Matriks Jordan Dan Aplikasinya Pada Sistem Linier Waktu Diskrit

Matrix is diagonalizable (similar with matrix diagonal) if and only if the sum of geometric multiplicities of its eigenvalues is n.If we search for an upper triangular form that is nearly diagonal as possible but is still attainable by similarity for every matrix, especially the sum of geometric multiplicities of its eigenvalues is less than n, the result is the Jordan canonical form, which is denoted by , and . In this paper, will be described how to get matrix S(in order to get matrix ) by using generalized eigenvector. In addition, it will also describe the Jordan canonical form and its properties, and some observation and application on discrete time linear system

QSPRs for estimating nematic transition temperatures of pyridine-containing liquid crystalline compounds

<p>Quantitative structure–property relationships were developed to predict the nematic–isotropic transition temperatures of 92 pyridine-containing liquid crystalline compounds using molecular descriptors calculated by CODESSA software and DRAGON software. The descriptors were also analysed by using principal component analysis. Essentials accounting for a reliable model were all considered carefully during model construction and assessment process. Five variables were selected out by stepwise forward regression analysis and were used as inputs to perform the multiple linear regression, support vector machine and projection pursuit regression (PPR) study. All models were validated through two ways, that is, internal cross-validation combined with a test set. Comparatively, the PPR model performs best both in the fitness and in the prediction capacity. For the test set, it gave a predictive correlation coefficient (<i>R</i>) of 0.991, root mean square error of 11.799 and absolute average relative deviation of 5.456, respectively. The relationships between the descriptors and the nematic–isotropic transition temperature of compounds were also discussed. The odd–even effect in the transition temperatures of mesogens in the same homologous series was also discussed.</p

Molecular Modeling Study on the Allosteric Inhibition Mechanism of HIV-1 Integrase by LEDGF/p75 Binding Site Inhibitors

<div><p>HIV-1 integrase (IN) is essential for the integration of viral DNA into the host genome and an attractive therapeutic target for developing antiretroviral inhibitors. LEDGINs are a class of allosteric inhibitors targeting LEDGF/p75 binding site of HIV-1 IN. Yet, the detailed binding mode and allosteric inhibition mechanism of LEDGINs to HIV-1 IN is only partially understood, which hinders the structure-based design of more potent anti-HIV agents. A molecular modeling study combining molecular docking, molecular dynamics simulation, and binding free energy calculation were performed to investigate the interaction details of HIV-1 IN catalytic core domain (CCD) with two recently discovered LEDGINs BI-1001 and CX14442, as well as the LEDGF/p75 protein. Simulation results demonstrated the hydrophobic domain of BI-1001 and CX14442 engages one subunit of HIV-1 IN CCD dimer through hydrophobic interactions, and the hydrophilic group forms hydrogen bonds with HIV-1 IN CCD residues from other subunit. CX14442 has a larger <i>tert</i>-butyl group than the methyl of BI-1001, and forms better interactions with the highly hydrophobic binding pocket of HIV-1 IN CCD dimer interface, which can explain the stronger affinity of CX14442 than BI-1001. Analysis of the binding mode of LEDGF/p75 with HIV-1 IN CCD reveals that the LEDGF/p75 integrase binding domain residues Ile365, Asp366, Phe406 and Val408 have significant contributions to the binding of the LEDGF/p75 to HIV1-IN. Remarkably, we found that binding of BI-1001 and CX14442 to HIV-1 IN CCD induced the structural rearrangements of the 140 s loop and oration displacements of the side chains of the three conserved catalytic residues Asp64, Asp116, and Glu152 located at the active site. These results we obtained will be valuable not only for understanding the allosteric inhibition mechanism of LEDGINs but also for the rational design of allosteric inhibitors of HIV-1 IN targeting LEDGF/p75 binding site.</p></div

Scheme of the active site DDE motif (Asp64, Asp116, and Glu152) models for (A) LEDGF/p75, (B) BI-1001 and (C) CX14442 bound HIV-1 IN complexes.

<p>The measured distances between the centroid of the side chains of the three conserved catalytic residues were labeled in each model.</p

The aligned representative conformations of BI-1001, CX14442, and LEDGF/p75 bound HIV-1 IN CCD dimer models.

<p>The averaged structures extracted from the MD trajectories were used. The BI-1001, CX14442, and LEDGF/p75 bound form are shown in yellow, cyan and gray, respectively. HIV-1 IN active site residues (Asp64, Asp116, and Glu152) are shown in stick. The LEDGINs and LEDGF/p75 are represented in stick and carton, respectively.</p

The LEDGF/p75 protein residues contribution to the total binding free energy of the LEDGF/p75 bound HIV-1 IN CCD complex.

<p>The LEDGF/p75 protein residues contribution to the total binding free energy of the LEDGF/p75 bound HIV-1 IN CCD complex.</p

Per-residue interaction spectrum of the residues of HIV-1 IN CCD with (A) BI-1001, (B) CX14442, and (C) LEDGF/p75 in complex with the HIV-1 IN CCD dimer from MM/GBSA free energy decomposition analysis.

<p>Per-residue interaction spectrum of the residues of HIV-1 IN CCD with (A) BI-1001, (B) CX14442, and (C) LEDGF/p75 in complex with the HIV-1 IN CCD dimer from MM/GBSA free energy decomposition analysis.</p

Electrostatic potential surface of the allosteric binding pocket of HIV-1 IN CCD dimer in interaction with (A) BI-1001, (B) CX14442, and (C) LEDGF/p75.

<p>The positive charges are displayed in blue, negative charges are displayed in red, and neutral residues are displayed in white. Color intensity is proportional to the charge value. The BI-1001, CX14442 and side chain of the LEDGF/p75 key residues, whose carbon atoms are shown as green spheres and labeled as red. The residue Trp131 from monomer A of HIV-1 CCD dimer is also labeled (black).</p

The monitored RMSD of the backbone atoms of protein (black), backbone atoms of binding pocket residues around 5 Å of ligand (blue), and the heavy atoms in the ligand (red) for: (A) BI-1001 and (B) CX14442 bound HIV-1 IN complexes with respect to the initial structures as a function of time.

<p>(C) The monitored RMSD of the backbone atoms of HIV-1 IN and LEDGF/p75 (black), backbone atoms of HIV-1 IN (blue), and backbone atoms of LEDGF/p75 (red) for LEDGF/p75 bound HIV-1 IN complex with respect to the initial structures as a function of time.</p

The generated pharmacophore for HIV-1 IN LEDGINs based on the receptor-ligand interactions.

<p>The model consists of hydrophobic and hydrophilic features on LEDGINs as well as the key residue in HIV-1 IN allosteric site. The hydrophobic and hydrophilic domains are shown in green and red, respectively. The residues that participated in the interaction between LEDGINs and HIV-1 IN CCD are labeled in cyan and orange, while the potential residues used for further extension LEDGINs design are labeled in black and gray.</p