Abstract:
We developed a set of equations that can be used to translate the graph of any regulatory network into a continuous dynamical system. Furthermore, it is also possible to locate its stable steady states. The method is based on the construction of two dynamical systems for a given network, one discrete and one continuous. The stable steady states of the discrete system can be found analytically, so they are used to locate the stable steady states of the continuous system numerically. To provide an example of the applicability of the method, we used it to model the regulatory network controlling T helper cell differentiation.The proposed equations have a form that permit any regulatory network to be translated into a continuous dynamical system, and also find its steady stable states. We showed that by applying the method to the T helper regulatory network it is possible to find its known states of activation, which correspond the molecular profiles observed in the precursor and effector cell types.The increasing use of high throughput technologies in different areas of biology has generated vast amounts of molecular data. This has, in turn, fueled the drive to incorporate such data into pathways and networks of interactions, so as to provide a context within which molecules operate. As a result, a wealth of connectivity information is available for multiple biological systems, and this has been used to understand some global properties of biological networks, including connectivity distribution [1], recurring motifs [2] and modularity [3]. Such information, while valuable, provides only a static snapshot of a network. For a better understanding of the functionality of a given network it is important to study its dynamical properties. The consideration of dynamics allows us to answer questions related to the number, nature and stability of the possible patterns of activation, the contribution of individual molecules or interactions to establishing such patterns, and th

Abstract:
Genetic regulatory networks are usually modeled by systems of coupled differential equations and by finite state models, better known as logical networks, are also used. In this paper we consider a class of models of regulatory networks which present both discrete and continuous aspects. Our models consist of a network of units, whose states are quantified by a continuous real variable. The state of each unit in the network evolves according to a contractive transformation chosen from a finite collection of possible transformations, according to a rule which depends on the state of the neighboring units. As a first approximation to the complete description of the dynamics of this networks we focus on a global characteristic, the dynamical complexity, related to the proliferation of distinguishable temporal behaviors. In this work we give explicit conditions under which explicit relations between the topological structure of the regulatory network, and the growth rate of the dynamical complexity can be established. We illustrate our results by means of some biologically motivated examples.

Abstract:
Here we propose a new approach to modeling gene expression based on the theory of random dynamical systems (RDS) that provides a general coupling prescription between the nodes of any given regulatory network given the dynamics of each node is modeled by a RDS. The main virtues of this approach are the following: (i) it provides a natural way to obtain arbitrarily large networks by coupling together simple basic pieces, thus revealing the modularity of regulatory networks; (ii) the assumptions about the stochastic processes used in the modeling are fairly general, in the sense that the only requirement is stationarity; (iii) there is a well developed mathematical theory, which is a blend of smooth dynamical systems theory, ergodic theory and stochastic analysis that allows one to extract relevant dynamical and statistical information without solving the system; (iv) one may obtain the classical rate equations form the corresponding stochastic version by averaging the dynamic random variables. It is important to emphasize that unlike the deterministic case, where coupling two equations is a trivial matter, coupling two RDS is non-trivial, specially in our case, where the coupling is performed between a state variable of one gene and the switching stochastic process of another gene and, hence, it is not a priori true that the resulting coupled system will satisfy the definition of a random dynamical system. We shall provide the necessary arguments that ensure that our coupling prescription does indeed furnish a notion of coupled random dynamical system. Finally, we illustrate our framework with three simple examples of "single-gene dynamics", which are the build blocks of our networks.

Abstract:
We study the dynamical properties of small regulatory networks treated as non autonomous dynamical systems called modules when working inside larger networks or, equivalently when subject to external signal inputs. Particular emphasis is put on the interplay between the internal properties of the open systems and the different possible inputs on them to deduce new functionalities of the modules. We use discrete-time, piecewise-affine and piecewise-contracting models with interactions of a regulatory nature to perform our study.

Abstract:
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop algorithms for analysis of compatibility and construction of canonical decompositions of such systems. To illustrate these techniques we describe their application to some cellular automata. Much attention is paid to study symmetries of the systems. In the case of deterministic systems, we reveale some important relations between symmetries and dynamics. We demonstrate that moving soliton-like structures arise inevitably in deterministic dynamical system whose symmetry group splits the set of states into a finite number of group orbits. We develop algorithms and programs exploiting discrete symmetries to study microcanonical ensembles and search phase transitions in mesoscopic lattice models. We propose an approach to quantization of discrete systems based on introduction of gauge connection with values in unitary representations of finite groups --- the elements of the connection are interpreted as amplitudes of quantum transitions. We discuss properties of a quantum description of finite systems. In particular, we demonstrate that a finite quantum system can be embedded into a larger classical system. Computer algebra and computational group theory methods were useful tools in our study.

Abstract:
We study the dynamics of discrete-time regulatory networks on random digraphs. For this we define ensembles of deterministic orbits of random regulatory networks, and introduce some statistical indicators related to the long-term dynamics of the system. We prove that, in a random regulatory network, initial conditions converge almost surely to a periodic attractor. We study the subnetworks, which we call modules, where the periodic asymptotic oscillations are concentrated. We proof that those modules are dynamically equivalent to independent regulatory networks.

Abstract:
Biological phenomena differ significantly from physical phenomena. At the heart of this distinction is the fact that biological entities have computational abilities and thus they are inherently difficult to predict. This is the reason why simplified models that provide the minimal requirements for computation turn out to be very useful to study networks of many components. In this chapter, we briefly review the dynamical aspects of models of regulatory networks, discussing their most salient features, and we also show how these models can give clues about the way in which networks may organize their capacity to evolve, by providing simple examples of the implementation of robustness and modularity.

Abstract:
We propose a method to decompose a multivariate dynamical system into weakly-coupled modules based on the idea that module boundaries constrain the spread of perturbations. Using a novel quality function called 'perturbation modularity', we find system coarse-grainings that optimally separate the dynamics of perturbation spreading into fast intra-modular and slow inter-modular components. Our method is defined directly in terms of system dynamics, unlike approaches that find communities in networks (whether in structural networks or 'functional networks' of statistical dependencies) or that impose arbitrary dynamics onto graphs. Due to this, we are able to capture the variation of modular organization across states, timescales, and in response to different perturbations, aspects of modularity which are all relevant to real-world dynamical systems. However, in certain cases, mappings exist between perturbation modularity and community detection methods of `Markov stability' and Newman's modularity. Our approach is demonstrated on several examples of coupled logistic maps. It uncovers hierarchical modular organization present in a system's coupling matrix. It also identifies the onset of a self-organized modular regime in coupled map lattices, where it is used to explore dependence of modularity on system state, parameters, and perturbations.

Abstract:
We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise affine contracting mappings whose variables represent gene expression levels. When compared to other models of regulatory networks, these models have an additional parameter which is identified as quantifying interaction delays. In spite of their simplicity, their dynamics presents a rich variety of behaviours. This phenomenology is not limited to piecewise affine model but extends to smooth nonlinear discrete time models of regulatory networks. In a first step, our analysis concerns general properties of networks on arbitrary graphs (characterisation of the attractor, symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc). In a second step, focus is made on simple circuits for which the attractor and its changes with parameters are described. In the negative circuit of 2 genes, a thorough study is presented which concern stable (quasi-)periodic oscillations governed by rotations on the unit circle -- with a rotation number depending continuously and monotonically on threshold parameters. These regular oscillations exist in negative circuits with arbitrary number of genes where they are most likely to be observed in genetic systems with non-negligible delay effects.

Abstract:
Elucidating the structure and/or dynamics of gene regulatory networks from experimental data is a major goal of systems biology. Stochastic models have the potential to absorb noise, account for un-certainty, and help avoid data overfitting. Within the frame work of probabilistic polynomial dynamical systems, we present an algorithm for the reverse engineering of any gene regulatory network as a discrete, probabilistic polynomial dynamical system. The resulting stochastic model is assembled from all minimal models in the model space and the probability assignment is based on partitioning the model space according to the likeliness with which a minimal model explains the observed data. We used this method to identify stochastic models for two published synthetic network models. In both cases, the generated model retains the key features of the original model and compares favorably to the resulting models from other algorithms.