Presentation1_Pseudo-nullclines enable the analysis and prediction of signaling model dynamics.pdf

A powerful method to qualitatively analyze a 2D system is the use of nullclines, curves which separate regions of the plane where the sign of the time derivatives is constant, with their intersections corresponding to steady states. As a quick way to sketch the phase portrait of the system, they can be sufficient to understand the qualitative dynamics at play without integrating the differential equations. While it cannot be extended straightforwardly for dimensions higher than 2, sometimes the phase portrait can still be projected onto a 2-dimensional subspace, with some curves becoming pseudo-nullclines. In this work, we study cell signaling models of dimension higher than 2 with behaviors such as oscillations and bistability. Pseudo-nullclines are defined and used to qualitatively analyze the dynamics involved. Our method applies when a system can be decomposed into 2 modules, mutually coupled through 2 scalar variables. At the same time, it helps track bifurcations in a quick and efficient manner, key for understanding the different behaviors. Our results are both consistent with the expected dynamics, and also lead to new responses like excitability. Further work could test the method for other regions of parameter space and determine how to extend it to three-module systems.</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

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

Additional file 4 of Signaling cascades transmit information downstream and upstream but unlikely simultaneously

Threshold robustness. (PDF 249 kb

Additional file 2 of Signaling cascades transmit information downstream and upstream but unlikely simultaneously

Likelihood curves and maxima. (PDF 483 kb