Catálogo de publicaciones - libros

A transition network (TN) is a graph-theoretical concept describing the transitions between (meta)stable states of dynamical systems. Here, we review methods to generate and analyze a TN for molecular systems. The appropriate identification of states and transitions from the potential energy surface of the molecule is discussed. We describe a formalism transforming a TN on a static energy surface into a time-dependent dynamic TN that yields the population probabilities for each system state and the interstate transition rates. Three analysis methods that help in understanding the dynamics of the molecular system based on the TN are discussed: (1) Disconnectivity graphs allow important features of the energy surface captured in a static TN to be visualized, (2) graph-theoretical methods enable the computation of the best transition paths between two predefined states of the TN, and (3) statistical methods from complex network analysis identify important features of the TN topology. A broad review of the literature is given, and some open research directions are discussed.

Pigmentation patterns in butterfly wings are one of the most spectacular and vivid examples of pattern formation in biology. In this chapter, we devote our attention to the mechanisms for generating global patterns. We focus on the relationship between pattern forming mechanisms for the fore- and hindwing patterns. Through mathematical modeling and computational analysis of and , our results indicate that the patterns formed on the forewing need not correlate to those of the hindwing in the sense that the formation mechanism is the same for both patterns. The independence of pattern formation mechanisms means that the coordination of united patterns of fore- and hindwings is accidental. This is remarkable, because from Oudemans’s principle [10], patterns appearing on the exposed surface of fore- and hindwing at the natural resting position are often integrated to form a composite and united adaptive pattern with their surrounding environment.

Cells of the embryonic vertebrate limb in high-density culture undergo chondrogenic pattern formation, which results in the formation of regularly spaced “islands” of cartilage analogous to the cartilage primordia of the developing limb skeleton. In this chapter a discrete, multiscale agent-based stochastic model is described, which is based on an extended cell representation coupled with biologically motivated reaction-diffusion processes and cell-matrix adhesion, for studying the behavior of limb bud precartilage mesenchymal cells. The model is calibrated using experimental data, and the sensitivity of key parameters is studied.

Swarming pattern formation of self-propelled entities is a prominent example of collective behavior in biology. Here we focus on bacterial swarming and show that the rod shape of self-propelled individuals is able to drive swarm formation without any kind of signaling.

The underlying mechanism we propose is purely mechanical and is evidenced through two different mathematical approaches: an on-lattice and an off-lattice individual-based model. The intensities of swarm formation we obtain in both approaches uncover that the length-width aspect ratio controls swarm formation. Moreover we show that there is an optimal aspect ratio that favors swarming.

Free boundary problems associated with biological tissue growing under conditions of nutrient limitation are formulated. Analysis by linear-stability methods, clarifying the models’ stability properties, is then described.

We present a modified first-order backward Euler finite difference scheme to solve advection-reaction-diffusion systems on fixed and continuously deforming domains. We compare our scheme to the second-order semi-implicit backward finite differentiation formula and conclude that for the type of equations considered, the first-order scheme has a larger region of stability for the time step than that of the second-order scheme (at least by a factor of ten) and therefore the first-order scheme becomes a natural choice when solving advection-reactiondiffusion systems on growing domains.

A simple mathematical model is proposed to study the influence of cell fission on transport. The model describes fractional tumor development, which is a one-dimensional continuous time random walk (CTRW). An answer to the question of how the malignant neoplasm cells appear at an arbitrary distance from the primary tumor is proposed. The model is a possible consideration for diffusive cancers as well. A chemotherapy influence on the CTRW is studied by an observation of stationary solutions.

In this chapter we briefly discuss the results of a mathematical model formulated in [22] that incorporates many processes associated with tumour growth. The deterministic model, a system of coupled non-linear partial differential equations, is a combination of two previous models that describe the tumour-host interactions in the initial stages of growth [11] and the tumour angiogenic process [6]. Combining these models enables us to investigate combination therapies that target different aspects of tumour growth. Numerical simulations show that the model captures both the avascular and vascular growth phases. Furthermore, we recover a number of characteristic features of vascular tumour growth such as the rate of growth of the tumour and invasion speed. We also show how our model can be used to investigate the effect of different anti-cancer therapies.

Glioblastoma is the most malignant form of brain cancer. It is extremely invasive; the mechanisms that govern invasion are not well understood. To better understand the process of invasion, we conducted an experiment in which a 3D tumor spheroid is implanted into a collagen gel. The paths of individual invasive cells were tracked. These cells were modeled as radially biased, persistent random walkers. The radial velocity bias was found to be 19.6 μm/hr.

A theoretical model based on the molecular interactions between a growing tumor and a dynamically evolving blood vessel network describes the transformation of the regular vasculature in normal tissues into a highly inhomogeneous tumor-specific capillary network. The emerging morphology, characterized by the compartmentalization of the tumor into several regions differing in vessel density, diameter and degree of tumor necrosis, is in accordance with experimental data for human melanoma. Vessel collapse, due to a combination of severely reduced blood flow and solid stress exerted by the tumor, leads to a correlated percolation process that is driven towards criticality by the mechanism of hydrodynamic vessel stabilization.