Dynamics of the Goodwin model combined with a “zero-order ultrasensitivity” phosphorylation module.

<p>The time series have been obtained by numerical integration of Eqs. (13), with the following parameter values: , , , , , . The kinetics rates of the phosphorylation/dephosphorylation reactions are as in Fig. 8 but multiplied by a factor 100. For clarity and to facilitate the comparison with the 3-variable model, variable has been multiplied by a factor 100. In panel B, a comparison of the limit cycle obtained for the present model (blue curve) and for the original 3-variable Goodwin model (black curve) shows a reasonable agreement between the two models. The period, about 40.7 a.u., is very close the period of the original Goodwin model. In panel D, the thin curve corresponds to the steady state of the phosphorylation module (cf. Fig. 8) and the thick curve is the trajectory of the present system.</p

Scheme of the Goodwin model combined with a single phosphorylation/dephosphorylation module.

<p>The phosphorylation/dephosphorylation module can generate sharp thresholds through “zero-order ultrasensitivity”.</p

Dynamics of the Goodwin model combined with the multisite phosphorylation module.

<p>These results have been obtained by numerical integration of Eqs. (11). Parameter values for the Goodwin model are as in Fig. 2. Kinetic parameter values for the phosphorylation module are as in Fig. 3 for but multiplied by a factor 100. Conservation parameter values, and , are the same as in Fig. 3. In panel A, the thin blue curve corresponds to at steady state, as obtained in Fig. 3, while the thick line denotes the trajectory of the present system. The inset is a zoom on the lower part of that curve. The oscillations are in very good agreement with the oscillations generated by the 3-variable model (Fig. 2, see also Fig. 7 for a comparison of the limit cycles). The period of the oscillations, about 40 a.u., is also consistent with the 3-variable model.</p

Scheme of the Goodwin model combined with the multisite phosphorylation module.

<p>In this model, variable is the kinase E which can be found in its free form or in a complex with any phosphoform of S. We also assume that all forms of free S, except , can induce the synthesis of X.</p

Scheme of the Goodwin model.

<p>In the original version of the model, the negative feedback exerted by Z on the synthesis of X is described by a non-linear Hill function.</p

Scheme of the Goodwin model combined with the double-phosphorylation module producing bistability.

<p>Scheme of the Goodwin model combined with the double-phosphorylation module producing bistability.</p

Bistability in the double-phosphorylation system (Eq. (17)).

<p>(A) Bifurcation diagram showing the steady state of as a function of . (B) Bistability domain in the parameter space. The other parameter values are: , , , , , , and . The bistable domain in panel A is delimited by two saddle-node bifurcation points at and (red points). The simulations have been carried out using XPP-AUTO software <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0069573#pone.0069573-Ermentout1" target="_blank">[49]</a>.</p

Comparison of the 3-variable Goodwin model with its variant based on multisite phosphorylation.

<p>Shown are the limit cycles obtained for the 3-variable model (black curve) and for its variant based on multisite phosphorylation, for various speeds of the phosphorylation module. The blue and red curves correspond to the cases illustrated in Figs. 5 and 6, respectively. The green curve is the limit cycle obtained when the kinetics rates of the multisite phosphorylation module are decreased by a factor 10 compared to the values given in Fig. 3. The violet dot is the stable steady state of the system when the kinetics rates of the multisite phosphorylation module are reduced by a factor 100.</p

Dynamics of the Goodwin model coupled to the bistable module (Eqs (18)).

<p>The parameter values for the Goodwin module are as in Fig. 11 except and . The parameter values for the bistable module are as in Fig. 13. In panel B, the black curve represents the limit cycle for the 3-variable model while the blue curve is the limit cycle for the present model. In panel D, the grey curve represents the steady state of the bistable module (as in Fig. 13) and the red curve is the trajectory of the present system.</p

Dynamics of Goodwin model.

<p>(A) Inhibitory Hill function (Eq. (4), with and ). (B) Limit cycle oscillations obtained by numerical integration of Eqs. (1)–(3) for the following parameter values (arbitrary units): , , , . The oscillation period is about 40 a.u. The dashed line indicates the Hill threshold .</p