Additional file 2: Figure S2. of Could remifentanil reduce duration of mechanical ventilation in comparison with other opioids for mechanically ventilated patients? A systematic review and meta-analysis

Secondary outcomes. There was no significant difference in hospital-LOS (a), costs (b), mortality (c) and agitation (d) in comparison with remifentanil and other opioids. (PDF 92 kb

The power iteration algorithm demonstrates how complicated dynamic structures arise from topologically connected elements of similar magnitude within the Jacobian matrix.

<p>(A) Power iteration can be used to calculate the dominant left eigenvector of the Jacobian matrix. The left eigenvectors are the modes of the metabolic network. The algorithm left multiplies the Jacobian matrix by a random vector (<i>u</i>_<i>i</i>), normalizes the resulting vector and repeats the process until the vector converges to the eigenvector. (B) Topologically connected Jacobian elements of similar magnitude determine complicated eigenvector structure. In this case study, we extracted a submatrix of <b>J</b> that corresponds to the nonzero elements of a certain eigenvector, which contains <i>G6PDH</i> enzyme forms. The four Jacobian elements (also the largest) that are key in determining this eigenvector structure are located in the 2^nd and 4^th rows, circled in black. Specifically, the structure of 2^nd or 4^th rows matches closely with that of the eigenvector, with similar ratios at the 2^nd and 4^th positions. Multiplying the Jacobian matrix by any non-orthogonal starting vector (<i>u</i>₁), for example the one shown, results in a vector (<i>u</i>₂) that has a structure more similar to the eigenvector. The contribution of those rows individually to eigenvector formation are further shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0189880#pone.0189880.g004" target="_blank">Fig 4</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0189880#pone.0189880.s006" target="_blank">S4 Fig</a>. For clear demonstration purposes, the comparison of relative colors only works for individual box (surrounded by black stroke) itself, but not across different boxes. (C) Principal component analysis on all power iteration vectors starting with 1000 different random vectors. We randomly picked 1000 starting vectors and multiplied them with the full Jacobian matrix (292 × 292). The starting vector is multiplied through several iterations (10 ~ 20) until it converges to the eigenvector (the dot product of the ending vector and the eigenvector is no greater than 1.0001 and no less than 0.9999). We then performed principal component analysis on all iteration vectors (including the starting vectors) and plotted each vector in terms of the contribution from the first two principal components. The first principal component corresponds to the leading eigenvector of the Jacobian matrix while the rest of components (less than 1% contribution each, only component 2 shown here) together explain the variation of the vector from the eigenvector. Ideally, the contribution of the rest of components will be 0 when the ending vector becomes the eigenvector. However, due to large order of magnitude differences between elements in <b>J</b> and the cutoff we set when comparing the ending vector with the eigenvector, we ended up with variations from the eigenvector (nonzero contribution of component 2 in the inset plot).</p

Diagonal dominance in the Jacobian matrix explains simple mode structures and corresponding eigenvalues with the help of Gershgorin circle theorem.

<p>(A) Example Jacobian matrix of the RBC metabolic network [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0189880#pone.0189880.ref023" target="_blank">23</a>] with different degrees of diagonal dominance. The Jacobian matrix of the metabolic network has a sparse structure, and the diagonal elements of the matrix are always negative due to the structure of the rate laws used. The matrix was extracted from the full concentration Jacobian matrix for illustrative purposes. (B) The entire set of eigenvalues of the Jacobian matrix is shown in the larger plot, with x-axis denoting the inverse of absolute eigenvalues at the log10 scale. In the inset, selected Gershgorin circles of the Jacobian matrix with circle centers ranging from -27 to -5 are shown for illustrative purposes. Eigenvalues greater than -27 are drawn together with the selected circles. The Gershgorin circles from rows with strong diagonal dominance have centers at -26.2 and -5.26 as shown, and the eigenvalues inside are -26.3 and -5.33. All eigenvalues are negative as the system is dynamically stable. The imaginary components of the eigenvalues are small and therefore are neglected. (C) The dynamic response of <i>GAPDH_T</i>, XMP, 5MDRU1P, compared to the respective modes dominated by these metabolites/enzymes, under an ATP hydrolysis perturbation. The dynamics of the mode dominated by a single metabolite coincide with the dynamics of that metabolite. These modes occur at fast, intermediate and slow timescales, showing that diagonal dominance can occur at any time as long as the structural properties of the Jacobian matrix allow.</p

<i>PGI</i> enzyme module and its associated matrices.

<p>(A) A schematic diagram of individual reaction steps associated with <i>PGI</i> enzyme module and its stoichiometric matrix. The PGI enzyme module consists of three reaction steps: binding of G6P (<i>PGI</i>1), conversion of G6P to F6P (<i>PGI</i>2) and release of F6P (<i>PGI</i>3). The enzyme form <i>PGI</i> is in italic. We use an “&” notation to denote that the enzyme form is bound with metabolite(s). (B) Graphical representation of the concept of half reaction. Here we demonstrate the half reaction associated with the binding/release process of G6P, which is held constant. To determine the equilibrium state of this half reaction, we are comparing the sensitivities associated with <i>PGI</i> (G6P<i>k</i>⁺_<i>PGI</i>1) and <i>PGI</i>&G6P (-<i>k</i>^-_<i>PGI</i>1). This comparison is equivalent to comparing G6P concentration and 1/K_{<i>eq</i>,<i>PGI</i>1}. (C) The gradient matrix of the PGI enzyme module. The gradient matrix (= <i>d</i><b>v</b>/<i>d</i><b>x</b>) is obtained from linearization of the reaction rates and represents reaction sensitivities to metabolite concentrations. (D) The cause of diagonal dominance demonstrated through the symbolic concentration Jacobian matrix of the <i>PGI</i> enzyme module. Using row 5 as a case study, we observe that, in the case of mass action rate law, diagonal dominance is determined by the distance from half-reaction equilibrium for individual half-reactions. When comparing the terms associated with <i>PGI</i>1 reaction between diagonal and off-diagonal positions, we are comparing the sensitivity of G6P (<i>PGIk</i>⁺_<i>PGI</i>1) and sensitivity of <i>PGI</i> (G6P<i>k</i>⁺_<i>PGI</i>1) with that of <i>PGI</i>&G6P (-<i>k</i>^-_<i>PGI</i>1). This comparison is equivalent to comparing the concentrations of <i>PGI</i> and G6P with K_{d,<i>PGI</i>1}(K_{d,<i>PGI</i>1} = <i>k</i>^-_<i>PGI</i>1/<i>k</i>⁺_<i>PGI</i>1), thus determining the distance from equilibrium for <i>PGI</i> and G6P binding/release half-reactions. The numerical values for each entry in row 5 is below the symbolic forms. Additionally, we can see clearly that column dominance cannot happen in the concentration Jacobian matrix due to the structure of mass action rate law. In the current case, we can see that the absolute sum of off-diagonal elements in a column is always at least as large as the absolute diagonal element, meaning that diagonal dominance does not occur across columns.</p

Topological and kinetic determinants of the modal matrices of dynamic models of metabolism

<div><p>Large-scale kinetic models of metabolism are becoming increasingly comprehensive and accurate. A key challenge is to understand the biochemical basis of the dynamic properties of these models. Linear analysis methods are well-established as useful tools for characterizing the dynamic response of metabolic networks. Central to linear analysis methods are two key matrices: the Jacobian matrix (<b>J)</b> and the modal matrix (<b>M</b>^-1) arising from its eigendecomposition. The modal matrix <b>M</b>^-1 contains dynamically independent motions of the kinetic model near a reference state, and it is sparse in practice for metabolic networks. However, connecting the structure of <b>M</b>^<b>-1</b> to the kinetic properties of the underlying reactions is non-trivial. In this study, we analyze the relationship between <b>J</b>, <b>M</b>^-1, and the kinetic properties of the underlying network for kinetic models of metabolism. Specifically, we describe the origin of mode sparsity structure based on features of the network stoichiometric matrix <b>S</b> and the reaction kinetic gradient matrix <b>G.</b> First, we show that due to the scaling of kinetic parameters in real networks, diagonal dominance occurs in a substantial fraction of the rows of <b>J</b>, resulting in simple modal structures with clear biological interpretations. Then, we show that more complicated modes originate from topologically-connected reactions that have similar reaction elasticities in <b>G</b>. These elasticities represent dynamic equilibrium balances within reactions and are key determinants of modal structure. The work presented should prove useful towards obtaining an understanding of the dynamics of kinetic models of metabolism, which are rooted in the network structure and the kinetic properties of reactions.</p></div

Analysis of complicated mode structure through power iteration with modified Jacobian matrix.

<p>We divide the vector multiplication with the Jacobian matrix into multiple steps. First of all, each row of the Jacobian matrix is multiplied by every element of the starting vector (Panel B solid black circles). We then sum up each column of the second matrix to obtain the resulting vector (Panel B dash black circles), which is normalized to give the ending vector. (A) The original Jacobian matrix and its leading left eigenvector. The matrix and the eigenvector are the same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0189880#pone.0189880.g003" target="_blank">Fig 3</a> and will be used for comparison with later panels. (B) Starting vector multiplied with the modified Jacobian matrix. We modified the Jacobian element at position (4, 4) to be the same value as the element at position (2, 2). The ending vector has a smaller ratio between the 2^nd and 4^th elements than that of the original eigenvector, as would be expected with a larger absolute value at position (4, 4). The eigenvector of this modified matrix is shown in the upper right of the panel. (C) Starting vector multiplied with a different modified Jacobian matrix. We further changed the modified Jacobian matrix in panel A to create a more symmetric structure, where the element at position (2, 4) is same as the element at position (4, 2). The ending vector has the same absolute values at the 2^nd and 4^th positions, showing that a fully symmetric Jacobian structure will create an equally weighted structure in eigenvector. The eigenvector of this modified matrix is shown in upper right. Overall, we demonstrate that changing the Jacobian element at either diagonal or off-diagonal position can alter the eigenvector of the matrix in a predictable manner, based on the topological pattern of the key elements determining the eigenvector structure. For clear demonstration purposes, the comparison of relative colors only works for individual box (surrounded by black stroke) itself, but not across different boxes.</p

The origin of complicated mode structure associated with <i>G6PDH</i> enzyme forms demonstrated through the associated matrices.

<p>The mode structure contains four enzyme forms (denoted as E1, E2, E3 and E4, full annotation at the bottom), with <i>G6PDH</i>&NADPH&6PGL and <i>G6PDH</i>&6PGL being the most dominant elements. We extracted the submatrices associated with those four enzyme forms and their related reactions. We show that three key reactions and their associated reaction sensitivities in <b>G</b> determine the mode structure. (A) The reaction steps for the biochemical reaction catalyzed by <i>G6PDH</i> enzyme. The four dominant enzyme forms in the mode are labeled with red circles. The reactions with their notations (R1 to R7) are labeled with blue rectangular boxes. The three key reactions determining the mode structure are circle with black rectangular boxes. (B) The stoichiometric matrix <b>S</b> for the four enzyme forms in the mode and their associated reactions. The <b>S</b> matrix describes the network topology of the enzyme forms and determines how they interact in the Jacobian matrix. (C) The symbolic and numerical gradient matrix <b>G</b> for the four enzyme forms in the mode and their associated reactions. The key reaction sensitivities determining the two largest elements in the mode are associated with reaction 6 and its corresponding enzyme forms. The key terms are <i>k</i>⁺₆ and NADPH<i>k</i>^-₆, which are similar in magnitude, due to the fact that NADPH concentration is similar to the equilibrium constant of the half reaction for NAPDH binding/release. (D) The symbolic and numerical Jacobian matrix <b>J</b> for the four enzyme forms in the mode. We found that the elements of reaction 6 in <b>G</b> dominate the topologically connected Jacobian elements that determine the mode structure. These elements are located at positions (2,2), (2,4), (4,2) and (4,4). Reaction 6 is connected to reaction 4 and 7, whose reaction sensitivities are much smaller in magnitude compared to that of reaction 6, resulting in very small coefficient for their associated elements in the mode (<i>G6PDH</i> and <i>G6PDH</i>&NADP&G6P).</p

Table_1.PDF

<p>The ability to learn from feedback is important for children’s adaptive behavior and school learning. Feedback has two main components, informative value and valence. How to disentangle these two components and what is the developmental neural correlates of using the informative value of feedback is still an open question. In this study, 23 children (7–10 years old) and 19 adults (19–22 years old) were asked to perform a rule induction task, in which they were required to find a rule, based on the informative value of feedback. Behavioral results indicated that the likelihood of correct searching behavior under negative feedback was low for children. Event-related potentials showed that (1) the effect of valence was processed in a wide time window, particularly in the N2 component; (2) the encoding process of the informative value of negative feedback began later for children than for adults; (3) a clear P300 was observed for adults; for children, however, P300 was absent in the frontal region; and (4) children processed the informative value of feedback chiefly in the left sites during the P300 time window, whereas adults did not show this laterality. These results suggested that children were less sensitive to the informative value of negative feedback possibly because of the immature brain.</p

Additional file 1: of Stress ulcer prophylaxis in intensive care unit patients receiving enteral nutrition: a systematic review and meta-analysis

S1. PICO question. S2. Excluded RCTs that did not provide sufficient information on EN. S3. Definitions of GI bleeding and nosocomial pneumonia in the included RCTs. S4. Risk of bias graph and summary of the included RCTs. (DOCX 39 kb

MOESM4 of Procalcitonin-guided antibiotic therapy in intensive care unit patients: a systematic review and meta-analysis

Additional file 4: Table S4. Non-compliance rate reported in the included RCTs