Υπολογισμός της εντροπίας διεμπλοκής σε συστήματα ταλαντωτών και στη θεωρία πεδίου

Στην παρούσα διπλωματική εργασία διαπραγματευόμαστε το θέμα της εντροπίας διεμπλοκής σε συστήματα συζευγμένων ταλαντωτών και σε βαθμωτή θεωρία πεδίου. Ξεκινάμε αναλύοντας την έννοια της κβαντικής διεμπλοκής και ένα μέτρο της, την εντροπία διεμπλοκής. Έπειτα, βρίσκουμε την εντροπία διεμπλοκής σε συστήματα 2 και N συζευγμένων ταλαντωτών με σκοπό να γενικευθούν αυτά τα αποτελέσματα στη θεωρία πεδίου. Στη συνέχεια βρίσκουμε την εντροπία διεμπλοκής στη θεωρία πεδίου και δείχνουμε ότι η κύρια συνεισφορά σε αυτήν είναι ανάλογη του εμβαδού της διαχωριστικής επιφάνειας που ορίζει τα δύο υπό μελέτη διεμπλεγμένα υποσυστήματα. Έπειτα μελετάμε ένα τριχοτομημένο σύστημα και βρίσκουμε ότι και σε αυτό η κύρια συνεισφορά στην εντροπία διεμπλοκής είναι ένας όρος εμβαδού, του οποίου ο συντελεστής είναι ο ίδιος με αυτόν του διχοτομημένου συστήματος που μελετήσαμε προηγουμένως. Μετά βρίσκουμε αριθμητικά τους όρους υποδεέστερης τάξης στο τριχοτομημένο σύστημα στις 2+1 διαστάσεις, προκειμένου να επιβεβαιώσουμε μία βασική ιδιότητα της εντροπίας διεμπλοκής, την ισχυρή υποαθροιστικότητα. Τέλος, αναφέρουμε κάποιες μελλοντικές προεκτάσεις που μπορούν να έχουν αυτά τα θέματα.In this thesis, we study entanglement entropy in systems of coupled harmonic oscillators and in scalar field theory. First we introduce the notion of quantum entanglement and a measure of it, namely the entanglement entropy. Then, we calculate entanglement entropy in systems of 2 and N coupled harmonic oscillators in order to generalize these results in field theory. Later, we calculate the entanglement entropy in field theory and show that the main contribution to it is proportional to the area of the separating surface that defines the two entangled subsystems under study. After that we investigate a tripartite system in 2+1 dimensions and find that the main contribution to entanglement entropy is again an area law term whose coefficient is identical to that of the bipartite system which we studied earlier. Finally, we numerically calculate the subleading terms in the tripartite system in order to verify a fundamental property of entanglement entropy, namely the strong subadditivity. We also discuss some possible extensions of this work

Activation of a vinculin-binding site in the talin rod involves re-arrangement of a five helix bundle

Talin is a conserved cytoskeletal protein consisting of two subdomains: the talin head and the talin rod. Talin physically connects integrins to the actin cytoskeleton and regulates the inside-out activation of integrins. Binding partners of the talin head include integrins, focal adhesion kinase (FAK) and phosphatidylinositol-4-phosphate 5 kinases type 1 gamma isoform (PIPK1gamma). Along the talin rod, two actin binding sites, one integrin binding site and a minimum of three vinculin-binding sites have so far been mapped.;Vinculin is a cytoskeletal protein, which localises to focal adhesions. It has a molecular weight of 120kDa and consists of two parts: the head and a C-terminal tail. The talin binding site is located in the vinculin head between residues 1-258 (Vh').;In this thesis, the crystal structures of talin rod fragments 482-655 and 482-789 are presented. The structures are classified and compared with structural homologues, demonstrating that the talin rod structural organisation resembles the vinculin tail, alpha-catenin and apolipoproteins A and E. Talin 482-655 is a five helix bundle and talin 656-789 a four helix bundle. All helices are amphipathic and the predicted vinculin-binding site 1 (VBS1) is found buried in the hydrophobic core of the 5-helix bundle of talin 482-655. With supporting evidence from the two structures, mutant binding studies, alanine substitution experiments, NMR spectroscopy, limited proteolysis and the crystal structure of Vh' in complex with talin 605-628, a mechanism involving extensive conformational changes for both talin and vinculin is proposed to be required, in order to achieve vinculin/talin interaction

The crystal structure of the platelet activator aggretin reveals a novel (alphabeta)2 dimeric structure

Aggretin is a C-type lectin purified from Calloselasma rhodostoma snake venom. It is a potent activator of platelets, resulting in a collagen-like response by binding and clustering platelet receptor CLEC-2. We present here the crystal structure of aggretin at 1.7 A which reveals a unique tetrameric quaternary structure. The two alphabeta heterodimers are arranged through 2-fold rotational symmetry, resulting in an antiparallel side-by-side arrangement. Aggretin thus presents two ligand binding sites on one surface and can therefore cluster ligands in a manner reminiscent of convulxin and flavocetin. To examine the molecular basis of the interaction with CLEC-2, we used a molecular modeling approach of docking the aggretin alphabeta structure with the CLEC-2 N-terminal domain (CLEC-2N). This model positions the CLEC-2N structure face down in the "saddle"-shaped binding site which lies between the aggretin alpha and beta lectin-like domains. A 2-fold rotation of this complex to generate the aggretin tetramer reveals dimer contacts for CLEC-2N which bring the N- and C-termini into the proximity of each other, and a series of contacts involving two interlocking beta-strands close to the N-terminus are described. A comparison with homologous lectin-like domains from the immunoreceptor family reveals a similar but not identical dimerization mode, suggesting this structure may represent the clustered form of CLEC-2 capable of signaling across the platelet membrane

Activation of a vinculin-binding site in the talin rod involves rearrangement of a five-helix bundle

The interaction between the cytoskeletal proteins talin and vinculin plays a key role in integrin-mediated cell adhesion and migration. We have determined the crystal structures of two domains from the talin rod spanning residues 482–789. Talin 482–655, which contains a vinculin-binding site (VBS), folds into a five-helix bundle whereas talin 656–789 is a four-helix bundle. We show that the VBS is composed of a hydrophobic surface spanning five turns of helix 4. All the key side chains from the VBS are buried and contribute to the hydrophobic core of the talin 482–655 fold. We demonstrate that the talin 482–655 five-helix bundle represents an inactive conformation, and mutations that disrupt the hydrophobic core or deletion of helix 5 are required to induce an active conformation in which the VBS is exposed. We also report the crystal structure of the N-terminal vinculin head domain in complex with an activated form of talin. Activation of the VBS in talin and the recruitment of vinculin may support the maturation of small integrin/talin complexes into more stable adhesions

Kinase Domain Insertions Define Distinct Roles of CLK Kinases in SR Protein Phosphorylation

Splicing requires reversible phosphorylation of serine/arginine-rich (SR) proteins, which direct splice site selection in eukaryotic mRNA. These phosphorylation events are dependent on SR protein (SRPK) and cdc2-like kinase (CLK) families. SRPK1 phosphorylation of splicing factors is restricted by a specific docking interaction whereas CLK activity is less constrained. To understand functional differences between splicing factor targeting kinases, we determined crystal structures of CLK1 and CLK3. Intriguingly, in CLKs the SRPK1 docking site is blocked by insertion of a previously unseen helix αH. In addition, substrate docking grooves present in related mitogen activating protein kinases (MAPKs) are inaccessible due to a CLK specific β7/8-hairpin insert. Thus, the unconstrained substrate interaction together with the determined active-site mediated substrate specificity allows CLKs to complete the functionally important hyperphosphorylation of splicing factors like ASF/SF2. In addition, despite high sequence conservation, we identified inhibitors with surprising isoform specificity for CLK1 over CLK3

Structure of PICK1 and other PDZ domains obtained with the help of self-binding C-terminal extensions

PDZ domains are protein–protein interaction modules that generally bind to the C termini of their target proteins. The C-terminal four amino acids of a prospective binding partner of a PDZ domain are typically the determinants of binding specificity. In an effort to determine the structures of a number of PDZ domains we have included appropriate four residue extensions on the C termini of PDZ domain truncation mutants, designed for self-binding. Multiple truncations of each PDZ domain were generated. The four residue extensions, which represent known specificity sequences of the target PDZ domains and cover both class I and II motifs, form intermolecular contacts in the expected manner for the interactions of PDZ domains with protein C termini for both classes. We present the structures of eight unique PDZ domains crystallized using this approach and focus on four which provide information on selectivity (PICK1 and the third PDZ domain of DLG2), binding site flexibility (the third PDZ domain of MPDZ), and peptide–domain interactions (MPDZ 12th PDZ domain). Analysis of our results shows a clear improvement in the chances of obtaining PDZ domain crystals by using this approach compared to similar truncations of the PDZ domains without the C-terminal four residue extensions