Nuclear-magnetic-resonance quantum calculations of the Jones polynomial

The repertoire of problems theoretically solvable by a quantum computer recently expanded to include the approximate evaluation of knot invariants, specifically the Jones polynomial. The experimental implementation of this evaluation, however, involves many known experimental challenges. Here we present experimental results for a small-scale approximate evaluation of the Jones polynomial by nuclear magnetic resonance (NMR); in addition, we show how to escape from the limitations of NMR approaches that employ pseudopure states. Specifically, we use two spin-1/2 nuclei of natural abundance chloroform and apply a sequence of unitary transforms representing the trefoil knot, the figure-eight knot, and the Borromean rings. After measuring the nuclear spin state of the molecule in each case, we are able to estimate the value of the Jones polynomial for each of the knots

Validating the structural role of Ser52 in HV1-69-sBnAbs.

<p><b>A</b>) F10 V-segment germline variants were analyzed for H5VN04 binding in the phage-Ab (5 scFv/phage) format by MSD ELISA <i>(left)</i> and for their ability to activate B-cell when expressed as B-cell receptors in the presence of H5VN04 <i>(right)</i>. <b>B</b>) HV1-69-sBnAb variants of S52I in F10 and A66, G52aP in CR6331, G17 and D8 were analyzed for H5VN04 reactivity by ELISA. <b>C</b>) Kinetic analysis by Biacore of F10 and A66 CDR-H2 variants against purified H5VN04. Residues colored in blue are non-germline amino acids. <b>D</b>) Circular dichroism measurement of F10 and the non-H5 reactive variant characterized by a germline configured CDR-H2 shows a highly similar CD profile for both constructs.</p

Characterization of HV1-69-sBnAbs VH domain.

<p><b>A</b>) Alignment of 38 published HV1-69-sBnAbs is shown with highlights referring to hydrophobic residues at position 53 (light plum), the conserved Phe54 (dark plum), the occurrence of CDR-H3-Tyr (pink) residues. Other highlights refer to panel <b>B</b>), which describes the result of a Fisher's exact test with Bonferroni adjustment that compared V-segment amino acid substitutions diversity and frequency of the 37 51p1 allele related HV1-69-sBnAbs with that of a reference <i>IGHV1-69</i> 51p1 allele related Ab dataset. 13 amino acid substitutions were determined to uniquely associate with the HV1-69-sBnAb dataset (P<0.05).</p

Understanding the structural role of the distinctive CDR-H2 amino acid substitutions in HV1-69-sBnAbs.

<p>A) VDW contact analysis (black lines) shows that Ser52 of F10 and CR9114 (orange), and Ile52 of CR6261(gray) make only intramolecular contacts; i.e., do not form contacts with their respective H5VN04s. Antibodies are shown in color; HA is in light gray. At far right, steric consequences of the germline Ile52 and the Ile52Ser substitutions are shown when the Abs are overlaid on their framework residues (RMSD ∼0.5 Å). Comparing structures of the HV1-69-sBnAbs, centered on Ile52 of CR6261 (green), with F10 (yellow) and CR9114 (cyan), the Ile52Ser mutation in F10 and CR9114 enables the 2 strands to come closer together, as indicated by the yellow and cyan arrows. Distances in red indicate hypothetical steric clashes (<3 Å) that would be created if Ile52 were present in CR9114 and F10. B) Comparison between the unbound (PDB 4FQH, left) and H5VN04-bound structures (PDB 4FQI, right) of CR9114, colored according to the magnitude of structural change after superposition on the main-chain of the VH domain (from blue = 0 Å, through white = 1 Å, to red = 1.8 Å). CDRs and side-chains of the major contact residues are shown, as depicted in <a href="http://www.plospathogens.org/article/info:doi/10.1371/journal.ppat.1004103#ppat-1004103-g001" target="_blank">Figure 1A</a>. Distances between the Cα and Cβ atoms of Phe54 and the Cα atom of CDR-H3 Tyr98 (shown as dashed lines) are indicated. Large rotations of the side chains of CDR-H3 Tyr98, CDR-H2 Phe54 and CDR-H2 Ile53 are also evident, as previously noted <a href="http://www.plospathogens.org/article/info:doi/10.1371/journal.ppat.1004103#ppat.1004103-Dreyfus1" target="_blank">[7]</a>.</p

Studying the location of SHM hotspots and nucleotide substitutions frequencies in the <i>IGHV1-69</i> reference Ab dataset.

<p><b>A</b>) Number of V-segment substitutions observed in 38 HV1-69-sBnAbs with color notations that designate the HV1-69-sBnAbs characterized by the distinctive CDR-H2 Ser52 or Ala/Gly52a and “other” which do not contain CDR-H2 Ser52 or Ala/Gly52a. (<b>B–D</b>) <b>Upper panel</b> – the common nucleotide substitutions that generated the distinctive amino acid substitutions in the respective CDR domain. <b>Main panel</b> - The <i>IGHV1-69</i> 51p1 Ab reference dataset was studied for substitution frequency of nucleotides in the V-segment CDRs and for location of AID and polη hotspots (AID = WR<u>C</u>Y yellow solid triangle/R<u>G</u>YW dark red solid triangle, Polη W<u>A</u> brown open triangle/<u>T</u>W blue empty triangle. The horizontal line shows the mean±SD (6.44±7.23) of non-germline nucleotide substitution frequency observed for FR regions to serve as a reference.</p

Semi-synthetic HV1-69 phage-Ab library yields potent anti-H5VN04/H1CA0409 Abs characterized by a minimal V-segment amino acid substitutions.

<p><b>A</b>) Characterization of binding activities of anti-H5VN04 and anti-H1CA0409 phage-Abs isolated from the semi-synthetic HV1-69 phage-display library. Sequences are detailed in <b>Figure S4 in <a href="http://www.plospathogens.org/article/info:doi/10.1371/journal.ppat.1004103#ppat.1004103.s001" target="_blank">Text S1</a></b>. <b>B</b>) Heavy chain CDR sequences of anti-H5VN04 phage Abs characterized by >95% neutralization activity against both H5VN04 and H1PR8 pseudotyped viruses. The 5 highlighted residues in the CDRs refer to panel <b>C</b>) which describes the result of a Chi square statistical analysis approach used to identify residues that were significantly enriched as compared to their frequency in the library (P<0.05). Also highlighted are Tyr99 and position73 in the <i>IGHV1-69</i> germline sequences.</p