Delete or merge regressors for linear model selection

We consider a problem of linear model selection in the presence of both continuous and categorical predictors. Feasible models consist of subsets of numerical variables and partitions of levels of factors. A new algorithm called delete or merge regressors (DMR) is presented which is a stepwise backward procedure involving ranking the predictors according to squared t-statistics and choosing the final model minimizing BIC. In the article we prove consistency of DMR when the number of predictors tends to infinity with the sample size and describe a simulation study using a pertaining R package. The results indicate significant advantage in time complexity and selection accuracy of our algorithm over Lasso-based methods described in the literature. Moreover, a version of DMR for generalized linear models is proposed

In silico Modeling of Itk Activation Kinetics in Thymocytes Suggests Competing Positive and Negative IP4 Mediated Feedbacks Increase Robustness

The inositol-phosphate messenger inositol(1,3,4,5)tetrakisphosphate (IP4) is essential for thymocyte positive selection by regulating plasma-membrane association of the protein tyrosine kinase Itk downstream of the T cell receptor (TCR). IP4 can act as a soluble analog of the phosphoinositide 3-kinase (PI3K) membrane lipid product phosphatidylinositol(3,4,5)trisphosphate (PIP3). PIP3 recruits signaling proteins such as Itk to cellular membranes by binding to PH and other domains. In thymocytes, low-dose IP4 binding to the Itk PH domain surprisingly promoted and high-dose IP4 inhibited PIP3 binding of Itk PH domains. However, the mechanisms that underlie the regulation of membrane recruitment of Itk by IP4 and PIP3 remain unclear. The distinct Itk PH domain ability to oligomerize is consistent with a cooperative-allosteric mode of IP4 action. However, other possibilities cannot be ruled out due to difficulties in quantitatively measuring the interactions between Itk, IP4 and PIP3, and in generating non-oligomerizing Itk PH domain mutants. This has hindered a full mechanistic understanding of how IP4 controls Itk function. By combining experimentally measured kinetics of PLC{\gamma}1 phosphorylation by Itk with in silico modeling of multiple Itk signaling circuits and a maximum entropy (MaxEnt) based computational approach, we show that those in silico models which are most robust against variations of protein and lipid expression levels and kinetic rates at the single cell level share a cooperative-allosteric mode of Itk regulation by IP4 involving oligomeric Itk PH domains at the plasma membrane. This identifies MaxEnt as an excellent tool for quantifying robustness for complex TCR signaling circuits and provides testable predictions to further elucidate a controversial mechanism of PIP3 signaling

Different molecular interactions in models M1–M7 produce different temporal profiles of PIP₃ binding to Itk.

<p>(<b>A</b>) Kinetics of PIP₃ association of Itk for fixed initial PIP₃ and Itk concentrations (100 and 370 molecules, respectively) in models with feedbacks (M1–M4, and M7, left panel) and no feedbacks (M5–M6, right panel). (B) The shapes of the temporal profiles can be characterized by the parameters peak time (<i>τ</i>_p), peak width (<i>τ</i>_w), and peak value or amplitude (<i>A</i>). The dimensionless asymmetry ratio <i>R</i> = <i>τ</i>_w/<i>τ</i>_p quantifies how symmetric the shape of the time profile is. A larger R value indicates larger asymmetry. (C) Variations in R in models M1–M7 for different initial concentrations of Itk and PIP₃. Color scales for R values are shown on the right of each panel.</p

Models containing Itk dimers and dueling feedbacks also show higher robustness for polyclonal T cells stimulated by anti-CD3 antibodies.

<p>PLCγ1 phosphorylation kinetics in <i>MHC^−/−</i> T cells stimulated by antibodies against (A) CD3 or (B) CD3 and CD4 at 1 µg/ml versus 5 µg/ml. (C) Variation of D_KL with R for the <i>in silico models</i> M1–M3 (blue, red and black, respectively), M7 (yellow), and M5–M6 (purple and maroon, respectively) at initial Itk (Itk⁰ = 100 molecules) and PIP₃ concentrations (PIP₃⁰ = 370 molecules) at <i>τ</i>_p = 1 min and <i>A</i>_avg = 60 molecules, representing anti-CD3 stimulation at 5 µg/ml. The orange bar indicates R<i>^expt</i>. Note we use <i>A</i>_avg to represent the amplitude A^expt in experiments measuring fold change in Itk phosphorylation (see the main text for further details). (D) Variation of D_KL with R for anti-CD3/CD4 stimulation at 5 µg/ml at <i>τ</i>_p = 1 min and <i>A</i>_avg = 80 molecules. The initial Itk (Itk⁰ = 140 molecules) and PIP₃ concentrations (PIP₃⁰ = 530 molecules) were used. The orange bar indicates R<i>^expt</i>. (E) and (F) show maps of the most robust models (with the lowest D_KL) as R<i>^expt</i> and <i>A</i> (shown as <i>A</i>_avg) are varied for the same parameters as in (C) and (D), respectively.</p

Relevant basic interactions between Itk, PIP₃ and IP₄.

<p>Following TCR-pMHC binding, Itk molecules are bound by the LAT signalosome via SLP-76 (not shown). Itk molecules (monomers or dimers, blue diamonds), bind the membrane lipid PIP₃ with low affinity through their PH domains. PIP₃ bound Itk phosphorylates and thereby activates LAT-bound PLCγ1. Activated PLCγ1 then hydrolyzes the membrane lipid PIP₂ into the soluble second messenger IP₃, a key mediator of Ca²⁺ mobilization. IP₃ 3-kinase B (ItpkB) converts IP₃ into IP₄ (red filled circle). For our <i>in silico</i> models, we simplified this series of reactions, encircled by the orange oval, into a single second order reaction where PIP₃ bound Itk converts PIP₂ into IP₄. In models M1–M4 and M7, IP₄ modifies the Itk PH domain (denoted as Itk^C, purple diamonds) to promote PIP₃ and IP₄ binding to the Itk PH domain. At the onset of the signaling, when the concentration of IP₄ is smaller than that of PIP₃, IP₄ helps Itk^C to bind to PIP₃ (left lower panel). However, as the concentration of IP₄ is increased at later times, IP₄ outcompetes PIP₃ for binding to Itk^C and sequesters Itk^C to the cytosol (right lower panel). In models M5/M6, IP₄ and PIP₃ do not augment each other’s binding to Itk. However, IP₄ still outcompetes PIP₃ for Itk PH domain binding when the number of IP₄ molecules becomes much larger than that of PIP₃ molecules at later times.</p

Experimentally measured PLCγ1activation kinetics in DP thymocytes stimulated with TCR ligands of different affinities and robustness of <i>in silico</i> models.

<p>(A) Immunoblots showing Y₇₈₃-phosphorylated (upper panels) and total (lower panels) PLCγ1 protein amounts in <i>RAG2^−/−MHC^−/− OT1 TCR-transgenic</i> DP thymocytes stimulated for the indicated times with MHCI tetramers presenting the indicated altered peptide ligands (APL). (B) Phospho-PLCγ1 levels normalized to total PLCγ1 protein amounts plotted over time for the indicated APLs. Their TCR affinity decreases in the order OVA (black)>Q4R7 (red)>Q4H7 (blue)>G4 (green). Band intensities were quantified via scanning and analysis with <i>ImageJ</i> software. Representative of several independent experiments. (C) Variation of the Kulback-Leibler distance D_KL with <i>R</i> for models M1–M3 (blue, red and black, respectively), M7 (yellow), and M4–M6 (orange, purple, and maroon, respectively) at high initial Itk (Itk⁰ = 140 molecules) and PIP₃ concentrations (PIP₃⁰ = 530 molecules), representing high-affinity OVA stimulation for <i>τ</i>_p = 2 min and <i>A</i> (shown as <i>A</i>_avg) = 40 molecules. Note we use <i>A</i> to represent the amplitude <i>A</i>^expt in experiments measuring fold change in Itk phosphorylation (see the main text for further details). The vertical orange bar indicates R<i>^expt</i> for OVA. Color legend in (D). (D) The color map shows which model is most robust (has the lowest D_KL) as <i>R^expt</i> and <i>A</i> (shown as <i>A</i>_avg) are varied for the same parameters as in (C). The color legend is depicted on the right.</p