Avalanche of bifurcations and hysteresis in a model of cellular differentiation

A mathematical model of the biochemical network underlying left-right asymmetry establishment in mammals

The expression of the TGF-β protein Nodal on the left side of vertebrate embryos is a determining event in the development of internal-organ asymmetry. We present a mathematical model for the control of the expression of Nodal and its antagonist Lefty consisting entirely of realistic elementary reactions. We analyze the model in the absence of Lefty and find a wide range of parameters over which bistability (two stable steady states) is observed, with one stable steady state a low-Nodal state corresponding to the right-hand developmental fate, and the other a high-Nodal state corresponding to the left. We find that bistability requires a transcription factor containing two molecules of phosphorylated Smad2. A numerical survey of the full model, including Lefty, shows the effects of Lefty on the potential for bistability, and on the conditions that lead to the system reaching one or the other steady state.

Continuous description of lattice discreteness effects in front propagation

Models describing microscopic or mesoscopic phenomena in physics are inherently discrete, where the lattice spacing between fundamental components, such as in the case of atomic sites, is a fundamental physical parameter. The effect of spatial discreteness over front propagation phenomenon in an overdamped one-dimensional periodic lattice is studied. We show here that the study of front propagation leads in a discrete description to different conclusions that in the case of its, respectively, continuous description, and also that the results of the discrete model, can be inferred by effective continuous equations with a supplementary spatially periodic term that we have denominated Peierls-Nabarro drift, which describes the bifurcation diagram of the front speed, the appearance of particle-type solutions and their snaking bifurcation diagram. Numerical simulations of the discrete equation show quite good agreement with the phenomenological description.

Properties of a "phase transition" induced by antiangiogenetic therapeutical protocols

Inhibiting angiogenesis has been found to be an interesting therapeutical strategy against cancer. In fact, the success of tumor growth is subordinated to the corresponding increase of the vascular system feeding the neoplasm. However, optimization and design of proper antiangiogenetic therapeutical strategies is still an open problem. We apply a recently developed angiogenesis model to study how variations in the relevant parameters, e.g., induced by chemicals, may cause a "phase transition" to a region in the parameter space in which angiogenesis is not succesful. To demonstrate the reliability of our approach and its usefulness, we will study some specific drugs and use our model to investigate the influence of the main variables involved in a clinical treatment: the administration time, the duration of the drug effect, and the drug dose.

Effects of anatomical constraints on tumor growth

Competition for available nutrients and the presence of anatomical barriers are major determinants of tumor growth in vivo. We extend a model recently proposed to simulate the growth of neoplasms in real tissues to include geometrical constraints mimicking pressure effects on the tumor surface induced by the presence of rigid or semirigid structures. Different tissues have different diffusivities for nutrients and cells. Despite the simplicity of the approach, based on a few inherently local mechanisms, the numerical results agree qualitatively with clinical data (computed tomography scans of neoplasms) for the larynx and the oral cavity.

Local interaction simulation approach for the response of the vascular system to metabolic changes of cell behavior

The self-regulatory interactions between cells and the vascular system are mediated by signals propagating at a finite speed. In order to build up a physical model of these processes, several features, such as storing of internal energy, nonclassical nonlinear behavior, and delay and threshold effects, have to be taken into account. Considering cells as particles in different metabolic states according to their internal energy, we have developed a model based on the local interaction simulation approach. Several numerical results, in qualitative agreement with biological observations, illustrate the applicability of the model and the method to implement it.

Noise sustained propagation: local versus global noise

We expand on prior results on noise supported signal propagation in arrays of coupled bistable elements. We present and compare experimental and numerical results for kink propagation under the influence of local and global fluctuations. As demonstrated previously for local noise, an optimum range of global noise power exists for which the medium acts as a reliable transmission "channel." We discuss implications for propagation failure in a model of cardiac tissue, and present a general theoretical framework based on discrete kink statistics. Valid for generic bistable chains, the theory captures the essential features observed in our experiments and numerical simulations.