On the validity of the local Fourier analysis

Local Fourier analysis (LFA) is a useful tool in predicting the convergence factors of geometric multigrid methods (GMG). As is well known, on rectangular domains with periodic boundary conditions this analysis gives the exact convergence factors of such methods. In this work, using the Fourier method, we extend these results by proving that such analysis yields the exact convergence factors for a wider class of problems

p53 bifurcation diagram.

<p>Red curves represent stable steady states and the black curves represent unstable steady states. The bifurcation diagram shows a tri-stability with hysteresis behavior. The ‘Going up’ means to turn ‘ON’ p53 and the ‘Coming down’ means to turn ‘OFF’ p53.</p

Dynamical analysis of cellular ageing by modeling of gene regulatory network based attractor landscape

<div><p>Ageing is a natural phenomenon that is inherently complex and remains a mystery. Conceptual model of cellular ageing landscape was proposed for computational studies of ageing. However, there is a lack of quantitative model of cellular ageing landscape. This study aims to investigate the mechanism of cellular ageing in a theoretical model using the framework of Waddington’s epigenetic landscape. We construct an ageing gene regulatory network (GRN) consisting of the core cell cycle regulatory genes (including p53). A model parameter (activation rate) is used as a measure of the accumulation of DNA damage. Using the bifurcation diagrams to estimate the parameter values that lead to multi-stability, we obtained a conceptual model for capturing three distinct stable steady states (or attractors) corresponding to homeostasis, cell cycle arrest, and senescence or apoptosis. In addition, we applied a Monte Carlo computational method to quantify the potential landscape, which displays: I) one homeostasis attractor for low accumulation of DNA damage; II) two attractors for cell cycle arrest and senescence (or apoptosis) in response to high accumulation of DNA damage. Using the Waddington’s epigenetic landscape framework, the process of ageing can be characterized by state transitions from landscape I to II. By <i>in silico</i> perturbations, we identified the potential landscape of a perturbed network (inactivation of p53), and thereby demonstrated the emergence of a cancer attractor. The simulated dynamics of the perturbed network displays a landscape with four basins of attraction: homeostasis, cell cycle arrest, senescence (or apoptosis) and cancer. Our analysis also showed that for the same perturbed network with low DNA damage, the landscape displays only the homeostasis attractor. The mechanistic model offers theoretical insights that can facilitate discovery of potential strategies for network medicine of ageing-related diseases such as cancer.</p></div

The ageing network model.

<p>The core GRN for ageing (arrows represent activations and bar arrows represent inhibitions).</p

Top view of landscape II in Fig 6(B).

<p>There is a line connecting the two attractors. The line is formed by two unstable manifolds representing a kinetic path between the cell cycle arrest and senescence (or apoptosis) basins of attraction.</p

No cancer attractor for low accumulation of DNA damage.

<p>In response to the inactivation of p53 with the deletion of two interactions to p53 (the activation of p53 by ATM and by ARF respectively), the landscape displays only one attractor for homeostasis and no cancer attractor for <i>a</i> = 0.5 and <i>b</i> = 0.05, which might correspond to young cells with low accumulation of DNA damage.</p

Disappearance of cancer attractor.

<p>(A) The landscape with the cancer attractor (<i>a</i> = 1.5, <i>b</i> = 0.05). (B) For <i>a</i> = 1.5, the cancer attractor disappears when the inhibition rate <i>b</i> is increased to 1.2.</p

Bistable activation of E2F.

<p>Bifurcation diagram of E2F (y-axis) with respect to growth signals (x-axis).</p

Potential landscapes for cell fate attractors (measured by p53 and ATM).

<p>(A) Landscape I: For low activation rate (<i>a</i> = 0.5, <i>b</i> = 0.05), the landscape displays one attractor (blue color) with low p53 protein corresponding to homeostasis. (B) Landscape II: For high activation rate (<i>a</i> = 1.5, <i>b</i> = 0.05), the landscape displays two attractors, one for cell cycle arrest and the other with high level of p53 protein corresponding to senescence or apoptosis. The senescence (or apoptosis) attractor is a dominant attractor of landscape II and thus a potential biomarker of ageing.</p

Two-parameter bifurcation diagram.

<p>Blue horizontal line indicates <i>b</i> = 0.05. Red curves represent stable steady states. This two-parameter bifurcation diagram shows that there are different combinations of values of <i>a</i> and <i>b</i> which can result in tri-stability.</p