Additional file 2 of Paracrine and autocrine regulation of gene expression by Wnt-inhibitor Dickkopf in wild-type and mutant hepatocytes

Matlab code. (ZIP 23 kb

Sensitivities for models of circadian and calcium oscillations.

<p>A-C: Circadian models. (A) Model for mammalian cells [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref037" target="_blank">37</a>] (compare <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.g001" target="_blank">Fig 1A and 1C</a> red triangles); (B) model for <i>D</i>. <i>melanogaster</i> [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref046" target="_blank">46</a>]; (C) model for <i>A</i>. <i>thaliana</i> [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref047" target="_blank">47</a>]. D-F: Calcium models. (D) Phenomenological model [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref038" target="_blank">38</a>] (compare <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.g001" target="_blank">Fig 1B and 1C</a> blue circles); (E) open-cell model [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref048" target="_blank">48</a>]; (F) closed-cell model [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref049" target="_blank">49</a>]. Black symbols denote median values, white symbols the sensitivities for the parameter set published together with the model. G, H: Box-plots of the period (G) and the amplitude (H) sensitivity distributions for the models from A-F. The schemes and further details of the models are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.g006" target="_blank">Fig 6</a>.</p

Work-flow of the analysis for the example of a calcium oscillations model [38].

<p>The analysis can be divided into eight parts. Details are described in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#sec010" target="_blank">Methods</a>.</p

Impact of the reaction kinetics on the sensitivity of the chain model.

<p>A-F: Schemes of the chain models with negative (A-C) or positive (D-F) feedback employing Michaelis-Menten kinetics in degradation reactions 3, 5, 7, 8 (A, D, indicated in gray, deg MM), in conversion reactions 2, 4, 6 (B, E, indicated in gray, conv MM), or in conversion and degradation reactions 2–8 (C, F, MM). G, H: Box-plots of the period sensitivities (G) or amplitude sensitivities (H) of the chain models with negative and positive feedback. I, J: Sensitivities of the negative feedback (I, neg fb, dark red dots) or positive feedback (J, pos fb, dark green dots) chain model with Michaelis-Menten kinetics in reactions 2–8 as shown in C or F, respectively. For comparison the data for the models with mass action kinetics (ma) in reactions 2–8 from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.g002" target="_blank">Fig 2D</a> are shown in addition.</p

Effect of structural characteristics on the sensitivities.

<p>Schematically, the effect of negative feedback (left) or positive feedback (right) and mass action kinetics (upper) or Michaelis-Menten kinetics (lower) on the period and amplitude sensitivities are depicted. The respective sensitivities are indicated by how much the period or amplitude of the perturbed system (oscillation for an example perturbation +Δ is shown in green) deviate from these characteristics in the unperturbed system (blue).</p

Feedback, Mass Conservation and Reaction Kinetics Impact the Robustness of Cellular Oscillations

<div><p>Oscillations occur in a wide variety of cellular processes, for example in calcium and p53 signaling responses, in metabolic pathways or within gene-regulatory networks, e.g. the circadian system. Since it is of central importance to understand the influence of perturbations on the dynamics of these systems a number of experimental and theoretical studies have examined their robustness. The period of circadian oscillations has been found to be very robust and to provide reliable timing. For intracellular calcium oscillations the period has been shown to be very sensitive and to allow for frequency-encoded signaling. We here apply a comprehensive computational approach to study the robustness of period and amplitude of oscillatory systems. We employ different prototype oscillator models and a large number of parameter sets obtained by random sampling. This framework is used to examine the effect of three design principles on the sensitivities towards perturbations of the kinetic parameters. We find that a prototype oscillator with negative feedback has lower period sensitivities than a prototype oscillator relying on positive feedback, but on average higher amplitude sensitivities. For both oscillator types, the use of Michaelis-Menten instead of mass action kinetics in all degradation and conversion reactions leads to an increase in period as well as amplitude sensitivities. We observe moderate changes in sensitivities if replacing mass conversion reactions by purely regulatory reactions. These insights are validated for a set of established models of various cellular rhythms. Overall, our work highlights the importance of reaction kinetics and feedback type for the variability of period and amplitude and therefore for the establishment of predictive models.</p></div

Sensitivities of models for various biological oscillations.

<p>A: Model of the repressilator [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref050" target="_blank">50</a>] (brown circles). B: Model of the MAPK pathway [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref051" target="_blank">51</a>] (yellow squares). C: Model of the glycolysis [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref052" target="_blank">52</a>,<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref053" target="_blank">53</a>] (olive stars). D: Model of the cell cycle [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref054" target="_blank">54</a>] (salmon triangles). White framed symbols denote median values. The schemes and further details of the models are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.s009" target="_blank">S9 Fig</a>. The originally published kinetics were altered employing Michaelis-Menten kinetics instead of mass action kinetics in A, C, D and mass action kinetics instead of Michaelis-Menten kinetics in B (black dots, black circle as median values). ma: mass action kinetics, MM: Michaelis-Menten kinetics.</p

Model structures of the circadian and calcium oscillations models examined in Fig 5.

<p>The column ‘Id’ gives the identifier of the model according to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.g005" target="_blank">Fig 5</a>. In the model structure column, black and gray arrows denote reactions in the models, reactions with Michaelis-Menten kinetics (MM) are thereby marked in gray. Dashed arrows represent regulated productions. Red lines ending in T-shape indicate negative regulations, green arrows denote positive regulations. Green or red arrows without source (in E, F) represent regulations by species S₁ (cytosolic calcium) on the according reaction.</p

Impact of mass conservation properties on the sensitivity of the chain model.

<p>A: Schemes and equations of a mass conversion and a regulated production rate (reaction 3). For a mass conversion, the reaction rate occurs in the equation of the source species as well as of the product species (highlighted by boxes). For a regulated production rate, only the equation of the product species is affected by the reaction (highlighted by box). Regulated production rates are in the following represented by dashed arrows, mass conversions by solid lines. B, C: Schemes of the chain model with negative (B) or positive (C) feedback in that mass conversion reactions 2, 4, 6 or 4, 6, respectively, have been replaced by regulated production rates. D, E: Sensitivities for the negative (D, neg fb, red squares) and positive (E, pos fb, dark green dots) feedback chain model as shown in B and C, respectively. For comparison, sensitivities considering mass conversion reaction rates only are shown (data from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.g002" target="_blank">Fig 2D</a>).</p

Sensitivity analysis of a circadian and a calcium oscillation model.

<p>A: Model scheme of the mammalian circadian oscillation model [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref037" target="_blank">37</a>]. The basic negative feedback is marked by an encircled minus sign, purely regulatory interactions without mass flow are indicated by dashed arrows. B: Model scheme of a phenomenological calcium oscillation model [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005298#pcbi.1005298.ref038" target="_blank">38</a>]. The basic positive feedback is indicated by an encircled plus sign. C: Period and amplitude sensitivities for the circadian model (red triangles) and the calcium model (blue circles). Each point gives the sensitivities obtained for one parameter set. Median values are indicated by black symbols.</p