Illustration of the smart Pareto filter.

<p>a) Indistinguishability region. b) Algorithm performance example.</p

Generic Pareto front.

<p>Full blue points indicate members of the pareto set. Point (a) is the optimum for objective function for a given value of (red points). Point (b) minimizes for another value of (compared to green points). For a member of the Pareto set, say (c), any attempt to improve a goal involves worsening the other, point (d) for comparison. Empty blue points are other possible solutions that are worse than those in the Pareto set.</p

Proposed algorithm for the multiobjective global optimization of metabolic networks.

<p>This method allows not only to generate a Pareto set, but also to systematically select the most promising subset of enzymatic profiles embedded therein.</p

14 solutions efficient of order 12 in decreasing order of .

<p>Recall that columns labeled as <i>K</i>_r represent indeed . Enzyme 1: Hexose transporters, enzyme 2: Glucokinase/Hexokinase, enzyme 3: Phosphofructokinase, enzyme 4: Trehalose 6-phosphate syntase complex (+Glycogen production), enzyme 5: Glyceraldehyde-3-phosphate dehydrogenase, enzyme 6: GOL (Glycerol production),enzyme 7: Pyruvate kynase, enzyme 8: ATPase, metabolite 1: Internal glucose, metabolite 2: Glucose-6-phosphate, metabolite 3: Fructose-1,6-diphosphate, metabolite 4: Phosphoenolpyruvate, metabolite 5: Adenosine triphosphate.</p

Illustrative example for the Pareto order of efficiency concept.

<p>Blue solution is efficient of order 5, whereas red solution is efficient of order 4 and green solution is efficient of order 3.</p

Pareto curve (blue circles) of the bi-criteria problem considering

<p><b> and </b><b> (Hexose transporters).</b> The other points represent projections of the same variables obtained during other bi-criteria optimization problems: - (red squares), - (magenta triangles), - (black stars), - (blue diamonds), - (red plus signs), - (magenta cross signs) and - (black asterisks). Fold-Change factors correspond to: : Hexose transporters, : Glucokinase/Hexokinase, : Phosphofructokinase, : Trehalose 6-phosphate syntase complex (+Glycogen production), : Glyceraldehyde-3-phosphate dehydrogenase, : GOL (Glycerol production), : Pyruvate kynase, : ATPase.</p

Box plot for the normalized Pareto set.

<p>In the bottom axis the fourteen objectives are represented. Objectives 1–8 correspond to – (see legend in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0043487#pone-0043487-g002" target="_blank">Figure 2</a>), objective 9 is indeed whereas the remaining 5 objectives represent –. : Internal glucose, : Glucose-6-phosphate, : Fructose-1,6-diphosphate, : Phosphoenolpyruvate, : Adenosine triphosphate.</p

Lower and upper bounds for objectives among the values attained by the set of Pareto solutions of order

<p><b>.</b> In particular, 611 solutions are efficient of order 14 (i.e., these are indeed the solutions obtained after applying the Smart filter); 214 solutions are efficient of order 13; and 14 solutions are efficient of order 12. Objectives are ordered as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0043487#pone-0043487-g003" target="_blank">Figure 3</a>. See legends in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0043487#pone-0043487-g002" target="_blank">Figure 2</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0043487#pone-0043487-g003" target="_blank">3</a>.</p

Percentage of parameter space where bistable responses are possible^a.

a<p>Some bidimensional sections of the multidimensional parameter space of bistability are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0031095#pone.0031095.s002" target="_blank">Figure S2</a>. The results show that in TCS with a bifunctional SK, both a TC_SK and a TC_RR cause a decrease in the size of the parametric region of bistability, with one exception: Model C has a larger parametric region of bistability when the signaling target is SK autophosphorylation (k₁). However, in systems with a monofunctional SK, a TCSK causes an increase and a TCRR causes a decrease in the size of the parametric region of bistability if the environment modulates the SK dephosphorylation (k₂). A|B stands for Model A controlled for Model B. A|C stands for Model A controlled for Model C.</p

Analyzed Two Component Systems modules.

<p>Model A represents a prototypical TCS. Model B represents a TCS with a SK-binding third component (TC_SK). Model C represents a TCS with a RR-binding third component (TC_RR). SK: sensor kinase; RR: response regulator; SKP: phosphorylated SK; RRP: phosphorylated RR; Ph: alternative phosphatase that dephosphorylates RRP; SKRR: dead-end complex, resulting from the binding of SK and RR; SKPRR: protein complex formed by the binding of SKP and RR; SKRRP: protein complex formed by the binding of SK and RRP; PhRRP: protein complex formed by the binding of Ph and RRP; SKTC and RRPTC: protein complexes formed by the binding of the third component to SK and RRP, respectively; (k₁, …, k₁₈): kinetic constants of the individual reactions. For simplicity, ATP and the release of inorganic phosphate are omitted. To analyze TCS modules with monofunctional sensors, k₈ is set to 0. To analyze TCS modules with bifunctional sensors, k₈ is set to be different from 0.</p