# Small World inside Large Metabolic Networks

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# Abstract

a graph theoretical analysis of the E. coli metabolic network and find that this network is a small-world graph.

the connectivity of the metabolites follows a power law

## Introduction

### One type of question

to ask for a given metabolic network concerns the availability and yield of transformation routes from nutrients to end-products.

This is traditionally answered by consideration of the presence or absence of all the necessary steps of the classical biochemical pathways, but it can be argued that interpretation of metabolic networks in terms of these classical pathways fails to reveal the full potential of the network.

methods to enumerate the full repertoire of potential pathways, such as elementary modes analysis.

### Another type of question

concerns the identity of the key intermediary metabolites that must be generated by catabolism for use in anabolism, and that therefore define the centre of metabolism dividing catabolism from anabolism.

### thus

the aim of this study was to characterize the structure of this particular metabolic network, to determine whether it can be objectively said to have a centre, and if so, to determine the identity of the central metabolites.

## Theory and methods

characteristic path length $L$

clustering coefficients $C(v)$

the benchmark case of a random graph with the same number of vertices $n$ and mean degree $\bar k$ by exploiting the available statistical theory of random graphs.

in connected sparse random graphs with $n$ nodes and average degree $\bar k$, $(\bar k « n)$, the probability $p$ of two vertices being connected is given by $p=\bar k/(n-1)$. Such graphs show

- a binomial distribution of vertex degree $k$
- a very small clustering coefficient, close to the theoretically attainable minimum of zero for large n
- a characteristic path length that is also close to the theoretically attainable minimum

Among all graphs with the same number of vertices and edges, random graphs are among the most rapidly traversed.

## Results

In a random graph with n nodes and probability p of two nodes being connected the degree of each vertex follows a binomial distribution with variance $(n-1)p(1-p)$. The variance in the degree of the metabolic graphs, however, is up to 20-fold greater than that of the corresponding random graph with $p=\bar k/(n-1)$, implying that some vertices in metabolic graphs have many more, and others many fewer, neighbors that vertices for a random graph.

key metabolites or key reactions

the degree distribution of a substrate graph is consistent with a power law, i.e, the probability P(k) of finding a vertex with degree k is proportional to $k^{-\tau}$.

the small-worldness is known as “six degrees of separation”.

The formal definition of a small-world graph is that it is sparse but much more highly clustered than an equally sparse random graph, and its characteristic path length L is close to the theoretical minimum shown by a random graph. “small-worldness” is a global graph property that cannot be found by studying local graph properties.