Regularizing capacity of metabolic networks

Abstract

Despite their topological complexity almost all functional properties of metabolic networks can be derived from steady-state dynamics. Indeed, many theoretical investigations (like flux-balance analysis) rely on extracting function from steady states. This leads to the interesting question, how metabolic networks avoid complex dynamics and maintain a steady-state behavior. Here, we expose metabolic network topologies to binary dynamics generated by simple local rules. We find that the networks' response is highly specific: Complex dynamics are systematically reduced on metabolic networks compared to randomized networks with identical degree sequences. Already small topological modifications substantially enhance the capacity of a network to host complex dynamic behavior and thus reduce its regularizing potential. This exceptionally pronounced regularization of dynamics encoded in the topology may explain, why steady-state behavior is ubiquitous in metabolism.

Outer-totalistic cellular automata on graphs

“…For all these rules, the entropy signature of real graphs is significantly smaller compared to the null model topologies. This is also true for hierarchized and anti-hierarchized null models, as well as for all other species investigated in [12]. We believe that the application of dynamic probes is a particularly helpful tool for studying dynamical constraints imposed by topology.…”

“…There we implemented and studied only a single rule, namely (+, −) as a dynamic probe and interpreted the enhanced regularizing capacity of real networks compared to randomized null models as a possible topological contribution to the reliable establishment of metabolic steadystates and to the effective dampening of fluctuations. To show that the results presented in [12] are valid over the whole range of dynamics discussed in the present paper, we now implement all possible rules of the form (α, γ) on substrate graphs and analyze the entropy signature differences. Figure 4 shows the results for Homo sapiens, Saccharomyces cerevisiae, Escherichia coli, and Bacillus subtilis.…”

“…On graphs, this classification inevitably fails because of a lacking natural node order. Instead, we apply two entropy-like measures, the Shannon entropy S and the word entropy W , which we have previously shown to provide a feasible framework for the quantification of pattern complexity [10,11,12]. The Shannon entropy S serves as a measure for the homogeneity of the spatio-temporal pattern, by averaging over all nodes: We compare 10 3 random initial conditions on the regular architecture with 10 3 samples of a randomized graph, where the number of incoming and outgoing links of every node is preserved and kept to d = 2, but the link architecture has been randomized [13].…”

“…Substrate graphs, where nodes represent metabolites and links represent a reaction between connected substrates can be generated from genomic data [22]. In [12], we recently studied the impact of the topology of metabolic networks on a specific dynamics. There we implemented and studied only a single rule, namely (+, −) as a dynamic probe and interpreted the enhanced regularizing capacity of real networks compared to randomized null models as a possible topological contribution to the reliable establishment of metabolic steadystates and to the effective dampening of fluctuations.…”

Modeling Regulatory Networks to Understand Plant Development: Small Is Beautiful

“…More complex still is the coordination of these ingredients to make an organism, which for plants involves graded responses to stimuli and cell fate decisions (switches), coupled to regulation of the cell cycle (oscillators) and cell growth. Note that there need not be a correlation between complex behavior and the complexity of a network; it is possible for quite simple networks to behave in a highly complex manner, while extremely complex networks can behave, due to their specify topology, in a regular and tightly controlled way (Csete and Doyle, 2004;Marr et al, 2007). Given a dynamical model (of whatever size or complexity), it is of crucial interest to learn as much as possible about the model steady states (where all components of the network, for example, concentrations of various proteins or mRNAs, are in equilibrium so that they do not change in time) and their stability (whether or not the system moves toward the steady state when given a small perturbation away from it) as determinants of system dynamics.…”