Maximino Aldana

Boolean dynamics of networks with scale-free topology

The dynamics of Boolean networks with scale-free topology are studied. The existence of a phase transition from ordered to chaotic dynamics, governed by the value of the scale-free exponent of the network, is shown analytically by analyzing the overlap between two distinct trajectories. The phase diagram shows that the phase transition occurs for values of the scale-free exponent in the open interval (2, 2.5). Since the Boolean networks under study are directed graphs, the scale-free topology of the input connections and that of the output connections are studied separately. Ultimately these two topologies are shown to be equivalent. A numerical study of the attractor structure of the configuration space reveals that this structure is similar in both networks with scale-free topologies and networks with homogeneous random topologies. However, an important result of this work is that the fine-tuning usually required to achieve stability in the dynamics of networks with homogeneous random topologies is no longer necessary when the network topology is scale-free. Finally, based on the results presented in this work, it is hypothesized that the scale-free topology favors the evolution and adaptation of network functioning from a biological perspective.

Boolean Dynamics with Random Couplings

This paper reviews a class of generic dissipative dynamical systems called N-K models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends upon K elements in the previous time step. We review the work of many authors on the behavior of the models, looking particularly at the structure and lengths of their cycles, the sizes of their basins of attraction, and the flow of information through the systems. In the limit of infinite N, there is a phase transition between a chaotic and an ordered phase, with a critical phase in between. We argue that the behavior of this system depends significantly on the topology of the network connections. If the elements are placed upon a lattice with dimension d,the system shows correlations related to the standard percolation or directed percolation phase transition on such a lattice. On the other hand, a very different behavior is seen in the Kauffman net in which all spins are equally likely to be coupled to a given spin. In this situation, coupling loops are mostly suppressed, and the behavior of the system is much more like that of a mean field theory. We also describe possible applications of the models to, for example, genetic networks, cell differentiation, evolution, democracy in social systems and neural networks.

A natural class of robust networks

As biological studies shift from molecular description to system analysis we need to identify the design principles of large intracellular networks. In particular, without knowing the molecular details, we want to determine how cells reliably perform essential intracellular tasks. Recent analyses of signaling pathways and regulatory transcription networks have revealed a common network architecture, termed scale-free topology. Although the structural properties of such networks have been thoroughly studied, their dynamical properties remain largely unexplored. We present a prototype for the study of dynamical systems to predict the functional robustness of intracellular networks against variations of their internal parameters. We demonstrate that the dynamical robustness of these complex networks is a direct consequence of their scale-free topology. By contrast, networks with homogeneous random topologies require fine-tuning of their internal parameters to sustain stable dynamical activity. Considering the ubiquity of scale-free networks in nature, we hypothesize that this topology is not only the result of aggregation processes such as preferential attachment; it may also be the result of evolutionary selective processes.

Gene expression dynamics in the macrophage exhibit criticality

Cells are dynamical systems of biomolecular interactions that process information from their environment to mount diverse yet specific responses. A key property of many self-organized systems is that of criticality: a state of a system in which, on average, perturbations are neither dampened nor amplified, but are propagated over long temporal or spatial scales. Criticality enables the coordination of complex macroscopic behaviors that strike an optimal balance between stability and adaptability. It has long been hypothesized that biological systems are critical. Here, we address this hypothesis experimentally for system-wide gene expression dynamics in the macrophage. To this end, we have developed a method, based on algorithmic information theory, to assess macrophage criticality, and we have validated the method on networks with known properties. Using global gene expression data from macrophages stimulated with a variety of Toll-like receptor agonists, we found that macrophage dynamics are indeed critical, providing the most compelling evidence to date for this general principle of dynamics in biological systems.

Critical Dynamics in Genetic Regulatory Networks: Examples from Four Kingdoms

The coordinated expression of the different genes in an organism is essential to sustain functionality under the random external perturbations to which the organism might be subjected. To cope with such external variability, the global dynamics of the genetic network must possess two central properties. (a) It must be robust enough as to guarantee stability under a broad range of external conditions, and (b) it must be flexible enough to recognize and integrate specific external signals that may help the organism to change and adapt to different environments. This compromise between robustness and adaptability has been observed in dynamical systems operating at the brink of a phase transition between order and chaos. Such systems are termed critical. Thus, criticality, a precise, measurable, and well characterized property of dynamical systems, makes it possible for robustness and adaptability to coexist in living organisms. In this work we investigate the dynamical properties of the gene transcription networks reported for S. cerevisiae, E. coli, and B. subtilis, as well as the network of segment polarity genes of D. melanogaster, and the network of flower development of A. thaliana. We use hundreds of microarray experiments to infer the nature of the regulatory interactions among genes, and implement these data into the Boolean models of the genetic networks. Our results show that, to the best of the current experimental data available, the five networks under study indeed operate close to criticality. The generality of this result suggests that criticality at the genetic level might constitute a fundamental evolutionary mechanism that generates the great diversity of dynamically robust living forms that we observe around us.

Floral Morphogenesis: Stochastic Explorations of a Gene Network Epigenetic Landscape

In contrast to the classical view of development as a preprogrammed and deterministic process, recent studies have demonstrated that stochastic perturbations of highly non-linear systems may underlie the emergence and stability of biological patterns. Herein, we address the question of whether noise contributes to the generation of the stereotypical temporal pattern in gene expression during flower development. We modeled the regulatory network of organ identity genes in the Arabidopsis thaliana flower as a stochastic system. This network has previously been shown to converge to ten fixed-point attractors, each with gene expression arrays that characterize inflorescence cells and primordial cells of sepals, petals, stamens, and carpels. The network used is binary, and the logical rules that govern its dynamics are grounded in experimental evidence. We introduced different levels of uncertainty in the updating rules of the network. Interestingly, for a level of noise of around 0.5–10%, the system exhibited a sequence of transitions among attractors that mimics the sequence of gene activation configurations observed in real flowers. We also implemented the gene regulatory network as a continuous system using the Glass model of differential equations, that can be considered as a first approximation of kinetic-reaction equations, but which are not necessarily equivalent to the Boolean model. Interestingly, the Glass dynamics recover a temporal sequence of attractors, that is qualitatively similar, although not identical, to that obtained using the Boolean model. Thus, time ordering in the emergence of cell-fate patterns is not an artifact of synchronous updating in the Boolean model. Therefore, our model provides a novel explanation for the emergence and robustness of the ubiquitous temporal pattern of floral organ specification. It also constitutes a new approach to understanding morphogenesis, providing predictions on the population dynamics of cells with different genetic configurations during development.

Phase transitions in Self-Driven many-particle systems and related non-equilibrium models: A network approach

We investigate the conditions that produce a phase transition from an ordered to a disordered state in a family of models of two-dimensional elements with a ferromagnetic-like interaction. This family is defined to contain under the same framework, among others, the XY-model and the Self-Driven Particles Model introduced by Vicsek et al. Each model is distinguished only by the rules that determine the set of elements with which each element interacts. We propose a new member of the family: the vectorial network model, in which a given fraction of the elements interact through direct random connections. This model is analogous to an XY-system on a network, and as such can be of interest for a wide range of problems. It captures the main aspects of the interaction dynamics that produce the phase transition in other models of the family. The network approach allows us to show analytically the existence of a phase transition in this vectorial network model, and to compute its relevant parameters for the case in which all elements are randomly connected. Finally we study numerically the conditions required for a phase transition to exist for different members of the family. Our results show that a qualitatively equivalent phase transition appears whenever even a small amount of long-range interactions are present (or built over time), regardless of other equilibrium or non-equilibrium properties of the system.

Numerical and Theoretical Studies of Noise Effects in the Kauffman Model

In this work we analyze the stochastic dynamics of the Kauffman model evolving under the influence of noise. By considering the average crossing time between two distinct trajectories, we show that different Kauffman models exhibit a similar kind of behavior, even when the structure of their basins of attraction is quite different. This can be considered as a robust property of these models. We present numerical results for the full range of noise level and obtain approximate analytic expressions for the above crossing time as a function of the noise in the limit cases of small and large noise levels.

Discrete Dynamics Model for the Speract-Activated Ca2+ Signaling Network Relevant to Sperm Motility

Understanding how spermatozoa approach the egg is a central biological issue. Recently a considerable amount of experimental evidence has accumulated on the relation between oscillations in intracellular calcium ion concentration ([Ca]) in the sea urchin sperm flagellum, triggered by peptides secreted from the egg, and sperm motility. Determination of the structure and dynamics of the signaling pathway leading to these oscillations is a fundamental problem. However, a biochemically based formulation for the comprehension of the molecular mechanisms operating in the axoneme as a response to external stimulus is still lacking. Based on experiments on the S. purpuratus sea urchin spermatozoa, we propose a signaling network model where nodes are discrete variables corresponding to the pathway elements and the signal transmission takes place at discrete time intervals according to logical rules. The validity of this model is corroborated by reproducing previous empirically determined signaling features. Prompted by the model predictions we performed experiments which identified novel characteristics of the signaling pathway. We uncovered the role of a high voltage-activated channel as a regulator of the delay in the onset of fluctuations after activation of the signaling cascade. This delay time has recently been shown to be an important regulatory factor for sea urchin sperm reorientation. Another finding is the participation of a voltage-dependent calcium-activated channel in the determination of the period of the fluctuations. Furthermore, by analyzing the spread of network perturbations we find that it operates in a dynamically critical regime. Our work demonstrates that a coarse-grained approach to the dynamics of the signaling pathway is capable of revealing regulatory sperm navigation elements and provides insight, in terms of criticality, on the concurrence of the high robustness and adaptability that the reproduction processes are predicted to have developed throughout evolution.

Regulatory design governing progression of population growth phases in bacteria.

It has long been noted that batch cultures inoculated with resting bacteria exhibit a progression of growth phases traditionally labeled lag, exponential, pre-stationary and stationary. However, a detailed molecular description of the mechanisms controlling the transitions between these phases is lacking. A core circuit, formed by a subset of regulatory interactions involving five global transcription factors (FIS, HNS, IHF, RpoS and GadX), has been identified by correlating information from the well- established transcriptional regulatory network of Escherichia coli and genome-wide expression data from cultures in these different growth phases. We propose a functional role for this circuit in controlling progression through these phases. Two alternative hypotheses for controlling the transition between the growth phases are first, a continuous graded adjustment to changing environmental conditions, and second, a discontinuous hysteretic switch at critical thresholds between growth phases. We formulate a simple mathematical model of the core circuit, consisting of differential equations based on the power-law formalism, and show by mathematical and computer-assisted analysis that there are critical conditions among the parameters of the model that can lead to hysteretic switch behavior, which – if validated experimentally – would suggest that the transitions between different growth phases might be analogous to cellular differentiation. Based on these provocative results, we propose experiments to test the alternative hypotheses.