# Yaron Goldstein:

## Analysis of metabolic networks with lattice theory

### Abstract

This thesis aims to describe metabolic networks and reaction pathways in the cell using an algebraic structure. On this basis we want to establish analytical methods that can be used for a variety of approaches in qualitative modeling. To analyze the metabolism, the steady state flux cone is presently the most commonly used mathematical model. Concepts such as elementary flux modes, flux coupling analysis (FCA) and minimal cut sets have been introduced based on this flux cone. This thesis identifies modeling approaches and their applications based on the developed abstract algebraic structure -- thus introducing a general framework for qualitative analysis of the metabolism. Reaction sets that can be simultaneously active in a cell are represented as the elements of a join (semi)lattice. A qualitative description of elementary flux modes is given and the concepts used are translated in lattice theory. Partially based on existing FCA algorithms, a method is presented that can rapidly identify the maximum of a lattice. This maximum is then used to define an abstract term of flux coupling that can be applied to a diversity of qualitative models. The implemented algorithm is applied to simulate a complete reaction double knockout (EFCA) in different cell models. These models are based on a steady state assumption that can be combined with further thermodynamic constraints. Additionally a concept of target prediction is introduced, utilizing the calculation of lattice maxima, and related to (minimal) cut sets. Taken as a whole, it can be said that the most known analytical approaches for the metabolism correspond perfectly well to the most special elements of join lattices. We are able to apply lattice theory to both the standard model based on the flux cone, pathways through hyper graphs and a model using logic formulas. It becomes apparent that this logic model is a relaxation of the standard model. Depending on the application, the elementary flux modes are shown to be either the minimal or the irreducible elements of a lattice. The Starbucks lemma proves that both these sets can be easily worked with in many qualitative models. It further proves our FCA algorithm is correct for literally all kinds of qualitative models. We will show that our implementation, even though much easier to adapt to other models, has a running time comparable to well-established FCA algorithms. As a matter of fact we will see that further improvements of speed are most likely reachable only by more efficient ways of programming, not by more effective theoretical approaches. Using lattice theory it is much easier to answer many open questions about the cell's metabolism. One reason for this is that it becomes easier to apply known results from certain models on other ones. Hence, the general framework of lattice theory is perfectly suited for qualitative modeling and to develop or optimize analytical methods.