Power Laws in Network Utilization

Despite their successes, purely topologic approaches have important intrinsic limitations. For example, the activity of the various metabolic reactions or regulatory interactions differs widely, some being highly active under most growth conditions while others are switched on only for some rare environmental circumstances. Therefore, an ultimate description of cellular networks requires us to consider the intensity (i.e., strength), the direction (when applicable) and the temporal aspects of the interactions. While so far we know little about the temporal aspects of the various cellular interactions, recent results have shed light on how the strength of the interactions is organized in metabolic and genetic-regulatory networks.4

In metabolic networks the flux of a given metabolic reaction, representing the amount of substrate being converted to a product within unit time, offers the best measure of interaction strength. Recent metabolic flux-balance approaches (FBA)18"20'38'59 that allow us to calculate the flux for each reaction, have significantly improved our ability to generate quantitative predictions on the relative importance of the various reactions, leading to experimentally testable hypotheses. Starting from a stoichiometric matrix of the K12 MG1655 strain of E. coli, containing 537 metabolites and 739 reactions,18"20'38 the steady state concentrations of all metabolites satisfy, j-[Ai]=JtSijVj=0 (5)

where S¡j is the stoichiometric coefficient of metabolite A\ in reaction j and Vj is the flux of reaction/ We use the convention that if metabolite A; is a substrate (product) in reaction j, S¡j < 0 (S,j > 0 ) and we constrain all fluxes to be positive by dividing each reversible reaction into two "forward" reactions with positive fluxes. Any vector of positive fluxes {Vj ( which satisfies Eq. (5) corresponds to a state of the metabolic network, and hence, a potential state of operation of the cell.

Assuming that cellular metabolism is in a steady state and optimized for the maximal growth rate,18'38 FBA allows us to calculate the flux for each reaction using linear optimization, providing a measure of each reactions relative activity.4 A striking feature of the flux distribution of E. coli is its overall inhomogeneity: reactions with fluxes spanning several orders of magnitude coexist under the same conditions (Fig. 5A). This is captured by the flux distribution for E. coli, which follows (the by now familiar) power law where the probability that a reaction has flux V is given by P(v) - (v + Vo)~". The flux exponent is predicted to be a = 1.5 by FBA methods.4 In a recent experiment21 the strength of the various fluxes of the central metabolism was measured, revealing4 the power-law flux dependence P(v) - v~" with a=l (Fig. 5B). This power law behavior indicates that the vast majority of reactions have quite small fluxes, while coexisting with a few reactions with extremely large flux values.

The observed flux distribution is compatible with two quite different potential local flux structures.4 A homogeneous local organization would imply that all reactions producing (consuming) a given metabolite have comparable fluxes. On the other hand, a more delocalized "hot backbone" is expected if the local flux organization is heterogeneous, such that each metabolite has a dominant source (consuming) reaction. To distinguish between these two scenarios for each metabolite i produced (consumed) by k reactions, we define the measure7'14 - \2

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