Fig 1.
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 (Omax), respectively. The Hill working range, HWR, is the input range relevant for the calculation of the system’s nH. 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.
Fig 2.
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 (O1,max ≫ EC502), while in (B), it is lower than that value (O1,max < EC502).
Fig 3.
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.
Fig 4.
Schematic representations of Goldbeter-Koshland dose-response curves with K1 ≳ 1 and K2 ≪ 1 (see equation in S1 Text) shown in log-linear scale (A) and in log-log scale (B). The corresponding response coefficient (C) shows no local ultrasensitivity for low input values (i.e. R ∼ 1), but displays high local ultrasensitivity, even larger than the module’s Hill coefficient nH, for intermediate input regions.
Fig 5.
Schematic response function diagrams for two different compositions of two GK ultrasensitive modules are shown in panels (A) and (B).
Axes were arranged as explained in Fig 2’s caption. In panel (A) O1,max ≫ EC502, and module-1’s HWR covers the input region below EC501, a region in which the curve shows no local ultrasensitivity (R1 = 1). In panel (B) we show a special scenario where the O2,max/EC502 ratio was tuned in order to set module-1’s HWR in its most ultrasensitive region.
Fig 6.
O’Shaughnessy et al. cascades scheme.
The three models of the mammalian MAPK cascade expressed in yeast. Represented with a dual-step phosphorylation (A), with the Raf and MEK layers replaced by a Hill Function (B) whose parameters were obtained by fitting the function to the active MEK dose-response, and a MAPK cascade with a single-step phosphorylation (C). In each case, estradiol is the input and dually phosphorylated ERK is the output.
Fig 7.
Dose-response analysis for the dual step phosphorylation model.
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 X10i and X90i values of each layer (i.e. mechanistic scheme), while red solid vertical lines mark the layer’s X10i and X90i associated to the composition of response curves of each module (i.e. Fnon—seq).
Fig 8.
Fitting by a Hill function may obscure relevant behaviors.
Dose-response curve of active MEK in O’Shaughnessy model compared with its fit by a Hill function (A). Respective response coefficient (B). Even though the dose-responses of active MEK and the Hill function appear to be similar (A), there are strong differences in their local ultrasensitivity.
Fig 9.
Equivalence between a single-step layer in O’Shaughnessy model and a covalent modification cycle.
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) (K1 = 0.04 and K2 = 1000, see S1 Text).