Misure fisiche e sistemi dinamici stocastici

Lo studio dell'andamento asintotico di sistemi dinamici stocastici generati da equazioni differenziali stocastiche, come quelle associate a modelli della fluidodinamica, ha messo in luce la presenza di particolari oggetti, come attrattori e misure invarianti, di difficile descrizione. Uno strumento che potrebbe avere un ruolo particolare nella descrizione di alcune proprietà asintotiche di questi sistemi è l'equilibrio statistico, una misura aleatoria invariante costruita mediante lo stessa costruzione pullback che ha rilevato la presenza di attrattori stocastici. Consideriamo un sistema dinamico stocastico φ su uno spazio Polacco (X,B), dato un sistema dinamico metrico (Ω, F, P, (θt)t ∈ ℜ}). Se il moto a un punto associato a φ è un processo di Markov, e il semigruppo markoviano associato possiede una misura invariante ρ, l'equilibrio statistico (μω)ω ∈ Ω di φ si trova come limite debole, per quasi ogni ω, della misura φ(t,θ-tω)ρ, per t tendente a infinito. Esso descrive il comportamento asintotico di orbite scelte casualmente secondo la misura ρ. Nella tesi ci chiediamo se, data una misura di probabilità λ su (X,B), possa aver luogo, per quasi ogni ω, la convergenza debole di φ(t,θ-tω)λ a μω. Proprietà di questo tipo sono oggetto di interesse sia nel caso di sistemi deterministici che stocastici, in quanto mettono in evidenza la presenza di misure che, permettendo una descrizione della distribuzione asintotica di orbite uscenti da generiche condizioni iniziali, hanno un particolare significato fisico. Per la descrizione della proprietà che consideriamo, è necessaria una analisi delle strutture peculiari dei sistemi dinamici stocastici, come le misure aleatorie, gli attrattori stocastici e i semigruppi di transizione; ad essi sono dedicati i primi capitoli. Viene poi considerato un possibile criterio per stabilire se la proprietà di convergenza debole presentata è soddisfatta. Le condizioni per l'applicabilità di questo criterio portano allo studio del moto a due punti associato al sistema dinamico stocastico, e in particolare all'ergodicità del semigruppo di Markov da esso generato. Nell'ultima parte della tesi viene affrontato il problema dell'ergodicità per i semigruppi di Markov associati al moto a un punto e al moto a due punti per un semplice modello stocastico di dinamica dei fluidi, e vengono riportati dei risultati parziali ottenuti nel tentativo di applicare il criterio considerato a questo sistema

Attractor identification in asynchronous Boolean dynamics with network reduction

Identification of attractors, that is, stable states and sustained oscillations, is an important step in the analysis of Boolean models and exploration of potential variants. We describe an approach to the search for asynchronous cyclic attractors of Boolean networks that exploits, in a novel way, the established technique of elimination of components. Computation of attractors of simplified networks allows the identification of a limited number of candidate attractor states, which are then screened with techniques of reachability analysis combined with trap space computation. An implementation that brings together recently developed Boolean network analysis tools, tested on biological models and random benchmark networks, shows the potential to significantly reduce running times.Comment: 13 page

Boolean analysis of lateral inhibition

We study Boolean networks which are simple spatial models of the highly conserved Delta–Notch system. The models assume the inhibition of Delta in each cell by Notch in the same cell, and the activation of Notch in presence of Delta in surrounding cells. We consider fully asynchronous dynamics over undirected graphs representing the neighbour relation between cells. In this framework, one can show that all attractors are fixed points for the system, independently of the neighbour relation, for instance by using known properties of simplified versions of the models, where only one species per cell is defined. The fixed points correspond to the so-called fine-grained “patterns” that emerge in discrete and continuous modelling of lateral inhibition. We study the reachability of fixed points, giving a characterisation of the trap spaces and the basins of attraction for both the full and the simplified models. In addition, we use a characterisation of the trap spaces to investigate the robustness of patterns to perturbations. The results of this qualitative analysis can complement and guide simulation-based approaches, and serve as a basis for the investigation of more complex mechanisms

Graph properties of biological interaction networks

This thesis considers two modelling frameworks for interaction networks in biology. The first models the interacting species qualitatively as discrete variables, with the regulatory graphs expressing their mutual influence. Circuits in the regulatory structure are known to be indicative of some asymptotic behaviours. We investigate the relationship between local negative circuits and sustained oscillations, presenting new examples of Boolean networks without local negative circuits and admitting a cyclic attractor. We then show how regulatory properties of Boolean networks can be investigated via satisfiability problems, and use the technique to examine the role of local negative circuits in networks of small dimension. To enable the application of Boolean techniques to the study of multivalued networks, a mapping of discrete networks to Boolean can be considered. The Boolean version, however, is defined only on a subset of the Boolean states. We propose a method for extending the Boolean version that preserves both the attractors and the regulatory structure of the network. Chemical reaction network theory models the dynamics of species concentrations via systems of ordinary differential equations, establishing connections between the network structure and the dynamics. Some results assume mass action kinetics, whereas biochemical models often adopt other rate forms. We propose algorithms for elimination of intermediate species, that can be used to find whether a mass action network simplifies to a given chemical system. We then consider the problem of identification of generalised mass action networks that give rise to a given mass action dynamics, while displaying useful structural properties, such as weak reversibility. In particular, we investigate systems obtained by preserving the reaction vectors of the mass action network, and outline a new algorithmic approach

Computing trap space-based control strategies for Boolean networks using answer set programming

Control of Boolean networks enables important medical and biological applications. At the core of many approaches is value percolation, by virtue of its simplicity and ease of implementation. Methods based uniquely on percolation can however miss many control strategies. We previously introduced a new method which, using the network's trap spaces, can uncover additional sets of interventions. In this work we present a highly efficient implementation of this methodology based on Answer Set Programming, allowing for simple and fast application to biological networks, as illustrated with some cases studies of cell reprogramming

Graph properties of biological interaction networks

This thesis considers two modelling frameworks for interaction networks in biology. The first models the interacting species qualitatively as discrete variables, with the regulatory graphs expressing their mutual influence. Circuits in the regulatory structure are known to be indicative of some asymptotic behaviours. We investigate the relationship between local negative circuits and sustained oscillations, presenting new examples of Boolean networks without local negative circuits and admitting a cyclic attractor. We then show how regulatory properties of Boolean networks can be investigated via satisfiability problems, and use the technique to examine the role of local negative circuits in networks of small dimension. To enable the application of Boolean techniques to the study of multivalued networks, a mapping of discrete networks to Boolean can be considered. The Boolean version, however, is defined only on a subset of the Boolean states. We propose a method for extending the Boolean version that preserves both the attractors and the regulatory structure of the network. Chemical reaction network theory models the dynamics of species concentrations via systems of ordinary differential equations, establishing connections between the network structure and the dynamics. Some results assume mass action kinetics, whereas biochemical models often adopt other rate forms. We propose algorithms for elimination of intermediate species, that can be used to find whether a mass action network simplifies to a given chemical system. We then consider the problem of identification of generalised mass action networks that give rise to a given mass action dynamics, while displaying useful structural properties, such as weak reversibility. In particular, we investigate systems obtained by preserving the reaction vectors of the mass action network, and outline a new algorithmic approach

Linear cuts in Boolean networks

Boolean networks are popular tools for the exploration of qualitative dynamical properties of biological systems. Several dynamical interpretations have been proposed based on the same logical structure that captures the interactions between Boolean components. They reproduce, in different degrees, the behaviours emerging in more quantitative models. In particular, regulatory conflicts can prevent the standard asynchronous dynamics from reproducing some trajectories that might be expected upon inspection of more detailed models. We introduce and study the class of networks with linear cuts, where linear components -- intermediates with a single regulator and a single target -- eliminate the aforementioned regulatory conflicts. The interaction graph of a Boolean network admits a linear cut when a linear component occurs in each cycle and in each path from components with multiple targets to components with multiple regulators. Under this structural condition the attractors are in one-to-one correspondence with the minimal trap spaces, and the reachability of attractors can also be easily characterized. Linear cuts provide the base for a new interpretation of the Boolean semantics that captures all behaviours of multi-valued refinements with regulatory thresholds that are uniquely defined for each interaction, and contribute a new approach for the investigation of behaviour of logical models

Realizations of kinetic differential equations

The induced kinetic differential equations of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a system of polynomial differential equations, is it possible to find a network which induces these equations; in other words: is it possible to find a kinetic realization of this system of differential equations? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc.? The constructive answers presented to a series of questions of the above type are useful when fitting differential equations to datasets, or when trying to find out the dynamic behavior of the solutions of differential equations. It turns out that some of these results can be applied when trying to solve seemingly unrelated mathematical problems, like the existence of positive solutions to algebraic equations