Combinatorial formulas for Kazhdan-Lusztig polynomials with respect to W-graph ideals

In \cite{y1} Yin generalized the definition of $W$-graph ideal $E_J$ in weighted Coxeter groups and introduced the weighted Kazhdan-Lusztig polynomials $\left \{ P_{x,y} \mid x,y\in E_J\right \}$, where $J$ is a subset of simple generators $S$. In this paper, we study the combinatorial formulas for those polynomials, which extend the results of Deodhar \cite{v3} and Tagawa \cite{h1}.Comment: 16 page

WB analysis showing the reactivity of <i>Aotus</i> immunised with different mixtures of mHABPs.

<p><b>A.</b> Reactivity with CSP-derived <b>25608</b>, <b>32958</b>, and TRAP <b>24246</b> mHABP mixtures 140 days after first immunisation (80 days after 3^rd or III80) showing reactivity in 2/8 <i>Aotus</i> with CSP-rI. <b>B.</b> Reactivity with CSP-derived <b>25608</b>, <b>32958</b>, and TRAP <b>24254</b> and <b>24246</b> mHABP mixtures evidencing reactivity with CSP-rII and with TRAP rII in 5/6 <i>Aotus</i> monkeys <b>C.</b> Reactivity with CSP-derived rI <b>25608</b>, rII <b>32958</b>, TRAP rII <b>24254</b>, MSP-2 <b>24112,</b> EBA-175 RI <b>13790</b> and rII <b>24292</b> and AMA-1 <b>10022</b>.</p

mHABP electron density. Side-chain atoms for p2, p3, p5, p7 and p8 in 25608.37 and 10022.43.

<p>The figure shows the side-chains for upwardly-oriented residues pointing to TCR-contacting residues. Polar amino acids present in p2, p5 and p8, displaying their non-bonding free electron pairs and π orbital surfaces shown in blurred red while σ orbitals for apolar ones present in p3 and p7 are shown by yellow surfaces. The φ angles (≤ -64°) in p7P (<b>25608.37</b>), p5P (<b>32958.2</b>), p7P (<b>24254.31</b>), p3P and p8P (<b>10022.43</b>) oriented this residue to make contact with the TCR.</p

Spz- and Mrz-derived mHABP fragments binding to the HLA-DRβ1* PBR.

<p>Side view of mHABPs binding to the HLA-DRβ1* PBR (as assessed by ¹H-NMR) displaying the residues according to the colour code: p1 (fuchsia), p2 (red), p3 (pale blue), p4 (dark blue), p5 (pink), p6 (orange), p7 (grey), p8 (yellow) and p9 (green). The dotted balls in light green represent the nonbonding free electron pairs able to establish H bonds with the HLA-DRβ1*PBR residues. The pink planes mark peptide bonds.</p

Steric-electron effects. 25608.37 (left panel) and 10022.43 (right panel) mHABP residues, displaying hybrid sigma (σ) orbitals (yellow), π and p orbitals perpendicular to them (red, blurred balloons).

<p>For <b>25608.37</b>: <b>A</b> and <b>B</b>. Phe1 displaying π resonance (red—bonds between p orbitals); Ser2 tetrahedron with the two free electron pairs (indicated) showing only the σ orbitals. <b>C</b>. Leu3 (green), showing the tetrahedron framing Cδ1 side-chain and orientation (pointing upwards), only Gly4 plane is shown. <b>D</b>. Glu5 (green), showing the tetrahedron framing Cɣ and trigon with Cδ and resonance between the two O and their corresponding Cδ from the side-chain (blurred red balloons); the electron charge is shown in blurred red orbitals. <b>E</b>. Asn 6 directed to Pocket 6, showing the tetrahedron, a trigon and the electron charge in blurred red. <b>F</b>. Pro 7 in grey and two trigons in green. <b>G</b>. Asn 8 directed toward the TCR with its corresponding p orbitals and its non-bonding free electron pair; Ala9 is also shown in green with a tetrahedron in the same colour. For <b>10022.43</b>: <b>H</b>. Phe 1 orientation, π resonance and its planes corresponding to peptide bonds. <b>I</b>. His2 showing the π resonance tiara. <b>J</b>. Pro3 cyclic structure with σ orbitals pointing upwards to contact the TCR, and also displaying the tetrahedron formed by the Ser4 side-chain pointing downwards. <b>K</b>. showing the tetrahedrons formed for Gly5 and Ser7 with its two free electron pairs. <b>L</b>. Pro 8 σ electrons pointing upwards and Val9, showing the two tetrahedrons framing Cδ1 and Cδ2 and their apolarity represented in σ-bonds.</p

Representative torsion angles (Φ, Ψ, χ1, χ2, χ3 and χ4) in mHABPs involved in mixtures immunised in <i>Aotus</i> monkeys.

<p>Grey shows the PPII_L region, yellow shows the β-turn region and green an α-helix conformation. Purple box, χ1 angles for p3 and p7, and χ2 angles for p5, highlighting their <i>gauche+</i> rotamer orientation. The colours of residues vertically displayed in each mHABP correspond to the code for <b><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0123249#pone.0123249.g003" target="_blank">Fig 3</a></b>.</p

IMPIPS: The <b>Im</b>mune <b>P</b>rotection-<b>I</b>nducing <b>P</b>rotein <b>S</b>tructure Concept in the Search for Steric-Electron and Topochemical Principles for Complete Fully-Protective Chemically Synthesised Vaccine Development

<div><p>Determining immune protection-inducing protein structures (IMPIPS) involves defining the stereo-electron and topochemical characteristics which are essential in MHC-p-TCR complex formation. Modified high activity binding peptides (mHABP) were thus synthesised to produce a large panel of IMPIPS measuring 26.5 ±3.5Å between the farthest atoms fitting into Pockets 1 to 9 of HLA-DRβ1* structures. They displayed a polyproline II-like (PPII_L) structure with their backbone O and N atoms orientated to establish H-bonds with specific residues from HLA-DRβ1*-peptide binding regions (PBR). Residues having specific charge and <i>gauche⁺</i> orientation regarding p3χ1, p5χ2, and p7χ1 angles determined appropriate rotamer orientation for perfectly fitting into the TCR to induce an appropriate immune response. Immunological assays in <i>Aotus</i> monkeys involving IMPIPS mixtures led to promising results; taken together with the aforementioned physicochemical principles, non-interfering, long-lasting, protection-inducing, multi-epitope, multistage, minimal subunit-based chemically-synthesised peptides can be designed against diseases scourging humankind.</p></div

Geographic localization of the five populations included in this study (modified from [49]).

<p>Geographic localization of the five populations included in this study (modified from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0014705#pone.0014705-Cervantes1" target="_blank">[49]</a>).</p

Number of Observed and Expected data according to a Negative Binomial distribution.

<p>*Data with p<0.05 are shown in bold types.</p><p>**Atypical squamous cells of undetermined significance (ASCUS).</p>†<p>Low squamous intraepithelial lesion (LSIL).</p>‡<p>High squamous intraepithelial lesion (HSIL).</p

Negative binomial regression model.

<p>*CI: Confidence Intervals.</p><p>**Percent change in expected count for unit increase in X.</p