Energetic landscape of the contributions of the catalytic and binding terms for ordered reaction mechanisms with different molecularities.

<p>Molecularities analysed include (A) Uni-Uni, (B) Bi-Bi and (C) Ter-Ter reaction mechanisms. In each case, the following Gibbs free energy differences were considered: -1, -5, -10, -20, -30, -50 and -80 kJ/mol and a reference flux of 1 mM/min was assumed. For this analysis, 10⁴ parameter sets were sampled. The blue and red lines denote the median of the distributions for the catalytic and binding terms of the reaction energetics, respectively, while the shaded areas represent the 95% confidence regions for the respective contributions. The black line represents the sum of the two contributions which yields Δ<i>G</i>_<i>r</i>/<i>RT</i>. As it can be observed, both contributions decrease linearly with decreasing Δ<i>G</i>_<i>r</i>/<i>RT</i>. In addition, the higher the molecularity the higher the relative importance of the binding term. The numbers showed above each line denote their slope and quantify the extent of their energetic contributions. Notably, the sum of all contributions must be one.</p

Revealing the impact of thermodynamics on enzyme kinetics.

<p>(A) Schematic representation of the bimolecular mechanisms considered for the analysis. (B) Substrate elasticities for three mechanisms considered: ordered (green), ping-pong (blue) and random-order (red) at different Gibbs free energy differences of reaction ranging from -1 to -80 kJ/mol. Elasticities were calculated every -1 kJ/mol interval by sampling 10⁴ instances each time. In this panel, each line represents the median of the respective elasticity distribution, while the shaded areas denote the 95% confidence regions. Depending on the chosen thermodynamic reference state, the elasticities can be almost constant for regions far from equilibrium (Δ<i>G</i>_<i>r</i> <-20 kJ/mol) or highly variable close to equilibrium (dashed blue line). The dashed green line represents the limit for the lineal regime of substrate elasticity variation, while the dashed red line denotes the zero limit. (C) Product elasticities for the same representative bimolecular mechanisms. The lines and shaded areas represent the same as for the substrate elasticities. Product inhibition is on average negligible for favourable thermodynamic conditions, <i>i</i>.<i>e</i>. Δ<i>G</i>_<i>r</i> < -30 kJ/mol (right side of the dashed blue line).</p

General framework for thermodynamically consistent parameterization and efficient sampling of metabolic reactions.

<p>(A) General Reaction Assembly and Sampling Platform (GRASP) workflow. The steps indicated by * are only required for parameterizing and sampling allosteric reactions. (B) Example of pattern constraints present in a random-order Uni-Bi mechanism with the formation of a ternary complex (<i>EPQ</i>). The intermediate <i>EPQ</i> splits to the <i>EP</i> and <i>EQ</i> enzyme intermediates. This behaviour can be modelled by uniformly sampling the solution space for the steady-state elementary net fluxes </p><p></p><p></p><p></p><p><mi>v</mi></p><p>elem</p><p>net</p><p></p><p></p><p></p><p></p>, which captures the stoichiometric properties of the reaction pattern. (C) Illustration of energetic constrains in the previous mechanism. There are 2 possible cycles converting <i>A</i> into <i>P+Q</i>, namely: <i>E</i> → <i>EA</i> → <i>EPQ</i> → <i>EP</i> → <i>E</i> and <i>E</i> → <i>EA</i> → <i>EPQ</i> → <i>EQ</i> → <i>E</i>. According to the principle of microscopic reversibility both pathways must be energetically equivalent as they execute the same reaction. Thermodynamic constraints for both paths are illustrated in the free energy graph. (D) Thermodynamic constraints on the equilibrium allosteric constant (<i>L</i>) within the generalized MWC model. The value of <i>L</i> depends on the ligand affinity of the active and tense states. Notably, in the absence of ligand the allosteric constant <i>L</i> favours the tense state, whereas with increasing concentrations of ligand the active state becomes more favoured (lower conformational energy).<p></p

Sampling monomeric cooperativity of mammalian glucokinase.

<p>(A) Probability distribution of the Hill coefficient for the glucokinase. Approximately 93% out of 10⁴ sampled kinetics displays cooperative behaviour. (B) Probability distribution for the ratio of the forward and reverse rate constants from the low- to the high-affinity enzyme state. The great majority (98.1%) of the models exhibiting positive cooperativity agrees with a slow transition from the low- to the high-affinity enzyme states. (C) Comparison of the kinetic space described by the model developed by Storer and Cornish-Bowden [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004195#pcbi.1004195.ref048" target="_blank">48</a>] (blue) and the sampled kinetics using the mnemonic model (red). Each line represents the real kinetic behaviour described by the two models, while the shaded areas denote one-standard-deviation confidence regions for each approach.</p

Schematic representation of the mnemonic model for the mammalian glucokinase.

<p>This model proposes a slow conformational transition from a low-affinity state (<i>E</i>^<i>*</i>) to a high-affinity state (<i>E</i>) that can be enhanced with increasing glucose concentration yielding the observed cooperative behaviour.</p

Complex allosteric regulatory interactions control the kinetic behaviour of phosphoenolpyruvate carboxylase (PEPC) in <i>E</i>. <i>coli</i>.

<p>(A) PEPC regulates anaplerotic metabolism in <i>E</i>. <i>coli</i> through several allosteric interactions. This enzyme carries out the conversion of phosphoenolpyruvate and bicarbonate into oxaloacetate and inorganic phosphate. All reactants are highlighted in black. The red and green dashed lines denote the inhibition and activation exerted by different effectors, respectively. (B) Ordered Bi-Bi mechanism for PEPC catalysis. Phosphoenolpyruvate, bicarbonate, oxaloacetate and orthophosphate are denoted by the abbreviations <i>pep</i>, <i>hco</i>_<i>3</i>^<i>−</i>, <i>oaa</i> and <i>p</i>_<i>i</i>, respectively. (C) Mechanism of synergistic activation of PEPC transitions proposed by Smith et al. [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004195#pcbi.1004195.ref057" target="_blank">57</a>]. This mechanism considers two separate binding sites for the two types of activators (allosteric sites) and another for the substrates (catalytic site), each of them being capable of independently interacting with different enzymatic complexes. Notably, this model assumes the existence of the relaxed enzyme (active) only in the presence of activators, one of which may be the substrate <i>pep</i>.</p

Analysis of Panther Pathways data.

<p>Output signals from the apoptosis and T-cell activation pathways described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0041977#pone.0041977-Thomas1" target="_blank">[5]</a> are shown with the minimum number of input signals required for activation, and the number of (non-trivial) input combinations that give rise to the given output. Certain outputs are created in the curation process (eg, Bim) and thus lack results in the uncurated data.</p

Data Properties and Statistics.

<p>The nature of the topology is best described by taking measurements of its representative graph (formulated as discussed in the Curation section of the <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0041977#s3" target="_blank">Methods</a>). Curation and SCC removal (by its nature) reduces connectivity within the graph, leading to the observed differences between the curated/uncurated pairs. Reactome, by contrast, is a much larger dataset with much greater complexity, as shown in number of connected components and the characteristic path length.</p

Logical loop structure, with an example from Reactome.

<p><b>a)</b> An example of a logical loop structure. There exist two satisfying assignments to this set of statements when J’ is set to 1 - one with all variables except I set to 1, and one with all variables set to 1. To exclude the former solution from the solution space, the statement describing the relationship represented as a dashed line is removed from the logical formulation. <b>b)</b> In one mechanism of ERK phosphorylation, MEK and ERK are activated by various upstream processes (inputs). These active signals then form a complex, with MEK acting as a catalytic subunit of the complex resulting in ERK phosphorylation. The complex dissociates post-phosphorylation to yield phosphorylated ERK and MEK. Conversion to the logical form of this reaction yields a set of logic statements with associated SCC that has topology similar to <b>a)</b>.</p

The main window of PATHLOGIC-S provides problem instantiation options.

<p>Custom objectives are supported through an objective builder, along with several presets. Target specification and activity settings are through the table presented, which can be restricted to show only system inputs and system outputs. At present, PATHLOGIC-S offers two methods of curation - one at the point of execution through dialog boxes, and one prior to execution through Cytoscape. Once the user clicks OK, the software executes the simulation, producing as output either a tab-delimited summary data file or a series of GML files specifying visualizations of the solutions found.</p