Juan Carlos Nuño

From time series to complex networks: The visibility graph

In this work we present a simple and fast computational method, the visibility algorithm, that converts a time series into a graph. The constructed graph inherits several properties of the series in its structure. Thereby, periodic series convert into regular graphs, and random series do so into random graphs. Moreover, fractal series convert into scale-free networks, enhancing the fact that power law degree distributions are related to fractality, something highly discussed recently. Some remarkable examples and analytical tools are outlined to test the methods reliability. Many different measures, recently developed in the complex network theory, could by means of this new approach characterize time series from a new point of view.

The visibility graph: A new method for estimating the Hurst exponent of fractional Brownian motion

Fractional Brownian motion (fBm) has been used as a theoretical framework to study real-time series appearing in diverse scientific fields. Because of its intrinsic nonstationarity and long-range dependence, its characterization via the Hurst parameter, H, requires sophisticated techniques that often yield ambiguous results. In this work we show that fBm series map into a scale-free visibility graph whose degree distribution is a function of H. Concretely, it is shown that the exponent of the power law degree distribution depends linearly on H. This also applies to fractional Gaussian noises (fGn) and generic f−β noises. Taking advantage of these facts, we propose a brand new methodology to quantify long-range dependence in these series. Its reliability is confirmed with extensive numerical simulations and analytical developments. Finally, we illustrate this method quantifying the persistent behavior of human gait dynamics.

Complex dynamics of a catalytic network having faulty replication into error-species

Abstract We examine the dynamics of catalytic networks when error is introduced through faulty self-replication into a mutant molecular species. The model consists of n species that individually self-replicate through noncatalytic and catalytic action, and catalyze the replication of other species. Faulty replication produces error mutants which are assumed to be kinetically indistinguishable from one another. This aggregate error-species (error-tail) undergoes noncatalyzed self-replication, but has no effect on the catalytic species. A constant-population criterion produces competition among all reactants. The time evolution of the catalytic species can be expressed by a set of ordinary differential equations. We provide a detailed parametric analysis of the dynamics in a computationally tractable reduced model. Kinetic constants K ji controlling the enzymatic reactions can be used as bifurcation parameters to generate a rich repertoire of periodic and complex chaotic dynamics. Except for changes in the parametric position of bifurcation points, system dynamics is stable in response to changes in the quality of replication Q , where 1- Q is the mutation rate, and in the amplification constant A for the catalytic species. At low values of Q , the system falls out of chaotic regimes and into a “random-replication” state at which there are no catalytic species present. There is a similar insensitivity to changes in the amplification factor for the error species, A e , except for A = 0, at which the chaotic regimes remain stable throughout the full range of Q . We discuss the behavior of our model against one in which error is handled by means of mutual intermulation between the catalytic species. Complex behavior in this intermutation model is extremely sensitive to the mutation rate. Because the error-tail is expressed only in terms of the catalytic species themselves rather than in variables representing the error-species, the error-tail model may provide a useful method with which to examine models of error-utilization in neuronal and other biological systems involving competitive interactions among their constituent parts.

Spatial dynamics of a model for prebiotic evolution

Abstract The selective properties of a population formed by RNA-like molecules (replicators) with catalytic capabilities are analyzed in an extended system. The population evolves in a closed reactor and is kept far from equilibrium by means of a recycling reaction that transforms the degradation products of the replicators into energy rich monomers, from which the species are built up. In the limit of infinite diffusion, for a particular set of parameters the system exhibits tristability between chaos and two fixed points. Under this setup, numerical techniques are used to prove the formation of spatial patterns when finite diffusion forces are taken into account in one- and two-dimensional spaces. Finally, the relevance of these results is discussed within a prebiotic framework.

Theories of Lethal Mutagenesis: From Error Catastrophe to Lethal Defection

RNA viruses get extinct in a process called lethal mutagenesis when subjected to an increase in their mutation rate, for instance, by the action of mutagenic drugs. Several approaches have been proposed to understand this phenomenon. The extinction of RNA viruses by increased mutational pressure was inspired by the concept of the error threshold. The now classic quasispecies model predicts the existence of a limit to the mutation rate beyond which the genetic information of the wild type could not be efficiently transmitted to the next generation. This limit was called the error threshold, and for mutation rates larger than this threshold, the quasispecies was said to enter into error catastrophe. This transition has been assumed to foster the extinction of the whole population. Alternative explanations of lethal mutagenesis have been proposed recently. In the first place, a distinction is made between the error threshold and the extinction threshold, the mutation rate beyond which a population gets extinct. Extinction is explained from the effect the mutation rate has, throughout the mutational load, on the reproductive ability of the whole population. Secondly, lethal defection takes also into account the effect of interactions within mutant spectra, which have been shown to be determinant for the understanding the extinction of RNA virus due to an augmented mutational pressure. Nonetheless, some relevant issues concerning lethal mutagenesis are not completely understood yet, as so survival of the flattest, i.e. the development of resistance to lethal mutagenesis by evolving towards mutationally more robust regions of sequence space, or sublethal mutagenesis, i.e., the increase of the mutation rate below the extinction threshold which may boost the adaptability of RNA virus, increasing their ability to develop resistance to drugs (including mutagens). A better design of antiviral therapies will still require an improvement of our knowledge about lethal mutagenesis.

Lifetimes of small catalytic networks

We analyse the stochastic properties of dynamical systems with finite populations of a few differentreplicator species. Our main interest is to evaluate the typicallifetime, i.e. the time for the extinction of the first species in the network, for different catalytic structures, as a function of the population size.

Non-uniformities and superimposed competition in a model of an autocatalytic network formed by error-prone self-replicative species

The particular dynamics of the previously proposed model of a catalytic network formed byn error-prone self-replicative species without and with superimposed competition is analysed. In the first case, two situations are studied in detail: a uniform network in which all the species are inter-coordinated in the same way, and a network with a species differentiated in its catalytic relation with the remaining elements. In the second case, the superimposed competition is introduced at two levels: first, as an asymmetry in one of the network species amplification factor considering a null self-catalytic vector, and secondly, as a non-null self-catalytic vector with no asymmetry in the other propertics of the species. This kind of system does not present complex behaviour and can be adequately deseribed by performing a standard linear analysis, which gives direct information on the asymptotic behaviour of the sytem. Finally, the biological implications of this analysis within the framework of biological evolution are discussed.

Fighting cheaters: How and how much to invest

Human societies are formed by different socio-economical classes which are characterized by their contribution to, and their share of, the common wealth available. Cheaters, defined as those individuals that do not contribute to the common wealth but benefit from it, have always existed, and are likely to be present in all societies in the foreseeable future. Their existence brings about serious problems since they act as sinks for the community wealth and deplete resources which are always limited and often scarce. To fight cheaters, a society can invest additional resources to pursue one or several aims. For instance, an improvement in social solidarity (e.g. by fostering education) may be sought. Alternatively, deterrence (e.g. by increasing police budget) may be enhanced. Then the following questions naturally arise: (i) how much to spend and (ii) how to allocate the expenditure between both strategies above. This paper addresses this general issue in a simplified setting, which however we believe of some interest. More precisely, we consider a society constituted by two productive classes and an unproductive one, the cheaters, and proposes a dynamical system that describes their evolution in time. We find it convenient to formulate our model as a three-dimensional ordinary differential equation (ODE) system whose variables are the cheater population, the total wealth and one of the productive social classes. The stationary values of the cheater population and the total wealth are studied in terms of the two parameters: phi (how much to invest) and s (how to distribute such expenditure). We show that it is not possible to simultaneously minimize the cheater population and maximize the total wealth with respect to phi and s. We then discuss the possibility of defining a compromise function to find suitable values of phi and s that optimize the response to cheating. In our opinion, this qualitative approach may be of some help to plan and implement social strategies against cheating.

Stoichiometric analysis of self-maintaining metabolisms ☆

This paper presents an extension of stoichiometric analysis in systems where the catalytic compounds (enzymes) are also intermediates of the metabolic network (dual property), so they are produced and degraded by the reaction network itself. To take this property into account, we introduce the definition of enzyme-maintaining mode, a set of reactions that produces its own catalyst and can operate at stationary state. Moreover, an enzyme-maintaining mode is defined as elementary with respect to a given reaction if the removal of any of the remaining reactions causes the cessation of any steady state flux through this reference reaction. These concepts are applied to determine the network structure of a simple self-maintaining system.

Evolutionary Daisyworld models: A new approach to studying complex adaptive systems

This paper presents a model of a population of error-prone self-replicative species (replicators) that interact with its environment. The population evolves by natural selection in an environment whose change is caused by the evolutionary process itself. For simplicity, the environment is described by a single scalar factor, i.e. its temperature. The formal formulation of the model extends two basic models of Ecology and Evolutionary Biology, namely, Daisyworld and Quasispecies models. It is also assumed that the environment can also change due to external perturbations that are summed up as an external noise. Unlike previous models, the population size self-regulates, so no ad hoc population constraints are involved. When species replication is error-free, i.e. without mutation, the system dynamics can be described by an (n + 1)-dimensional system of differential equations, one for each of the species initially present in the system, and another for the evolution of the environment temperature. Analytical results can be obtained straightforwardly in low-dimensional cases. In these examples, we show the stabilizing effect of thermal white noise on the system behavior. The error-prone self-replication, i.e. with mutation, is studied computationally. We assume that species can mutate two independent parameters: its optimal growth temperature and its influence on the environment temperature. For different mutation rates the system exhibits a large variety of behaviors. In particular, we show that a quasispecies distribution with an internal sub-distribution appears, facilitating species adaptation to new environments. Finally, this ecologically inspired evolutionary model is applied to study the origin and evolution of public opinion.