Consensus under Misaligned Orientations

This paper presents a consensus algorithm under misaligned orientations, which is defined as (i) misalignment to global coordinate frame of local coordinate frames, (ii) biases in control direction or sensing direction, or (iii) misaligned virtual global coordinate frames. After providing a mathematical formulation, we provide some sufficient conditions for consensus or for divergence. Besides the stability analysis, we also conduct some analysis for convergence characteristics in terms of locations of eigenvalues. Through a number of numerical simulations, we would attempt to understand the behaviors of misaligned consensus dynamics.Comment: 23 pages, 9 figure

The network of mating pathway in budding yeast deduced into minimal DoRA circuits.

<p>Left: the original mating pathway in budding yeast. Middle: the simplified pathway by coarse-graining. Right: the corresponding minimal DoRA circuits of two and three nodes in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0034727#pone-0034727-g004" target="_blank">Figure 4</a>.</p

Motifs and mechanisms that are sufficient for achieving DoRA function.

<p>The nodes labeled with “R”, “O”, and “M” are input node, output node, and middle buffering node, respectively. Dashed links are assumed as regulations from basal enzymes in the environment.</p

Screening for networks with DoRA function.

<p>For each network with two nodes and three nodes, 10,000 random sets of parameters were assigned. The corresponding kinetic equations were solved numerically to obtain the dose-response curves. Linear correlation coefficients were calculated subsequently from the relationship between the responses of the output node and the input node. The number of parameter sets that render good linear output-input dependence (Q-value) measures the ability for the corresponding network to achieve dose-response alignment.</p

Ranking of the Q-values.

<p>Q-values for all two-node networks (A) and all three networks (B), both showing exponential-like dependence on the ranking.</p

Analysis of 633 functional networks of three nodes with Q-value larger than 15.

<p>(a) Venn diagram of networks with three characters: input node directly regulates output node, input and output nodes regulate the middle buffering node with opposing signs, and both. (b) Topological clustering for DoRA networks with ONR or MNR or with both ONR and MNR. (c) Motif analysis of 633 robust DoRA networks.</p

Simplest functional networks.

<p>The minimal networks that were identified to have the DoRA function (a–d). The red links in a network are regulations that are confined to saturation or linear regions, i.e., motifs (denoted by M_i) that are listed in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0034727#pone-0034727-g005" target="_blank">Figure 5</a>. The red links in b(vi) is a variation of M1 motif (VM1). The number in the bracket is Q-value for the circuit. All two-node minimal networks were found to achieve DoRA by constraining the enzyme regulation of the output node (a, ONR). Three-node simplest networks can achieve DoRA by constrained regulations of either the output node A (b, ONR) or the middle buffering node B (c, MNR) or both (d, ONR/MNR). Two examples of non-minimal functional networks are illustrated in (e).</p

The basic characteristic of all SNP for ABCA1, APOE and HMGCR.

<p>The basic characteristic of all SNP for ABCA1, APOE and HMGCR.</p

The level of umbilical cord blood lipids in different SNPs of <i>ABCA1</i> rs2422493, <i>APOE</i> rs7412 and <i>HMGCR</i> rs12916(x¯ ± s, mmol/L).

<p>There is no significant difference in lipid level between different genotypes of <b><i>HMGCR</i></b> rs12916 and <b><i>ABCA1</i></b> rs2422493 and <b><i>APOE</i></b> rs7412 in cord blood.</p

Characteristics of the study population (n = 378).

<p>^a Lipid levels in maternal peripheral blood</p><p>^b Lipid levels in cord blood. <i>P</i>: Significance between preterm group and term group</p><p>Characteristics of the study population (n = 378).</p