Structural Kinetic Modeling of Metabolic Networks

To develop and investigate detailed mathematical models of cellular metabolic processes is one of the primary challenges in systems biology. However, despite considerable advance in the topological analysis of metabolic networks, explicit kinetic modeling based on differential equations is still often severely hampered by inadequate knowledge of the enzyme-kinetic rate laws and their associated parameter values. Here we propose a method that aims to give a detailed and quantitative account of the dynamical capabilities of metabolic systems, without requiring any explicit information about the particular functional form of the rate equations. Our approach is based on constructing a local linear model at each point in parameter space, such that each element of the model is either directly experimentally accessible, or amenable to a straightforward biochemical interpretation. This ensemble of local linear models, encompassing all possible explicit kinetic models, then allows for a systematic statistical exploration of the comprehensive parameter space. The method is applied to two paradigmatic examples: The glycolytic pathway of yeast and a realistic-scale representation of the photosynthetic Calvin cycle.Comment: 14 pages, 8 figures (color

Finite-size and correlation-induced effects in Mean-field Dynamics

The brain's activity is characterized by the interaction of a very large number of neurons that are strongly affected by noise. However, signals often arise at macroscopic scales integrating the effect of many neurons into a reliable pattern of activity. In order to study such large neuronal assemblies, one is often led to derive mean-field limits summarizing the effect of the interaction of a large number of neurons into an effective signal. Classical mean-field approaches consider the evolution of a deterministic variable, the mean activity, thus neglecting the stochastic nature of neural behavior. In this article, we build upon two recent approaches that include correlations and higher order moments in mean-field equations, and study how these stochastic effects influence the solutions of the mean-field equations, both in the limit of an infinite number of neurons and for large yet finite networks. We introduce a new model, the infinite model, which arises from both equations by a rescaling of the variables and, which is invertible for finite-size networks, and hence, provides equivalent equations to those previously derived models. The study of this model allows us to understand qualitative behavior of such large-scale networks. We show that, though the solutions of the deterministic mean-field equation constitute uncorrelated solutions of the new mean-field equations, the stability properties of limit cycles are modified by the presence of correlations, and additional non-trivial behaviors including periodic orbits appear when there were none in the mean field. The origin of all these behaviors is then explored in finite-size networks where interesting mesoscopic scale effects appear. This study leads us to show that the infinite-size system appears as a singular limit of the network equations, and for any finite network, the system will differ from the infinite system

The Dynamics of Hybrid Metabolic-Genetic Oscillators

The synthetic construction of intracellular circuits is frequently hindered by a poor knowledge of appropriate kinetics and precise rate parameters. Here, we use generalized modeling (GM) to study the dynamical behavior of topological models of a family of hybrid metabolic-genetic circuits known as "metabolators." Under mild assumptions on the kinetics, we use GM to analytically prove that all explicit kinetic models which are topologically analogous to one such circuit, the "core metabolator," cannot undergo Hopf bifurcations. Then, we examine more detailed models of the metabolator. Inspired by the experimental observation of a Hopf bifurcation in a synthetically constructed circuit related to the core metabolator, we apply GM to identify the critical components of the synthetically constructed metabolator which must be reintroduced in order to recover the Hopf bifurcation. Next, we study the dynamics of a re-wired version of the core metabolator, dubbed the "reverse" metabolator, and show that it exhibits a substantially richer set of dynamical behaviors, including both local and global oscillations. Prompted by the observation of relaxation oscillations in the reverse metabolator, we study the role that a separation of genetic and metabolic time scales may play in its dynamics, and find that widely separated time scales promote stability in the circuit. Our results illustrate a generic pipeline for vetting the potential success of a potential circuit design, simply by studying the dynamics of the corresponding generalized model

Model evaluation for glycolytic oscillations in yeast biotransformations of xenobiotics

Anaerobic glycolysis in yeast perturbed by the reduction of xenobiotic ketones is studied numerically in two models which possess the same topology but different levels of complexity. By comparing both models' predictions for concentrations and fluxes as well as steady or oscillatory temporal behavior we answer the question what phenomena require what kind of minimum model abstraction. While mean concentrations and fluxes are predicted in agreement by both models we observe different domains of oscillatory behavior in parameter space. Generic properties of the glycolytic response to ketones are discussed

Coexistence of amplitude and frequency modulations in intracellular calcium dynamics

The complex dynamics of intracellular calcium regulates cellular responses to information encoded in extracellular signals. Here, we study the encoding of these external signals in the context of the Li-Rinzel model. We show that by control of biophysical parameters the information can be encoded in amplitude modulation, frequency modulation or mixed (AM and FM) modulation. We briefly discuss the possible implications of this new role of information encoding for astrocytes.Comment: 4 pages, 4 figure

Scale-Free topologies and Activatory-Inhibitory interactions

A simple model of activatory-inhibitory interactions controlling the activity of agents (substrates) through a "saturated response" dynamical rule in a scale-free network is thoroughly studied. After discussing the most remarkable dynamical features of the model, namely fragmentation and multistability, we present a characterization of the temporal (periodic and chaotic) fluctuations of the quasi-stasis asymptotic states of network activity. The double (both structural and dynamical) source of entangled complexity of the system temporal fluctuations, as an important partial aspect of the Correlation Structure-Function problem, is further discussed to the light of the numerical results, with a view on potential applications of these general results.Comment: Revtex style, 12 pages and 12 figures. Enlarged manuscript with major revision and new results incorporated. To appear in Chaos (2006