L2 Domain Research Articles

Modeling the Epidermal Growth Factor—Epidermal Growth Factor Receptor L2 Domain Interaction: Implications for the Ligand Binding Process

Signaling from the epidermal growth factor (EGF) receptor is triggered by the binding of lig-ands such as EGF or transforming growth factor alpha (TGF-α) and subsequent receptor dimerization. An understanding of these processes has been hindered by the lack of structural information about the ligand-bound, dimerized EGF receptor. Using an NMR-derived structure of EGF and a homology model of the major ligand binding domain of the EGF receptor and experimental data, we modeled the binding of EGF to this EGF receptor fragment. In this low resolution model of the complex, EGF sits across the second face of the EGF receptor L2 domain and EGF residues 10–16, 36–37, 40–47 bind to this face. The structural model is largely consistent with previously published NMR data describing the residues of TGF-α which interact strongly with the EGF receptor. Other EGF residues implicated in receptor binding are accounted by our proposal that the ligand binding is a two-step process with the EGF binding to at least one other site of the receptor. This three-dimensional model is expected to be useful in the design of ligand-based antagonists of the receptor.

Interaction between the individual isoenzymes of pyruvate dehydrogenase kinase and the inner lipoyl-bearing domain of transacetylase component of pyruvate dehydrogenase complex

Protein-protein interactions play an important role in the regulation of enzymic activity of pyruvate dehydrogenase kinase (PDK). It is generally believed that the binding of PDK to the inner lipoyl-bearing domain L2 of the transacetylase component E2 of pyruvate dehydrogenase complex largely determines the level of kinase activity. In the present study, we characterized the interaction between the individual isoenzymes of PDK (PDK1-PDK4) and monomeric L2 domain of human E2, as well as the effect of this interaction on kinase activity. It was found that PDK isoenzymes are markedly different with respect to their affinities for L2. PDK3 demonstrated a very tight binding, which persisted during isolation of PDK3-L2 complexes using size-exclusion chromatography. Binding of PDK1 and PDK2 was readily reversible with the apparent dissociation constant of approx. 10 microM for both isoenzymes. PDK4 had a greatly reduced capacity for L2 binding (relative order PDK3>PDK1=PDK2>PDK4). Monomeric L2 domain alone had very little effect on the activities of either PDK1 or PDK2. In contrast, L2 caused a 3-fold increase in PDK3 activity and approx. 37% increase in PDK4 activity. These results strongly suggest that the interactions between the individual isoenzymes of PDK and L2 domain are isoenzyme-specific and might be among the major factors that determine the level of kinase activity of particular isoenzyme towards the pyruvate dehydrogenase complex.

Structural Requirements within the Lipoyl Domain for the Ca2+-dependent Binding and Activation of Pyruvate Dehydrogenase Phosphatase Isoform 1 or Its Catalytic Subunit

The inner lipoyl domain (L2) of the dihydrolipoyl acetyltransferase (E2) 60-mer forms a Ca(2+)-dependent complex with the pyruvate dehydrogenase phosphatase 1 (PDP1) or its catalytic subunit, PDP1c, in facilitating large enhancements of the activities of PDP1 (10-fold) or PDP1c (6-fold). L2 binding to PDP1 or PDP1c requires the lipoyl-lysine prosthetic group and specificity residues that distinguish L2 from the other lipoyl domains (L1 in E2 and L3 in the E3-binding component). The L2-surface structure contributing to binding was mapped by comparing the capacities of well folded mutant or lipoyl analog-substituted L2 domains to interfere with E2 activation by competitively binding to PDP1 or PDP1c. Our results reveal the critical importance of a regional set of residues near the lipoyl group and of the octanoyl but not the dithiolane ring structure of the lipoyl group. At the other end of the lipoyl domain, substitution of Glu(182) by alanine or glutamine removed L2 binding to PDP1 or PDP1c, and these substitutions for the neighboring Glu(179) also greatly hindered complex formation (E179A > E179Q). Among 11 substitutions in L2 at sites of major surface residue differences between the L1 and L2 domains, only the conversion of Val-Gln(181) located between the critical Glu(179) and Glu(182) to the aligned Ser-Leu sequence of the L1 domain greatly reduced L2 binding. Certain modified L2 altered E2 activation of PDP1 differently than PDP1c, supporting significant impact of the regulatory PDP1r subunit on PDP1 binding to L2. Our results indicate hydrophobic binding via the extended aliphatic structure of the lipoyl group and required adjacent L2 structure anchor PDP1 by acting in concert with an acidic cluster at the other end of the domain.

THE HEAT FLOW OF THE CCR ALGEBRA

Let Pf(x) =−if′(x) and Qf(x) = xf(x) be the canonical operators acting on an appropriate common dense domain in L2(ℝ). The derivations DP(A) = i(PA−AP) and DQ(A) = i(QA−AQ) act on the *-algebra [Ascr ] of all integral operators having smooth kernels of compact support, for example, and one may consider the noncommutative ‘Laplacian’, L = D2P+D2Q, as a linear mapping of [Ascr ] into itself.L generates a semigroup of normal completely positive linear maps on [Bscr ](L2(ℝ)), and this paper establishes some basic properties of this semigroup and its minimal dilation to an E0-semigroup. In particular, the author shows that its minimal dilation is pure and has no normal invariant states, and he discusses the significance of those facts for the interaction theory introduced in a previous paper.There are similar results for the canonical commutation relations with n degrees of freedom, where 1 [les ] n < 1.

Identification of a determinant of epidermal growth factor receptor ligand-binding specificity using a truncated, high-affinity form of the ectodomain.

Murine and human epidermal growth factor receptors (EGFRs) bind human EGF (hEGF), mouse EGF (mEGF), and human transforming growth factor alpha (hTGF-alpha) with high affinity despite the significant differences in the amino acid sequences of the ligands and the receptors. In contrast, the chicken EGFR can discriminate between mEGF (and hEGF) and hTGF-alpha and binds the EGFs with approximately 100-fold lower affinity. The regions responsible for this poor binding are known to be Arg(45) in hEGF and the L2 domain in the chicken EGFR. In this study we have produced a truncated form of the hEGFR ectodomain comprising residues 1-501 (sEGFR501), which, unlike the full-length hEGFR ectodomain (residues 1-621, sEGFR621), binds hEGF and hTGF-alpha with high affinity (K(D) = 13-21 and 35-40 nM, respectively). sEGFR501 was a competitive inhibitor of EGF-stimulated mitogenesis, being almost 10-fold more effective than the full-length EGFR ectodomain and three times more potent than the neutralizing anti-EGFR monoclonal antibody Mab528. Analytical ultracentrifugation showed that the primary EGF binding sites on sEGFR501 were saturated at an equimolar ratio of ligand and receptor, leading to the formation of a 2:2 EGF:sEGFR501 dimer complex. We have used sEGFR501 to generate three mutants with single position substitutions at Glu(367), Gly(441), or Glu(472) to Lys, the residue found in the corresponding positions in the chicken EGFR. All three mutants bound hTGF-alpha and were recognized by Mab528. However, mutant Gly(441)Lys showed markedly reduced binding to hEGF, implicating Gly(441), in the L2 domain, as part of the binding site that recognizes Arg(45) of hEGF.

Reaction Mechanism for Mammalian Pyruvate Dehydrogenase Using Natural Lipoyl Domain Substrates

The pyruvate dehydrogenase (E1) component of the pyruvate dehydrogenase complex (PDC) catalyzes a two-step reaction. Recombinant production of substrate amounts of the lipoyl domains of the dihydrolipoyl transacetylase (E2) component of the mammalian PDC allowed kinetic characterization of the rapid physiological reaction catalyzed by E1. Using either the N-terminal (L1) or the internal (L2) lipoyl domain of E2 as a substrate, analyses of steady state kinetic data support a ping pong mechanism. Using standard E1 preparations, Michaelis constants (Km) were 52 ± 14 μM for L1 and 24.8 ± 3.8 μM for pyruvate and kcat was 26.3 s−1. With less common, higher activity preparations of E1, the Km values were ≥160 μM for L1 and ≥35 μM for pyruvate and kcat was ≥70 s−1. Similar results were found with the L2 domain. The best synthetic lipoylated-peptide (L2 residues 163–177) was a much poorer substrate (Km ≥ 15 mM, kcat ≈ 5 s−1; kcat/Km decreased >1500-fold) than L1 or L2, but a far better substrate in the E1 reaction than free lipoamide (kcat/Km increased >500-fold). Each lipoate source was an effective substrate in the dihydrolipoyl dehydrogenase (E3) reaction, but E3 had a lower Km for the L2 domain than for lipoamide or the lipoylated peptides. In contrast to measurements with slow E1 model reactions that use artificial acceptors, we confirmed that the natural E1 reaction, using lipoyl domain acceptors, was completely inhibited (>99%) by phosphorylation of E1 and the phosphorylation strongly inhibited the reverse of the second step catalyzed by E1. The mechanisms by which phosphorylation interferes with E1 activity is interpreted based on accrued results and the location of phosphorylation sites mapped onto the 3-D structure of related α-keto acid dehydrogenases.

The homologous pairing domain of RecA also mediates the allosteric regulation of DNA binding and ATP hydrolysis: a remarkable concentration of functional residues

Switching between the active (ATP and DNA bound) and inactive conformations of the homologous recombination RecA protein is regulated by ATP hydrolysis. First, we use the homologous pairing domain of RecA derived from its mobile loop L2 to show that the interaction of this random coil peptide with the γ-phosphate of ATP results in a peptide β-conformation similar to that previously shown to be induced by DNA binding. Next, we show that in the whole RecA protein two residues in this L2 domain, Gln194 and Arg196, are catalytic amino acid residues for ATP hydrolysis and functionally resemble the corresponding residues engaged in GTP hydrolysis by two distinct classes of G proteins. Finally, we show that the role of DNA and high salt in the stimulation of the ATPase of RecA is to stabilize this highly mobile region involved in hydrolysis. This is a role similar to that described for RGSs in the activation of the GTPase of heterotrimeric G proteins. Therefore, (i) a prototypical DNA-dependent ATPase and ATP-stimulated DNA-binding protein, RecA, and eukaryotic signaling proteins share common stereochemical regulatory mechanisms; and (ii) in a remarkable example of parsimony, loop L2 is a molecular switch that controls both ATP promoted DNA binding and pairing reactions and DNA stimulated ATP hydrolysis.

Characterization of the L1-Neurocan-binding Site: IMPLICATIONS FOR L1-L1 HOMOPHILIC BINDING

The L1 adhesion molecule is a 200-220-kDa membrane glycoprotein of the Ig superfamily implicated in important neural processes including neuronal cell migration, axon outgrowth, learning, and memory formation. L1 supports homophilic L1-L1 binding that involves several Ig domains but can also bind with high affinity to the proteoglycan neurocan. It has been reported that neurocan can block homophilic binding; however, the mechanism of inhibition and the precise binding sites in both molecules have not been determined. By using fusion proteins, site-directed mutagenesis, and peptide blocking experiments, we have characterized the neurocan-binding site in the first Ig-like domain of human L1. Results from molecular modeling suggest that the sequences involved in neurocan binding are localized on the surface of the first Ig domain and largely overlap with the G-F-C beta-strands proposed to interact with the fourth Ig domain during homophilic binding. This suggests that neurocan may sterically hinder a proper alignment of L1 domains. We find that the C-terminal portion of neurocan is sufficient to mediate binding to the first Ig domain of L1, and we suggest that the sushi domain cooperates with a glycosaminoglycan side chain in forming the binding site for L1.

Specificity Determinants for the Pyruvate Dehydrogenase Component Reaction Mapped with Mutated and Prosthetic Group Modified Lipoyl Domains

Efficient catalysis in the second step of the pyruvate dehydrogenase (E1) component reaction requires a lipoyl group to be attached to a lipoyl domain that displays appropriately positioned specificity residues. As substrates, the human dihydrolipoyl acetyltransferase provides an N-terminal (L1) and an inner (L2) lipoyl domain. We evaluated the specificity requirements for the E1 reaction with 27 mutant L2 (including four substitutions for the lipoylated lysine, Lys(173)), with three analogs substituted for the lipoyl group on Lys(173), and with selected L1 mutants. Besides Lys(173) mutants, only E170Q mutation prevented lipoylation. Based on analysis of the structural stability of mutants by differential scanning calorimetry, alanine substitutions of residues with aromatic side chains in terminal regions outside the folded portion of the L2 domain significantly decreased the stability of mutant L2, suggesting specific interactions of these terminal regions with the folded domain. E1 reaction rates were markedly reduced by the following substitutions in the L2 domain (equivalent site-L1): L140A, S141A (S14A-L1), T143A, E162A, D172N, and E179A (E52A-L1). These mutants gave diverse changes in kinetic parameters. These residues are spread over >24 A on one side of the L2 structure, supporting extensive contact between E1 and L2 domain. Alignment of over 40 lipoyl domain sequences supports Ser(141), Thr(143), and Glu(179) serving as specificity residues for use by E1 from eukaryotic sources. Extensive interactions of the lipoyl-lysine prosthetic group within the active site are supported by the limited inhibition of E1 acetylation of native L2 by L2 domains altered either by mutation of Lys(173) or enzymatic addition of lipoate analogs to Lys(173). Thus, efficient use by mammalian E1 of cognate lipoyl domains derives from unique surface residues with critical interactions contributed by the universal lipoyl-lysine prosthetic group, key specificity residues, and some conserved residues, particularly Asp(172) adjacent to Lys(173).

Marked Differences between Two Isoforms of Human Pyruvate Dehydrogenase Kinase

Pyruvate dehydrogenase kinase (PDK) isoforms 2 and 3 were produced via co-expression with the chaperonins GroEL and GroES and purified with high specific activities in affinity tag-free forms. By using human components, we have evaluated how binding to the lipoyl domains of the dihydrolipoyl acetyltransferase (E2) produces the predominant changes in the rates of phosphorylation of the pyruvate dehydrogenase (E1) component by PDK2 and PDK3. E2 assembles as a 60-mer via its C-terminal domain and has mobile connections to an E1-binding domain and then two lipoyl domains, L2 and L1 at the N terminus. PDK3 was activated 17-fold by E2; the majority of this activation was facilitated by the free L2 domain (half-maximal activation at 3.3 microm L2). The direct activation of PDK3 by the L2 domain resulted in a 12.8-fold increase in k(cat) along with about a 2-fold decrease in the K(m) of PDK3 for E1. PDK3 was poorly inhibited by pyruvate or dichloroacetate (DCA). PDK3 activity was stimulated upon reductive acetylation of L1 and L2 when full activation of PDK3 by E2 was avoided (e.g. using free lipoyl domains or ADP-inhibited E2-activated PDK3). In marked contrast, PDK2 was not responsive to free lipoyl domains, but the E2-60-mer enhanced PDK2 activity by 10-fold. E2 activation of PDK2 resulted in a greatly enhanced sensitivity to inhibition by pyruvate or DCA; pyruvate was effective at significantly lower levels than DCA. E2-activated PDK2 activity was stimulated >/=3-fold by reductive acetylation of E2; stimulated PDK2 retained high sensitivity to inhibition by ADP and DCA. Thus, PDK3 is directly activated by the L2 domain, and fully activated PDK3 is relatively insensitive to feed-forward (pyruvate) and feed-back (acetylating) effectors. PDK2 was activated only by assembled E2, and this activated state beget high responsiveness to those effectors.

The insulin receptor: from protein sequence to structure.

Sequences of the insulin receptor (IR), the type-I insulin-like growth-factor receptor (IGFR) and the insulin-receptor-related receptor show that they belong to a homologous family but, until recently, have given few clues about their structures. Three repeats of fibronectin type III have been identified close to the membrane. Although the N-terminal L1, Cys-rich and L2 domains of the IGFR have been identified from their sequences and their structures determined by X-ray crystallography, little is known of their relative positions in the complete receptor in vivo. Here, we ask what can be learnt further from the analysis of sequences, about the structure, organization and function of the extracellular regions of the IR family.

Members of the Insulin Receptor Family Contain Three Fibronectin Type III Domains

The insulin receptor is a large transmembrane dimer, comprised of several domains. Detailed 3D structural information is available for the L1-cys-rich-L2 domains in the extracellular region (ectodomain) and for the tyrosine kinase catalytic domain in the cytoplasmic portion of the receptor. In addition, previous sequence analyses have identified two fibronectin type III domains in the C-terminal half of each ectodomain monomer. In this report, evidence is provided to show that a third fibronectin type III module exists between the L2 domain and the two previously described fibronectin type III domains.

Requirements for the Adaptor Protein Role of Dihydrolipoyl Acetyltransferase in the Up-regulated Function of the Pyruvate Dehydrogenase Kinase and Pyruvate Dehydrogenase Phosphatase

The dihydrolipoyl acetyltransferase (E2 component) is a 60-mer assembled via its COOH-terminal domain with exterior E1-binding domain and two lipoyl domains (L2 then L1) sequentially connected by mobile linker regions. E2 facilitates markedly enhanced function of the pyruvate dehydrogenase kinase (PDK) and pyruvate dehydrogenase phosphatase (PDP). Human E2 structures were prepared with only one lipoyl domain (L1 or L2) or with alanines substituted at the sites of lipoylation (Lys-46 in L1 or Lys-173 in L2). The L2 domain and its lipoyl group were shown to be essential for markedly enhanced PDP function and were required for greatly up-regulated PDK function. The complete absence of the L1 domain reduced the enhancements of both of these activities but not the maximal effector-stimulated PDK activity through acetylation of L2. With nonlipoylated L2 present, lipoylated L1 supported a lesser enhancement in PDK function with significant stimulation upon acetylation of L1. Prevention of L1 lipoylation in K46AE2 removed this competitive L1 role and enhanced L2-facilitated PDK activity beyond that of native E2 when PDK activity was measured in the absence or in the presence of stimulatory effectors. Thus, the E2-L2 domain has a paramount role in facilitating enhanced PDK and PDP function but inclusion of E2-L1 domain, even in a noninteracting (nonlipoylated) form, contributes to the marked elevation of these activities.

Comparative Phase Behavior About the L2 Phase of Ternary and Quaternary Systems of Triton X-100 and Its Separated p-tert-OPEn (n = 5, 7, and 9) Components in Cyclohexane

The nonionic surfactant Triton X-100 (TX-100) is separated by preparative column chromatography into its poly(oxyethylene tert-octylphenyl ether) (OPEn) components. The phase behavior in the neighborhood of the L2 phase is investigated for ternary (cyclohexane/H2O) and quaternary (cyclohexane/H2O/n-hexanol) systems of TX-100 and OPEn (n = 5, 7 and 9). The extent of the L2 domain and the maximum amount of water wo,max that can be solubilized by isotropic solutions strongly depend on the concentration and the length (n) of the polar chain of the surfactant, temperature, and the amount of n-hexanol present. Depending on these variables, with increasing wo (molar ratio of water to surfactant) some of the following phase transitions occur: L2 → LC → L → liquid/gel → gel. Except for the liquid crystalline LC phase, all phases involved are optically isotropic, and the liquid L phase is most likely bicontinuous. Since n-hexanol is a better solvent than cyclohexane for both the surfactant and water, it destabiliz...

Activated Function of the Pyruvate Dehydrogenase Phosphatase through Ca2+-facilitated Binding to the Inner Lipoyl Domain of the Dihydrolipoyl Acetyltransferase

Micromolar Ca2+ facilitates approximately 10-fold enhancement of pyruvate dehydrogenase phosphatase (PDP) activity by aiding the association of PDP with the dihydrolipoyl acetyltransferase (E2) component. Connected by linker regions, E2 consists of two lipoyl domains, the NH2-lipoyl domain (L1) and the interior lipoyl domain (L2), and a pyruvate dehydrogenase component binding domain surrounding a 60-mer inner core. Using recombinant constructs of L1 or L2, E2-enhanced PDP activity was markedly decreased by L2 but not by L1, effectively competing with intact E2 in Ca2+-dependent binding of PDP (half-maximal reduction at 2.0 microM L2 versus 6.7 microM E2 subunit). Using L2 fused to glutathione S-transferase resulted in direct Ca2+-dependent binding of PDP to L2 (Kd, approximately 1.7 microM L2). Affinity-bound glutathione S-transferase-L2 was used to purify PDP to homogeneity by selective binding and elution by Ca2+ chelation. The large activity enhancement of PDP by E2 was eliminated by enzymatic removal of lipoates from E2 and restored by their enzymatic reintroduction. The critical role of the L2 lipoate is not in binding of PDP to E2, since PDP was still bound by delipoylated L2, and delipoylated L2 inhibited E2-enhanced PDP activity, although lipoylated L2 was more effective in each of these tests. Thus, pyruvate dehydrogenase complex activity is increased by enhanced availability of PDP to its E2-bound, phosphorylated pyruvate dehydrogenase substrate as a consequence of the Ca2+-facilitated interchange of PDP among the mobile L2 domains and an essential (undetermined) step engaging the L2 lipoate.

Outline structure of the human L1 cell adhesion molecule and the sites where mutations cause neurological disorders.

The L1 cell adhesion molecule has six domains homologous to members of the immunoglobulin superfamily and five homologous to fibronectin type III domains. We determined the outline structure of the L1 domains by showing that they have, at the key sites that determine conformation, residues similar to those in proteins of known structure. The outline structure describes the relative positions of residues, the major secondary structures and residue solvent accessibility. We use the outline structure to investigate the likely effects of 22 mutations that cause neurological diseases. The mutations are not randomly distributed but cluster in a few regions of the structure. They can be divided into those that act mainly by changing conformation or denaturing their domain and those that alter its surface properties.

Lipoyl Domain-based Mechanism for the Integrated Feedback Control of the Pyruvate Dehydrogenase Complex by Enhancement of Pyruvate Dehydrogenase Kinase Activity

To conserve carbohydrate reserves, the reaction of the pyruvate dehydrogenase complex (PDC) must be down-regulated when the citric acid cycle is provided sufficient acetyl-CoA. PDC activity is reduced primarily through increased phosphorylation of its pyruvate dehydrogenase (E1) component due to E1 kinase activity being markedly enhanced by elevated intramitochondrial NADH:NAD+ and acetyl-CoA:CoA ratios. A mechanism is evaluated in which enhanced kinase activity is facilitated by the build-up of the reduced and acetylated forms of the lipoyl moieties of the dihydrolipoyl acetyltransferase (E2) component through using NADH and acetyl-CoA in the reverse of the downstream reactions of the complex. Using a peptide substrate, kinase activity was stimulated by these products, ruling out the possibility kinase activity is increased due to changes in the reaction state of its substrate, E1 (thiamin pyrophosphate). Each E2 subunit contains two lipoyl domains, an NH2-terminal (L1) and the inward lipoyl domain (L2), which were individually produced in fully lipoylated forms by recombinant techniques. Although reduction and acetylation of the L1 domain or free lipoamide increased kinase activity, those modifications of the lipoate of the kinase-binding L2 domain gave much greater enhancements of kinase activity. The large stimulation of the kinase generated by acetyl-CoA only occurred upon addition of the transacetylase-catalyzing (lipoyl domain-free) inner core portion of E2 plus a reduced lipoate source, affirming that acetylation of this prosthetic group is an essential mechanistic step for acetyl-CoA enhancing kinase activity. Similarly, the lesser stimulation of kinase activity by just NADH required a lipoate source, supporting the need for lipoate reduction by E3 catalysis. Complete enzymatic delipoylation of PDC, the E2-kinase subcomplex, or recombinant L2 abolished the stimulatory effects of NADH and acetyl-CoA. Retention of a small portion of PDC lipoates lowered kinase activity but allowed stimulation of this residual kinase activity by these products. Reintroduction of lipoyl moieties, using lipoyl protein ligase, restored the capacity of the E2 core to support high kinase activity along with stimulation of that activity up to 3-fold by NADH and acetyl-CoA. As suggested by those results, the enhancement of kinase activity is very responsive to reductive acetylation with a half-maximal stimulation achieved with approximately 20% of free L2 acetylated and, from an analysis of previous results, with acetylation of only 3-6 of the 60 L2 domains in intact PDC. Based on these findings, we suggest that kinase stimulation results from modification of the lipoate of an L2 domain that becomes specifically engaged in binding the kinase. In conclusion, kinase activity is attenuated through a substantial range in response to modest changes in the proportion of oxidized, reduced, and acetylated lipoyl moieties of the L2 domain of E2 produced by fluctuations in the NADH:NAD+ and acetyl-CoA:CoA ratios as translated by the rapid and reversible E3 and E2 reactions.

Gauge theory on a non-simply connected domain and representations of canonical commutation relations

A quantum system of a particle interacting with a (non-Abelian) gauge field on the non-simply connected domain M=R2\{an}Nn=1 is considered, where an, n=1,...,N, are fixed isolated points in R2. The gauge potential A of the gauge field is a p×p anti-Hermitian matrix-valued 1-form on M, and may be strongly singular at the points an, n=1,...,N. If A is flat, then the physical momentum and the position operators {Pj,qj}2j=1 of the particle satisfy the canonical commutation relations (CCR) with two degrees of freedom on a suitable dense domain of the Hilbert space L2(R2;Cp). A necessary and sufficient condition for this representation to be the Schrödinger 2-system is given in terms of the Wilson loops of the rectangles not intersecting an, n=1,...,N. This also gives a characterization for the representation to be non-Schrödinger. It is proven that, for a class of gauge potentials, which is not necessarily flat, Pj is essentially self-adjoint. Moreover, an example, which gives a class of non-Schrödinger representations of the CCR with two degrees of freedom, is discussed in some detail.

Some inequalities for sub-Markovian generators and their applications to the perturbation theory

We characterize the domain in L p {L^p} -space of generators of submarkovian semigroups in terms of the form domain in L 2 {L^2} and give the corresponding inequality. Using this inequality we obtain a criterion for the formal difference A − B A - B of such generators to be a generator of a contraction semigroup in L p {L^p} . The conditions on perturbation are expressed in terms of forms, i.e., in L 2 {L^2} -terms.

Fréchet vs. Gâteaux differentiability of Lipschitzian functions

Examples have been given of Lipschitzian functions that are Gâteaux-differentiable everywhere, but nowhere Fréchet-differentiable. One such example has been reported, mistakenly, in several papers as having domain in L 2 ( [ 0 , π ] ) {L^2}([0,\pi ]) , when it should have been L 1 ( [ 0 , π ] ) {L^1}([0,\pi ]) . We discuss this example.