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

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.

First-Order Reversal Curves of Sets of Bistable Magnetostrictive Microwires.

Amorphous microwires have attracted substantial attention in the past decade because of their useful technological applications. Their bistable magnetic response is determined by positive or negative magnetostriction, respectively. First-order reversal curves (FORC) are a powerful tool for analyzing the magnetization reversal processes of many-body ferromagnetic systems that are essential for a deeper understanding of those applications. After theoretical considerations about magnetostatic interactions among microwires, this work introduces a systematic experimental study and analysis of the FORC diagrams for magnetostrictive microwires exhibiting an individually bistable hysteresis loop, from a single microwire to sets of an increasing number of coupled microwires, the latter considered as an intermediate case to the standard many-body problem. We performed the study for sets of quasi-identical and different hysteretic microwires where we obtained the coercivity Hc and interaction Hu fields. In the cases with relevant magnetostatic interactions, FORC analysis supplies deeper information than standard hysteresis loops since the intrinsic fluctuations of the switching field generate a complex response. For sets of microwires with very different coercivity, the coercivity distributions of the individual microwires characterize the FORC diagram.

A model for the regulation of apoptosis intrinsic pathway: The potential role of the transcriptional regulator E2F in the point of no return.

Apoptosis has been extensively characterized by both experimental approaches and model simulations. However, it is still not fully understood how the regulation occurs, especially in the intrinsic pathway, which can be activated by a great variety of signals. In addition, the conditions in which a point of no return could be reached remain elusive. In this work, we use differential equations models to approach these issues. Our starting point was the model for caspase activation of Legewie et al. (Legewie S, et al., PLoS Computational Biology 2006, 2(9): e120), which exhibits irreversible bistability. We added an activation module to this model, with the main events related to mitochondrial outer membrane permeabilization, which includes cytochrome C release by the mitochondria and its effects on caspase activation and respiratory chain disruption. This "Extended Legewie Model" (ELM) uses BAK as the apoptotic stimulus and active caspase 3 as a measure of apoptosis activation. Unexpectedly, in the extended model, BAK cannot trigger apoptosis activation using physiologically sound initial values of the variables, due to limitations in apoptosome concentration increase. Therefore, the next step was to find a regulatory mechanism, allowing apoptosis activation in the ELM, starting from physiological initial concentrations. For this aim, we performed a sensitivity analysis on the 61 parameters of the system, finding that those producing the most relevant changes in the qualitative behaviour were the rates of synthesis of caspase 3, caspase 9 and XIAP. Based on these results, the transcription factor E2F was included in the ELM because it directly regulates the rate of synthesis of caspase 3 and 9. Depending on the concentration of E2F, the ELM shows different qualitative behaviours. On one hand, for low E2F apoptosis is impossible and for high E2F apoptosis is inevitable. Therefore, if E2F is sufficiently increased, the point of no return is crossed. On the other hand, for intermediate values of E2F there is a bistable region where the fate of the system also depends on the concentration of BAK and other signalling species.

Modeling the intracellular dynamics of the dengue viral infection and the innate immune response.

The interplay between the dengue virus and the innate immune response is not fully understood. Here, we use deterministic and stochastic approaches to investigate the dynamics of the interaction between the interferon-mediated innate immune response and the dengue virus. We aim to develop a quantitative representation of these complex interactions and predict their system-level dynamics. Our simulation results predict bimodal and bistable dynamics that represent viral clearance and virus-producing states. Under normal conditions, we determined that the viral infection outcome is modulated by the innate immune response and the positive-strand viral RNA concentration. Additionally, we tested system perturbations by external stimulation, such as the direct induction of the innate immune response by interferon, and a therapeutic intervention consisting of the direct application of mRNA encoding for several interferon-stimulated genes. Our simulation results suggest optimal regimes for the studied intervention approaches.

Toppling Pencils-Macroscopic Randomness from Microscopic Fluctuations.

We construct a microscopic model to study discrete randomness in bistable systems coupled to an environment comprising many degrees of freedom. A quartic double well is bilinearly coupled to a finite number N of harmonic oscillators. Solving the time-reversal invariant Hamiltonian equations of motion numerically, we show that for N=1, the system exhibits a transition with increasing coupling strength from integrable to chaotic motion, following the Kolmogorov-Arnol'd-Moser (KAM) scenario. Raising N to values of the order of 10 and higher, the dynamics crosses over to a quasi-relaxation, approaching either one of the stable equilibria at the two minima of the potential. We corroborate the irreversibility of this relaxation on other characteristic timescales of the system by recording the time dependences of autocorrelation, partial entropy, and the frequency of jumps between the wells as functions of N and other parameters. Preparing the central system in the unstable equilibrium at the top of the barrier and the bath in a random initial state drawn from a Gaussian distribution, symmetric under spatial reflection, we demonstrate that the decision whether to relax into the left or the right well is determined reproducibly by residual asymmetries in the initial positions and momenta of the bath oscillators. This result reconciles the randomness and spontaneous symmetry breaking of the asymptotic state with the conservation of entropy under canonical transformations and the manifest symmetry of potential and initial condition of the bistable system.

Edge fires drive the shape and stability of tropical forests.

In tropical regions, fires propagate readily in grasslands but typically consume only edges of forest patches. Thus, forest patches grow due to tree propagation and shrink by fires in surrounding grasslands. The interplay between these competing edge effects is unknown, but critical in determining the shape and stability of individual forest patches, as well the landscape-level spatial distribution and stability of forests. We analyze high-resolution remote-sensing data from protected Brazilian Cerrado areas and find that forest shapes obey a robust perimeter-area scaling relation across climatic zones. We explain this scaling by introducing a heterogeneous fire propagation model of tropical forest-grassland ecotones. Deviations from this perimeter-area relation determine the stability of individual forest patches. At a larger scale, our model predicts that the relative rates of tree growth due to propagative expansion and long-distance seed dispersal determine whether collapse of regional-scale tree cover is continuous or discontinuous as fire frequency changes.

Role reversal in a predator-prey interaction.

Predator-prey relationships are one of the most studied interactions in population ecology. However, little attention has been paid to the possibility of role exchange between species, despite firm field evidence of such phenomena in nature. In this paper, we build a mathematical model capable of reproducing the main phenomenological features of role reversal in a classical system and present results for both the temporal and spatio-temporal cases. We show that, depending on the choice of parameters, our role-reversal dynamical system exhibits excitable-like behaviour, generating waves of species' concentrations that propagate through space. Our findings fill a long-standing gap in modelling ecological interactions and can be applicable to better understanding ecological niche shifts and planning of sustainable ecosystems.