Retroactive signaling in multi-cycle pathways.

<p>M for all to , except for (A)–(C) M, and for (D) M. (A) is varied in the range [M] such that goes from to . M, M, M, MM, M. (B) same but is varied on the range and M. (C) identical to (A) except that cycle 2 is deactivated, with MM. (D) M, M, M, M, M, M, MM, MM.</p

Phosphorylated fraction of protein 2 as a function of 2 control parameters of the downstream cycle 1, namely and .

<p>The graphs are obtained by solving Eqs.(16)-(18) with the following parameters : , M, M; On the left figures (B,D,F,H) cycle 2 is assumed deactivated, with MM. These values are swapped for the right figures (C,E,G,I) where cycle 2 is assumed activated. Panels (B,C,F,G) : cycle 1 is either deactivated (M), or activated (M). On panels (D,E,H,I), phosphatase is varied from to (so that varies from 0 to 2). Panels (D,H) : for the upper curve the total protein 1 is M and for the lower curve M. Panel (E,I) : for the upper curve the total protein 1 is M and for the lower curve M.</p

Behaviors of cycle 1 as a function of , the total protein in cycle 1.

<p>The kinase for this cycle is denoted by and the phosphatase by . The abscissa are scaled by the characteristic range , cf. Eq. (1). A) Two cases are considered for cycle 1, which is said deactivated if and activated if . B-C) Increase of the intermediate complex when cycle 1 is respectively deactivated or activated. D-E) Variations of activated and non-activated proteins in the two cases and . The graphs were obtained by solving Eqs.(16)-(18) with the following parameters : , M, M, ; panels (B-D) : M.; panels (C-E) : M.</p

Motifs of short signaling pathways illustrating the concept of retroactive signaling in (A) a 2-cycle cascade and (B) in a 3-cycle cascade.

<p>Thick arrows indicate the direction of signaling.</p

Equivalence between a single-step layer in O’Shaughnessy model and a covalent modification cycle.

<p>O’Shaughnessy et al. single-step layer (A) and the equivalent covalent modification cycle (B). (C) Steady state transfer functions of ERK layer in isolation of the O’Shaughnessy single-step cascade (blue dashed line), compared to a centered Goldbeter-Function with equivalent parameters (red solid line) (<i>K</i>₁ = 0.04 and <i>K</i>₂ = 1000, see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180083#pone.0180083.s001" target="_blank">S1 Text</a>).</p

Schematic response function diagrams for two different compositions of two GK ultrasensitive modules are shown in panels (A) and (B).

<p>Axes were arranged as explained in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180083#pone.0180083.g002" target="_blank">Fig 2</a>’s caption. In panel (A) <i>O</i>_1,<i>max</i> ≫ <i>EC</i>50₂, and module-1’s HWR covers the input region below <i>EC</i>50₁, a region in which the curve shows no local ultrasensitivity (<i>R</i>₁ = 1). In panel (B) we show a special scenario where the <i>O</i>_2,<i>max</i>/<i>EC</i>50₂ ratio was tuned in order to set module-1’s HWR in its most ultrasensitive region.</p

Dose-response analysis for the dual step phosphorylation model.

<p>Transfer functions for each of the three layers of the MAPK cascade (A-C), obtained considering for each layer i) the isolated module (Is, dotted blue), ii) a mechanistic implementation of the model (Seq, dashed-turquoise) and iii) the mathematical composition of isolated response functions (Non-Seq, continous red). The corresponding response coefficient curves are shown in panels (D-F). Turquoise dashed vertical lines show the <i>X</i>10_<i>i</i> and <i>X</i>90_<i>i</i> values of each layer (i.e. mechanistic scheme), while red solid vertical lines mark the layer’s <i>X</i>10_<i>i</i> and <i>X</i>90_<i>i</i> associated to the composition of response curves of each module (i.e. <i>F</i>^{<i>non</i>—<i>seq</i>}).</p

Hill function dose-response.

<p>Schematic representation of Hill-type dose-response curves, in log-linear (A) and log-log scale (B). The EC10 and EC90 are the inputs needed to produce an output of 10% and 90% of the maximal response (<i>O</i>_<i>max</i>), respectively. The <i>Hill working range</i>, HWR, is the input range relevant for the calculation of the system’s <i>n</i>_<i>H</i>. For isolated modules, the HWR = [EC10, EC90]. Panel (C) displays the local ultrasensitivity (the response coefficient R) as a function of input. Note that for Hill functions, inputs much smaller than the EC50 have Rs around the Hill coefficient.</p

Schematic diagrams of the response function when composing a Hill function in module-1, with a linear function (in green) or a power function (in blue) in module-2.

<p>Schematic diagrams of the response function when composing a Hill function in module-1, with a linear function (in green) or a power function (in blue) in module-2.</p

Hill functions composition.

<p>Schematic response function diagrams for two different compositions of a pair of Hill-type ultrasensitive modules. In each panel, the dose-response function of the first module is displayed in the lower semi-plane: the downward vertical axis representing the first module’s input signal while its response function, which corresponds to the second module’s input, is displayed along the horizontal axis. The dose-response curve for the second module is displayed in the upper-plane. In (A) the maximum output of the first module is higher than the EC50 of the second module (<i>O</i>_1,<i>max</i> ≫ <i>EC</i>50₂), while in (B), it is lower than that value (<i>O</i>_1,<i>max</i> < <i>EC</i>50₂).</p