Scaling in complex systems: a link between the dynamics of networks and growing interfaces.

We consider growing interfaces as dynamical networks whose nodes are the discrete points of the interface and the edges the physical interactions among them. We map the points of the interface formed at each time into a graph by means of a visibility algorithm. As the corresponding interfaces grow, their visibility graphs change over time. We show that the visibility graphs are all scale free for each time. We use the variance of the node degrees as a measure of the dynamical properties of these graphs. This magnitude reveals an unexpected scaling behaviour of these graphs in both the number of nodes and time. This enables to define three robust exponents that characterize any type of dynamics with more detail than the classical scaling analysis applied directly to the physical interfaces. To check the feasibility of this approach we study and classify six different dynamical processes and estimate their critical exponents. We conclude that the dynamics of physical systems far from equilibrium can be determined by its corresponding visibility network. Indeed, this methodology is able to discern among dynamical processes that hitherto have been classified in the same universality class according to the scaling analysis of their interfaces.

Optimal stoichiometric designs of ATP-producing systems as determined by an evolutionary algorithm.

The design of metabolic pathways is thought to be the result of an optimization process such that the structure of contemporary metabolic routes maximizes a particular objective function. Recently, it has been shown that some essential stoichiometric properties of glycolysis can be explained on the basis of the requirement for a high ATP production rate. Because the number of stoichiometrically feasible designs increases strongly with the number of reactions involved, a systematic analysis of all the possibilities turns out to be inaccessible beyond a certain system size. We present, therefore, an alternative approach to compute in a more efficient way the optimal design of glycolysis interacting with an external ATP-consuming reaction. The algorithm is based on the laws of evolution by natural selection, and may be viewed as a particular version of evolutionary algorithms. The following conclusions are derived: (a) evolutionary algorithms are very useful search strategies in determining optimal stoichiometries of metabolic pathways. (b) Essential topological features of the glycolytic network may be explained on the basis of flux optimization. (c) There is a strong interrelation between the optimal stoichiometries and the thermodynamic and kinetic properties of the participating reactions. (d) Some subsequences of reactions in optimal pathways are strongly conserved at variation of system parameters, which may be understood by applying principles of metabolic control analysis.

Generalization of the theory of transition times in metabolic pathways: a geometrical approach.

Cell metabolism is able to respond to changes in both internal parameters and boundary constraints. The time any system variable takes to make this response has relevant implications for understanding the evolutionary optimization of metabolism as well as for biotechnological applications. This work is focused on estimating the magnitude of the average time taken by any observable of the system to reach a new state when either a perturbation or a persistent variation occurs. With this aim, a new variable, called characteristic time, based on geometric considerations, is introduced. It is stressed that this new definition is completely general, being useful for evaluating the response time, even in complex transitions involving periodic behavior. It is shown that, in some particular situations, this magnitude coincides with previously defined transition times but differs drastically in others. Finally, to illustrate the applicability of this approach, a model of a reaction mediated by an allosteric enzyme is analyzed.

METATOOL: for studying metabolic networks.

MOTIVATION: To reconstruct metabolic pathways from biochemical and/or genome sequence data, the stoichiometric and thermodynamic feasibility of the pathways has to be tested. This is achieved by characterizing the admissible region of flux distributions in steady state. This region is spanned by what can be called a convex basis. The concept of 'elementary flux modes' provides a mathematical tool to define all metabolic routes that are feasible in a given metabolic network. In addition, we define 'enzyme subsets' to be groups of enzymes that operate together in fixed flux proportions in all steady states of the system. RESULTS: Algorithms for computing the convex basis and elementary modes developed earlier are briefly reviewed. A newly developed algorithm for detecting all enzyme subsets in a given network is presented. All of these algorithms have been implemented in a novel computer program named METATOOL, whose features are outlined here. The algorithms are illustrated by an example taken from sugar metabolism. AVAILABILITY: METATOOL is available from ftp://bmsdarwin.brookes.ac. uk/pub/software/ibmpc/metatool. SUPPLEMENTARY INFORMATION: http://www. biologie.hu-berlin.de/biophysics/Theory/tpfeiffer/metatoo l.html

Transient times in linear metabolic pathways under constant affinity constraints.

In the early seventies, Easterby began the analytical study of transition times for linear reaction schemes [Easterby (1973) Biochim. Biophys. Acta 293, 552-558]. In this pioneer work and in subsequent papers, a state function (the transient time) was used to measure the period before the stationary state, for systems constrained to work under both constant and variable input flux, was reached. Despite the undoubted usefulness of this quantity to describe the time-dependent features of these kinds of systems, its application to the study of chemical reactions under other constraints is questionable. In the present work, a generalization of these magnitudes to linear metabolic pathways functioning under a constant-affinity constraint is carried out. It is proved that classical definitions of transient times do not reflect the actual properties of the transition to the steady state in systems evolving under this restriction. Alternatively, a more adequate framework for interpretation of the transient times for systems with both constant and variable input flux is suggested. Within this context, new definitions that reflect more accurately the transient characteristics of constant affinity systems are stated. Finally, the meaning of these transient times is discussed.

Network organization of cell metabolism: monosaccharide interconversion.

The structural properties of carbohydrate metabolism are being studied. The present contribution focuses mainly on those processes involving the transfer of carbon fragments among sugars. It is shown how enzymatic activities fix the way the system self-organizes stoichiometrically at the steady state. It is proven that there exists a specific correspondence between the set of all possible enzymic activities, the activity set, and the set of stoichiometrically compatible flux distributions through the pathway. On the one hand, there are enzymic activities that do not allow a stoichiometrically feasible coupling at the steady state of the reactions involved in the conversion. On the other hand, there are enzymic activities that are related to one or more flux distributions at the steady state (i.e. with one or several rate vectors respectively). For this latter group, it can be demonstrated that the structure of the system depends on other non-structural factors, such as boundary constraints and the kinetic parameters. As a consequence, it is suggested that this kind of metabolic process must be viewed as a complex reaction network instead of a sequential number of steps. Some implications of these derivations are illustrated for the particular conversion of CO2 --> C3. General remarks are also discussed within the framework of network models of cell metabolism.

The influence of mutation on autocatalytic reaction networks.

A particular class of ordinary differential equations (ODEs) describing catalyzed, template-induced, and erroneous replication is investigated. The ODEs can be split into a replicator part accounting for the correct replication and a mutation term accounting for all miscopying processes. The set of all species is divided into the catalytically active "viable" species and an error tail subsuming all other species. Neglecting both the intermutation among the viable species and the reflux from the error tail allows for an extensive analysis of the autocatalytic network. If mutation rates are small enough, a perturbation approach is feasible showing that mutation in general simplifies the qualitative behavior of the dynamical system. Special cases, such as Schlögl's model, the uniform model, and the hypercycle, show that the viable species become unstable beyond a critical mutation rate: There is an analogue to the error threshold of the quasi-species model also in nonlinear autocatalytic reaction networks with mutation.

Influence of the hypercycle on the error threshold: a stochastic approach.

The role of fluctuations on the error threshold of the hypercycle has been studied by a stochastic approach on a very simplified model. For this model, the master equation was derived and its unique steady state calculated. This state implies the extinction of the system. But the actual time necessary to reach the steady state may be astronomically long whereas for times of experimental interest the system could be near some quasi-stationary states. In order to explore this possibility a Gillespie simulation of the stochastic process has been carried out. These quasi-stationary states correspond to the deterministic steady states of the system. The error threshold shifts towards higher values of the quality factor Q. Moreover, information about the fluctuations around the quasi-stationary states is obtained. The results are discussed in relation to the deterministic states.