases shows that the unique N-terminal extension is shared by multiple insect lineages. An evolutionary perspective on L23a structure and function within insect lineages is discussed.

MOTIVIC INVARIANTS OF ARTIN STACKS AND 'STACK FUNCTIONS'

An invariant I of quasiprojective K-varieties X with values in a commutative ring R is if I(X)= I(Y)+I(X\Y) for Y closed in X, and I(X x Y)=I(X)I(Y). Examples include Euler characteristics chi and virtual Poincare and Hodge polynomials. We first define a unique extension I' of I to finite type Artin K-stacks F, which is motivic and satisfies I'([X/G])=I(X)/I(G) when X is a K-variety, G a K-group acting on X, and [X/G] is the quotient stack. This only works if I(G) is invertible in R for all special K-groups G, which excludes I=chi as chi(K*)=0. But we can extend the construction to get round this. Then we develop the theory of on Artin stacks. These are a universal generalization of constructible functions on Artin stacks, as studied in the author's paper math.AG/0403305. There are several versions of the construction: the basic one SF(F), and variants SF(F,I,R),... twisted by motivic invariants. We associate a Q-vector space SF(F) or an R-module SF(F,I,R) to each Artin stack F, with functorial operations of multiplication, pullbacks phi^* and pushforwards phi_* under 1-morphisms phi : F --> G, and so on. They will be important tools in the author's series on Configurations in abelian categories, math.AG/0312190, math.AG/0503029, math.AG/0410267 and math.AG/0410268.

Generalized Latin squares and their defining sets

A generalized Latin square of type ( n , k ) is an n × n array of symbols 1 , 2 , … , k such that each of these symbols occurs at most once in each row and each column. Let d ( n , k ) denote the cardinality of the minimal set S of given entries of an n × n array such that there exists a unique extension of S to a generalized Latin square of type ( n , k ) . In this paper we discuss the properties of d ( n , k ) for k = 2 n - 1 and k = 2 n - 2 . We give an alternate proof of the identity d ( n , 2 n - 1 ) = n 2 - n , which holds for even n , and we establish the new result d ( n , 2 n - 2 ) ⩾ n 2 - ⌊ 8 n 5 ⌋ . We also show that the latter bound is tight for n divisible by 10.

β-Subunit appendages promote 20S proteasome assembly by overcoming an Ump1-dependent checkpoint

Proteasomes are responsible for most intracellular protein degradation in eukaryotes. The 20S proteasome comprises a dyad-symmetric stack of four heptameric rings made from 14 distinct subunits. How it assembles is not understood. Most subunits in the central pair of beta-subunit rings are synthesized in precursor form. Normally, the beta5 (Doa3) propeptide is essential for yeast proteasome biogenesis, but overproduction of beta7 (Pre4) bypasses this requirement. Bypass depends on a unique beta7 extension, which contacts the opposing beta ring. The resulting proteasomes appear normal but assemble inefficiently, facilitating identification of assembly intermediates. Assembly occurs stepwise into precursor dimers, and intermediates contain the Ump1 assembly factor and a novel complex, Pba1-Pba2. beta7 incorporation occurs late and is closely linked to the association of two half-proteasomes. We propose that dimerization is normally driven by the beta5 propeptide, an intramolecular chaperone, but beta7 addition overcomes an Ump1-dependent assembly checkpoint and stabilizes the precursor dimer.

Fractal interpolation on the Sierpinski Gasket

We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V 0 = { p 1 , p 2 , p 3 } be the set of vertices of SG and u i ( x ) = 1 2 ( x + p i ) the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations u w = u w 1 u w 2 ⋯ u w n for any sequence w = ( w 1 , w 2 , … , w n ) ∈ { 1 , 2 , 3 } n . The union of the images of V 0 under these iterations is the set of nth stage vertices V n of SG. Let F : V n → R be any function. Given any numbers α w ( w ∈ { 1 , 2 , 3 } n ) with 0 < | α w | < 1 , there exists a unique continuous extension f : SG → R of F, such that f ( u w ( x ) ) = α w f ( x ) + h w ( x ) for x ∈ SG , where h w are harmonic functions on SG for w ∈ { 1 , 2 , 3 } n . Interpreting the harmonic functions as the “degree 1 polynomials” on SG is thus a self-similar interpolation obtained for any start function F : V n → R .

M. tuberculosis Ser/Thr Protein Kinase D Phosphorylates an Anti-Anti–Sigma Factor Homolog

Receptor Ser/Thr protein kinases are candidates for sensors that govern developmental changes and disease processes of Mycobacterium tuberculosis (Mtb), but the functions of these kinases are not established. Here, we show that Mtb protein kinase (Pkn) D overexpression alters transcription of numerous bacterial genes, including Rv0516c, a putative anti-anti–sigma factor, and genes regulated by sigma factor F. The PknD kinase domain directly phosphorylated Rv0516c, but no other sigma factor regulator, in vitro. In contrast, the purified PknB and PknE kinase domains phosphorylated distinct sigma regulators. Rather than modifying a consensus site, PknD phosphorylated Rv0516c in vitro and in vivo on Thr2 in a unique N-terminal extension. This phosphorylation inhibited Rv0516c binding in vitro to a homologous anti-anti–sigma factor, Rv2638. These results support a model in which signals transmitted through PknD alter the transcriptional program of Mtb by stimulating phosphorylation of a sigma factor regulator at an unprecedented control site.

On the extensions of homogeneous polynomials

We investigate the problem of the uniqueness of the extension of n-homogeneous polynomials in Banach spaces. We show in particular that in a nonreflexive Banach space X that admits contractive projection of finite rank of at least dimension 2, for every n > 3 there exists an n-homogeneous polynomial on X that has infinitely many extensions to X**. We also prove that under some geometric conditions imposed on the norm of a complex Banach lattice E, for instance when E satisfies an upper p-estimate with constant one for some p > 2, any 2-homogeneous polynomial on E attaining its norm at x ∈ E with a finite rank band projection P x , has a unique extension to its bidual E**. We apply these results in a class of Orlicz sequence spaces.

Obstacles to factorization of partial differential operators into several factors

In the paper, construction of algorithms for factoring linear partial differential operators is studied. New theoretical concepts--obstacle and ring of obstacles--are introduced. These are invariants and posses other interesting properties. A theorem on unique factorization extension starting from some moment, which is important from the standpoint of construction of such algorithms, is proved. Obstacles in the case of the second and third degrees are found.

Second-order superintegrable quantum systems

A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n − 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, then the system is second-order superintegrable, the most tractable case and the one we study here. Such systems have remarkable properties: multi-integrability and separability, a quadratic algebra of symmetries whose representation theory yields spectral information about the Schrodinger operator, and deep connections with expansion formulas relating classes of special functions. For n = 2 and for conformally flat spaces when n = 3, we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here, we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension.