Recent Advances in Mathematical Programming with Semi-continuous Variables and Cardinality Constraint

Abstract Mathematical programming problems with semi-continuous variables and cardinality constraint have many applications, including production planning, portfolio selection, compressed sensing and subset selection in regression. This class of problems can be modeled as mixed-integer programs with special structures and are in general NP-hard. In the past few years, based on new reformulations, approximation and relaxation techniques, promising exact and approximate methods have been developed. We survey in this paper these recent developments for this challenging class of mathematical programming problems.

TCR:pMHC measurements.

<p>DRMs were purified from total CD8⁺ T cells sorted from seven chronically-infected SIVsmE660-infected monkeys. The DRMs were evaluated for specific binding, measured in resonance units (RU), to pMHC monomers constructed with p11C, p54E660, and p68A epitope peptides and Mamu-A*01. Data are representative of the binding observed from all seven monkeys evaluated. A) Initial experiments to detect p11C, p54E660, and p68A monomer binding to DRMs. 100 μg/mL p11C (red), 100 μg/mL p54E660 (blue), and 150 μg/mL p68A (green) pMHC monomer binding are overlaid from experiments conducted on separate Biacore Chips. Binding of the p68A:Mamu-A*01 monomer to the DRMs was not observed at any concentration of monomer tested (25–200 μg/mL). B) Titrations of p11C (top) and p54E660 (bottom) peptide:Mamu-A*01 monomers for calculation of binding kinetics and affinity. Overlaid sensograms of the binding of p11C and p54E660 pMHC monomers to DRMs purified from total CD8⁺ T cells are shown. Monomer binding was evaluated using pMHC concentrations ranging from 25 to 200 μg/mL. The black curve shows the Langmuir fitted curve that was used to calculate kinetics. C) Detection of p68A peptide:Mamu-A*01 monomer binding to DRMs. p68A-specific CD8⁺ T cells were collected from multiple tetramer-specific flow cytometric cell sorts and pooled for DRM purification. Titrations of p68A pMHC monomers were performed at concentrations ranging from 150 to 1000 μg/mL. Binding to DRMs from monkey ZD57 at 1000 μg/mL is shown and is representative of the four monkeys that were evaluated. All readings have been normalized by subtracting the binding of the control monomer TL8 run at the same concentrations as the experimental monomers.</p

Frequencies of p11C- and p54AS-specific CD8⁺ T cells and plasma viral loads during primary infection.

<p>A) Mean frequencies of the p11C- and p54AS-specific CD8⁺ T cells in peripheral blood. * significant at <i>p</i>≤0.05. B) Plasma SIV RNA levels in the peripheral blood.</p

Expression of exhaustion-associated genes.

<p>* p≤0.05.</p

TCR:pMHC binding values.

<p>Values are the median (range) from seven monkeys.</p><p><i>k_on</i>, association rate.</p><p><i>k_off</i>, dissociation rate.</p><p>K_d, equilibrium dissociation constant. Calculated using the equation <i>k_off</i>/<i>k_on</i>.</p><p>* <i>k_on</i> of the p68A monomers was estimated as being at least as fast as the fastest measured monomer, p11C.</p>‡<p><i>k_off</i> of p68A monomers was estimated as being at least as fast as the limit of detection of the Biacore instrument, approx.1/s.</p>†<p>K_d of p68A monomer binding calculated using estimated values of the <i>k_on</i> and <i>k_off</i> using the equation <i>k_off</i>/<i>k_on</i>.</p

Genes compared between dominant and subdominant epitope-specific CD8⁺ T cells.

<p>Genes compared between dominant and subdominant epitope-specific CD8⁺ T cells.</p

Dominant p11C-specific population contained lower frequency of cytokine- and chemokine-producing cells than subdominant epitope-specific populations.

<p>PBMCs from monkeys chronically-infected with either SIVmac251 (n = 4) or SIVsmE660 (n = 5) were stimulated with either p11C, p54AS/E660, or p68A peptides and intracellular staining was used to assess production of the chemokine MIP-1β and cytokines IFNγ, TNFα, and IL-2. Bars represent mean ± SEM. A) Individual cytokine and chemokine production. Top, SIVmac251. Bottom, SIVsmE660. B) Polyfunctional analysis. Positivity for each cytokine/chemokine is indicated by the dots below the bar graph. Vertical bars are grouped into 4, 3, 2 or 1 function (indicated by the pink, light blue, purple, and orange horizontal bars, respectively). C) Polyfunctionality charts. Left, SIVmac251. Right, SIVsmE660. Each slice represents the percentage of tetramer-positive cells expressing between 1 and 4 functions. Data were collected between weeks 36–42 for SIVmac251 and 14–25 for SIVsmE660.</p

Phenotype and cytotoxic potential of dominant and subdominant epitope-specific cells during chronic SIVmac251 and SIVsmE660 infection.

<p>The frequency, cell surface phenotype, and <i>ex vivo</i> perforin and granzyme B content were evaluated for p11C-, p54AS/E660-, and p68A-specific cells, identified with Mamu-A*01 tetramers, from peripheral blood of chronically infected SIVmac251- and SIVsmE660- infected rhesus monkeys. A) Frequencies of the p11C-, p54AS/E660-, and p68A-specific cells from (top) SIVmac251- and (bottom) SIVsmE660- infected rhesus monkeys. Error bars indicate median ± interquartile range. SIVmac251 frequencies measured between weeks 37–50, except for 133-06 which died early and data presented are from week 18. SIVsmE660 frequencies are measured between weeks 19–22. B) Phenotyping of cells from SIVmac251- (top, n = 4) and SIVsmE660-infected monkeys (bottom, n = 8) based on cell surface expression of CCR7 and CD28. Positivity for CCR7 and CD28 is indicated by the + and − signs below the bar graph. Cells were categorized as central memory (CCR7⁺CD28⁺), transitional memory (CCR7⁻CD28⁺), or effector memory (CCR7⁻CD28⁻). All tetramer-positive cells were CD95⁺. Phenotyping was conducted between weeks 44–78 for SIVmac251 and 33–46 for SIVsmE660. C) Measurement of percent of <i>ex vivo</i> perforin⁺granzyme B⁺ cells from SIVmac251- (Left, n = 5) or SIVsmE660-infected monkeys (right, n = 8). D) Per-cell expression of perforin and granzyme B, measured by geometric mean of fluorescence (GMF) of perforin and granzyme B staining on cells from SIVmac251- (left, n = 5) and SIVsmE660-infected monkeys (right, n = 8). Perforin and granzyme B measurements were conducted between weeks 63–83 for SIVmac251 and 41–49 for SIVsmE660. In parts B–D, error bars represent mean ± SEM. * and **, significant at <i>p</i>≤0.05 and 0.01, respectively, after Bonferroni correction. NS, not significant.</p

Genes differentially expressed between dominant p11C- and subdominant p54AS-specific CD8⁺ T cells.

<p>The median of the normalized expression values, measured in fluorescence units, of A) maturation, B) cytotoxicity, and C) proliferation and apoptosis genes that were found to be differentially expressed between the p11C- and p54AS-specific cells on at least one time point are shown for each timepoint evaluated during the first 70 days following SIVmac251 infection. The expression of these genes by total naïve CD8⁺ T cells, measured on day 0, is also shown. For those genes with more than one probe on the BeadChip, the numerical probe IDs are included in the gene name. * indicates that expression met the criteria for differential expression on that timepoint.</p

The dominant p11C-specific cells exhibited decreased antigen-specific expansion compared to subdominant epitope-specific cells.

<p>PBMCs from monkeys chronically-infected with either SIVmac251 (A) or SIVsmE660 (B) were stimulated <i>in vitro</i> with either p11C (red), p54E660/AS (blue), or p68A (green) peptide, harvested on days 3, 4, 5, 6, 8, 10, 12, and 14 following stimulation, and measured by flow cytometry to calculate the percent of tetramer-positive CD8⁺ T cells. Expansion was calculated as the fold change of the percent of each tetramer-positive population on each day, relative to day 0. Data from three SIVmac251- and three SIVsmE660-infected monkeys are shown. Measurements were conducted between weeks 40–52 for SIVmac251 and 31–44 for SIVsmE660.</p