Exploiting Equalities in Polynomial Programming

We propose a novel solution approach for polynomial programming problems with equality constraints. By means of a generic transformation, we show that solution schemes for the, typically simpler, problem without equalities can be used to address the problem with equalities. In particular, we propose new solution schemes for mixed binary programs, pure 0–1 quadratic programs, and the stable set problem

Logconcave Random Graphs

We propose the following model of a random graph on n vertices. Let F be a distribution in Rn(n−1)/2+ with a coordinate for every pair ij with 1≤i,j≤n. Then GF,p is the distribution on graphs with n vertices obtained by picking a random point X from F and defining a graph on nvertices whose edges are pairs ij for which Xij≤p. The standard Erdős-Rényi model is the special case when F is uniform on the 0-1 unit cube. We examine basic properties such as the connectivity threshold for quite general distributions. We also consider cases where the Xij are the edge weights in some random instance of a combinatorial optimization problem. By choosing suitable distributions, we can capture random graphs with interesting properties such as triangle-free random graphs and weighted random graphs with bounded total weight.</p

Static-Arbitrage Bounds on the Prices of Basket Options via Linear Programming

We show that the problem of computing sharp upper and lower static-arbitrage bounds on the price of a European basket option, given the prices of other similar options, can be cast as a linear program (LP). The LP formulations readily yield super-replicating (subreplicating) strategies for the upper (lower) bound problem. The dual counterparts of the LP formulations in turn yield underlying asset price distributions that replicate the given option prices, and the bound on the new basket option’s price. In the special case when the given option prices are those of vanilla options on the underlying assets, we show that the LP formulations admit further simplifications. In particular, for the upper bound problem we derive closed-form formulas for the basket’s price bound, and for the corresponding superreplicating strategy. In addition, our LP approach admits efficient modeling of additional features such as basket options with negative weights, bid/ask spreads, transaction costs, and diversification constraints. We provide numerical experiments to illustrate some of our results

Analysis of procollagen (A) and collagen (B) molecules synthesized by the patient (P) and control (C) cells and mutation identification (C-E).

<p>A,B, SDS-PAGE run with 2 M urea to enhance the difference in migration between overmodified and normal chains. Compared to control cells, proα1(I) and α1(I) chains from patient were more heterogeneous and some of the chains had delayed electrophoretic migration due to posttranslational overmodification. C, The shapes of electrophoretic bands of CNBr (CB) peptides from normal (large solid bands) and overmodified (upward streaks marked by arrows) collagen chains. The main bands might be slanted (e.g. control α1(I)) due to imperfect shape/orientation of the band cut out from the original collagen gel for CNBr digestion. The streaks of overmodified chains in the top left corner of the main bands are distinguished by their different orientation. The presence of overmodified α1(I)CB3, α1(I)CB7 and α1(I)CB8, but not α1(I)CB6, is consistent with a defect in folding within residues 551–821 of the triple helical domain (see panel E), that is, amino-terminal to the beginning of α1(I)CB6. D, The heterozygous mutation, c.2873G>T in exon 41 of one <i>COL1A1</i> allele, resulted in substitution of the arginine by leucine at position 958 of the protein (p.Arg958Leu), which is position 780 of the triple helical domain (Arg780Leu). E, The figure of CB-peptides shows the location of the methionyl residues in the α1(I) chain (vertical bars). The open circles (○) represent the electrophoretic mobility of the peptides derived from the normal chains relative to the streaks of overmodified peptides from the mutant chains (closed circles, ●).</p

Model of effects of substitution of arginine by leucine at position 780 of the triple helix in the α1(I) chain.

<p>A stretch of 14 residues are shown in three chains which in α1(I) are Pro773-Gln-Gly-Ile-Ala-Gly-Gln-Arg-Gly-Val-Val-Gly-Leu-Pro786. The glycines at position 775 are represented by the space filling models in which cyan (chain A) and magenta (chain B) are the α1(I) chains and the α2(I) chain is in yellow (chain C). The chain order is assumed to be α1(I)- α2(I)- α1(I) in the helical domain. A, The normal sequence in which the residue at position 780 of the triple helix is arginine shown in magenta and cyan on their respective chains. The hydrogen bonds are shown by the dotted lines. B, The mutant sequence in which both residues at position 780 are occupied by leucine. No hydrogen bonds form to stabilize the molecule.</p

Delay in type I collagen folding (A) and intracellular retention (B) caused by substitutions for glycine and arginine in the triple helical domain of proα1(I) chains.

<p>A, Collagen triple helix folding measured by incorporation of azidohomoalanine (Aha) into newly synthetized collagen chains after 10 min Aha pulse, followed by chymotrypsin-trypsin digestion, labeling with DIBO dye, and quantifying α1(I) chains resistant to chymotrypsin-trypsin digestions by SDS-PAGE. The fraction of folded (chymotrypsin-trypsin resistant) molecules was measured from the intensity of Aha-DIBO-labeled α1(I) bands on SDS-PAGE relative to the intensity of these bands after 2 h chase. B, Residence time of intracellular Aha-DIBO-labeled collagen after 2 h Aha-pulse. C, Visualization of a shift in migration of overmodified mutant molecules vs. normal (NC) molecules on SDS-PAGE by double fluorescent labeling. Binary mixtures of AlexaFluor488-labeled NC, G763S, G766C, R780L, or R780C (green) with Cy5-labeld NC (red) were used to compare the migration of mutant vs normal control in the same SDS-PAGE lane. Relative to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0200264#pone.0200264.g002" target="_blank">Fig 2B</a>, the shift is less pronounced since urea was not used to enhance the separation, but it is still clearly visualized by the separation of red and green colors. D, Amino acid analysis shows higher fraction of hydroxylysines in all mutant collagens. The standard deviation was measured in NC (5 samples) and assumed to have similar relative value in other collagens.</p

Clinical summary for α1(I)-R780L.

<p>A, Family pedigree; the proband (II-1) is marked with arrow. B, radiographs of the proband at 4 months (left) and 1 year (right) of age.</p

Cellular stress caused by glycine and arginine mutations in type I collagen.

<p>A, Markers of integrated cell stress response (p-EIF2α) and unfolded protein response (BIP) as well as ER chaperones PDI, calnexin and calreticulin were measured in biological-triplicate by Western blotting relative to β-actin. B, Unfolded protein response (<i>HSPA5</i>/BiP and <i>XBP1(S)</i>) and integrated cell stress response (<i>DDIT3</i>/CHOP and <i>CRYAB</i>/αB-crystallin) mRNA markers were measured in technical-triplicate by qPCR (ΔΔC_T method) using a geometric mean of HPRT1, B2M and GAPDH as an endogenous control.</p

Type I collagen triple helix destabilization caused by glycine and arginine mutations.

<p>A, DSC denaturation thermograms of pepsin-purified collagen (solid lines) and procollagen (dotted lines) secreted into media by normal control (NC) and patient cells. B, Denaturation thermograms of pepsin-purified collagen from media (secreted, solid lines), cell layer (intracellular, dashed line), and extracellular matrix deposited by cells (matrix, dash-dotted lines). C,D, Analysis of α1(I)-R780C collagen thermal stability by enzymatic digestion and gel electrophoresis. In A and B, each thermogram peak represents denaturation of molecules with or without mutant chains. The peak maximum is the corresponding apparent denaturation temperature T_m, which is the same in collagen and procollagen molecules (for NC as well as most mutants) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0200264#pone.0200264.ref027" target="_blank">27</a>]. A heterozygous Gly substitution in the α1(I) chain might result in up to 3 peaks on the denaturation thermogram, representing molecules with 2, 1, and 0 mutant chains. In α1(I)-G766C collagen, the molecules with 1 and 2 mutant chains have the same T_m, producing one peak at ≈39.5 °C; the molecules without mutant chains produce the second peak at ≈42 °C. The same is true for α1(I)-G763S collagen. In α1(I)-R780L or α1(I)-R780C, molecules with 1 and 2 mutant chains produce a peak at ≈41 °C; the 42 °C peak of molecules without the mutant chains is too close to 41 °C, so that it is not resolved. The area under each peak is proportional to the fraction of the corresponding molecules. A change in this fraction upon secretion from cells or incorporation into matrix alters the shape of the denaturation thermogram [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0200264#pone.0200264.ref027" target="_blank">27</a>]. A reduced intensity of the 39.5 °C peak in matrix vs. secreted collagen indicated that only 25% of α1(I)-G766C molecules in extracellular matrix contained mutant chains, but no effects of other mutations on collagen secretion or matrix incorporation were detected. In C and D, thermal stability of secreted collagen was measured by 2 min equilibration at different temperatures followed by 5 min at room temperature, 1 min digestion with trypsin/chymotrypsin mixture, and separation of chains by gel electrophoresis at non-reducing conditions as described in [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0200264#pone.0200264.ref031" target="_blank">31</a>]. Panel C shows quantitative analysis of gel electrophoresis (D), each point representing an average of 4 experiments.</p

Cleavage of mutant collagens by MMP1, MMP2, and catalytic domain of MMP1 (MMP1(ΔC)).

<p>A, cleavage kinetics of collagens by MMP1 in molecules with Cys for α1(I) Gly or Cys for α1(I) Arg substitutions. Each time point represents an average of two experiments. In one, the mutant labeled with Alexa Fluor 488 (AF488) was mixed with normal control labelled with Cy5 and treated with MMP1 within the same sample tube. In the other, the fluorescent labels were switched. The error bar represents the corresponding standard deviation. Gray circles show the kinetics of cleavage for disulfide bonded chains formed in molecules containing two α1(I) chains with Cys for Gly or Cys for Arg substitutions. B, relative cleavage rates for all studied mutants determined as the ratio of the initial cleavage rate of all α1(I) chains in the mutant to the initial cleavage rate of α1(I) chains in the normal control within the same sample tube. The initial cleavage rates were determined by linear regression of the first 3–4 data points in the kinetic experiments shown in A and similar experiments for the other mutants and enzymes. The cleavage rates of α1(I)-R780L with MMP1(ΔC) were not measured. The cleavage rates of all mutants were larger than in normal control with high statistical significance (<i>p</i><0.001). For instance, the cleavage rate by MMP2 was 0.18±0.01 in α1(I)-R780L vs. 0.01±0.006 in NC (in the same units), yielding high statistical significance for (0.18±0.01)/(0.01±0.006) > 1 despite noticeable uncertainty in the absolute value of this ratio.</p