Networks in Chemistry and Biology

Biological evolution and evolutionary computation

Ivo Hofacker, Bärbel M.R. Stadler, Andreas Wagner, Christoph Flamm, Alan Lepedes, Peter F. Stadler, Christian V. Forst, Homayoun Bagheri-Chaichian, Fernando Fontanari (from left to right)

From left to right:
Ivo Hofacker, Bärbel M.R. Stadler, Andreas Wagner, Christoph Flamm, Alan Lepedes, Peter F. Stadler, Christian V. Forst, Homayoun Bagheri-Chaichian, Fernando Fontanari.

The reason for the gathering of the workgroup on "Networks in Chemistry and Biology" in Bielefeld, May 2001, was to identify avenues for future research on naturally occurring networks, in particular for metabolic networks and gene regulation networks.

 

A Mathematical Framework for Biological Networks

Despite a number of promising approaches there does not seem to be a generally accepted description of chemical reaction networks, let alone of more general classes of biologically relevant networks such as gene-regulation networks. It was one of the purposes of this workgroup at ZIF to initiate an investigation into these direction.

While chemical reaction networks can be represented as bipartite graph with one class of nodes representing the reactions and the other class of nodes standing for the reactants (see e.g. Temkin, 1996, and the figure below) this framework does not seem satisfactory because the standard graph-theoretical concepts such as path and cycle do not appear to match well with chemical properties of the network. On the other hand, the routes central to metabolic flux analysis (MFA, see e.g., Fell, 1997; Schilling 2000), are defined as extremal rays of a certain convex cone associated with the stoichiometric matrix S of the network, i.e., in purely algebraic terms. A particular type of directed hypergraph which is closely related to S (Zeigarnik, 2000) is of particular interest in this context. It has become clear that a more geometric interpretation of the algebraic approach is desirable. Petri nets are another promising approach that has a distinctly algebraic flavor. See e.g. (Hofestädt 1998).

Not surprisingly, a static description of network structures will be insufficient to understand their functionality, however. Hence a comprehensive formalism that encapsulates both metabolic fluxes and its regulation at the genetic level will be necessary.

 

Comparative Network Genomics

The analysis of physicochemical properties of metabolic networks is a well established research area that derived from the field of Origin of Life.

Multi-level approaches

With the advent of post-genomic research and the exponentially growing number of completely sequenced genomes suggest the use of new multi-level approaches that combine genome information with non-genomic network connection.

The conventional sequence comparison and phylogenetic analysis of individual sequences can be extended to metabolic networks (Forst 1999). For this purpose we have developed a method that combines distance information of aligned sequences with network information of metabolic networks.

Connectivity information

Connectivity information of metabolic networks is coded into an adjacency matrix and combined with alignments of corresponding enzymes that function in the network. The distance matrices of individual enzymes are then combined by a direct sum, considering gap-distances for missing connections in the network and different weights for ortholog and paralog network presentations.

An illustrative example is the trp/ser Biosynthesis Network that shows an interesting correlation between operon conservation and pathway similarity (Forst 2001).

Connectivity information

Connectivity information of metabolic networks is coded into an adjacency matrix and combined with alignments of corresponding enzymes that function in the network. The distance matrices of individual enzymes are then combined by a direct sum, considering gap-distances for missing connections in the network and different weights for ortholog and paralog network presentations.

 

The Structure of Chemical Reaction Networks

Recent work on the over-all structure of metabolic networks (Jeong 2000, Wagner 2000) has revealed that they exhibit generic features of small world networks such as small diameters and a power-law degree distribution. Recently some of us have investigated the distributions of short cycles in these large metabolic network. We find that both metabolic network and models for the chemical reaction networks of planetary atmospheres have a particularly large number of triangles and a deficit in large cycles (Gleiss 2001).

Simplified chemical network of athomsphere of the Iovian satellite Io

Simplified chemical network of athomsphere of the Iovian satellite Io

Short cycles reduce the length of detours when a connection is clipped, so we propose that long cycles in metabolism may have been selected against in order to shorten transition times and reduce the likelihood of oscillations in response to external perturbations.

 

Limits of the Static Description

The following example, due to Homayoun Bagheri-Chaichian (2001), shows how far-reaching, and unpredictable from a "static" point of view, the consequences of network dynamics can be.

Metabolic control analysis (Kacser & Burns, 1973, Heinrich & Rapoport, 1974) was developed for the understanding of multi-enzyme systems. At the core of this approach is the flux summation theorem. This theorem implies that there is an invariant relationship between the control coefficients of enzymes in a pathway. One of the main conclusions that has been derived from the summation theorem is phenotypic robustness to mutation (e.g. dominance) is an inherent property of metabolic systems and hence does not require an evolutionary explanation (Kacser & Burns, 1981; Porteous, 1996).

Sum of the control coefficents as a function of enzyme concentrations in two-step pathway

Sum of the control coefficents as a function of enzyme concentrations in two-step pathway

Bagheri-Chaichian (2001) showed that for mutations involving discrete changes (of any magnitude) in enzyme concentration the flux summation theorem does not hold. The scenarios examined are two-enzyme pathways with a diffusion barrier, two enzyme pathways that allow for enzyme saturation and two enzyme pathways that have both saturable enzymes and a diffusion barrier. The results are extensible to sequential pathways with any number of enzymes. The fact that the flux summation theorem cannot hold in sequential pathways casts serious doubts on the claim that robustness with respect to mutations is an inherent property of metabolic systems.

 

References