A guide to the Choquard equation

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$-\Delta u + V(x)u = \bigl(|x|^{-(N-\alpha)} * |u|^p\bigr)|u|^{p - 2} u \qquad \text{in $\mathbb{R}^N$},$$ and some of its variants and extensions.Comment: 39 page

On uniqueness for the generalized choquard equation

We consider the generalized Choquard equation describing trapped electron gas in three dimensional case. The study of orbital stability of the energy minimizers (known as ground states) depends essentially in the local uniqueness of these minimizers. The uniqueness of the minimizers for the case p = 2, i.e. for the case of Hartree–Choquard is well known. The main difficulty for the case p ≠ 2 is connected with possible lack of control on the Lp norm of the minimizers. Our main result treats the local uniqueness of radial positive minimizers for p ∈ (5∕3, 7∕3)

Correlation between tumor regression and T cell responses in melanoma patients vaccinated with a MAGE antigen

The cancer-germline gene MAGE-3 codes for tumor-specific antigens recognized on many tumors by T lymphocytes. A MAGE-3 antigen presented by HLA-A1 has been used in several vaccination trials on metastatic melanoma patients. Only a small minority of patients have shown evidence of tumor regression. Attempts to correlate the tumor rejections with the cytotoxic T lymphocyte (CTL) response against the vaccine have been hampered by the low level of these responses. In noncancerous individuals, the frequency of the T cell precursors against antigen MAGE-3.A1 is ≈4 × 10(-7) CD8 T cells. The diversity of the T cell receptor repertoire of these anti-MAGE-3.A1 precursors was analyzed in one individual. The results indicate that it is very likely that the repertoire comprises >100 clonotypes. On this basis, it is possible to use not only the frequency of CTL precursors in the blood but also the presence of dominant clonotypes to ascertain in patients the existence of anti-MAGE-3.A1 responses as low as 10(-6) of CD8. With this approach, we observed a correlation between tumor regression and anti-MAGE-3.A1 CTL responses in patients vaccinated with a recombinant virus encoding the antigen and also in patients vaccinated with peptide-pulsed dendritic cells. In contrast, for patients showing tumor regression after vaccination with peptide alone, CTL responses were almost never observed. It is possible that even those CTL responses that are below our present detection level can trigger a sequence of events that leads to tumor regression

Cytotoxic T lymphocytes directed against a tumor-specific mutated antigen display similar HLA tetramer binding but distinct functional avidity and tissue distribution

We have previously identified an antigen (Ag) recognized on a human large cell carcinoma of the lung by a tumor-specific cytotoxic T lymphocyte clone derived from autologous tumor infiltrating lymphocytes (TILs). The antigenic peptide is presented by HLA-A2 molecules and is encoded by a mutated α-actinin-4 (ACTN4) gene. In the present report, we have isolated two anti-α-actinin-4 T cell clones from the same patient TIL and from his peripheral blood lymphocytes (PBLs) by using tetramers of soluble HLA-A2 molecules loaded with the mutated peptide. Although all of the clones displayed similar tetramer labeling, those isolated from PBL showed lower avidity of Ag recognition and killed the specific target much less efficiently, indicating that tetramer staining does not correlate with clone avidity/tumor reactivity. T cell receptor (TCR) analysis revealed that α-actinin-4-reactive clones used distinct α and β chain rearrangements, demonstrating TCR repertoire diversity. Interestingly, TCRβ chain gene usage indicated that only Ag-specific clones with high functional avidity were expanded at the tumor site, whereas a low-avidity clone was exclusively amplified in patient peripheral blood. Our results point to the existence of distinct but overlapping antitumor TCR repertoires in TIL and PBL and suggest a selective in situ expansion of tumor-specific cytotoxic T lymphocyte with high avidity/tumor reactivity

A monoclonal cytolytic T-lymphocyte response observed in a melanoma patient vaccinated with a tumor-specific antigenic peptide encoded by gene MAGE-3

Vaccination of melanoma patients with tumor-specific antigens recognized by cytolytic T lymphocytes (CTL) produces significant tumor regressions in a minority of patients. These regressions appear to occur in the absence of massive CTL responses. To detect low-level responses, we resorted to antigenic stimulation of blood lymphocyte cultures in limiting dilution conditions, followed by tetramer analysis, cloning of the tetramer-positive cells, and T-cell receptor (TCR) sequence analysis of the CTL clones that showed strict specificity for the tumor antigen. A monoclonal CTL response against a MAGE-3 antigen was observed in a melanoma patient, who showed partial rejection of a large metastasis after treatment with a vaccine containing only the tumor-specific antigenic peptide. Tetramer analysis after in vitro restimulation indicated that about 1/40,000 postimmunization CD8(+) blood lymphocytes were directed against the antigen. The same TCR was present in all of the positive microcultures. TCR evaluation carried out directly on blood lymphocytes by PCR amplification led to a similar frequency estimate after immunization, whereas the TCR was not found among 2.5 × 10(6) CD8(+) lymphocytes collected before immunization. Our results prove unambiguously that vaccines containing only a tumor-specific antigenic peptide can elicit a CTL response. Even though they provide no information about the effector mechanisms responsible for the observed reduction in tumor mass in this patient, they would suggest that low-level CTL responses can initiate tumor rejection