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2 Extreme Pathway Lengths and Reaction Participation in Genome Scale Metabolic Networks Jason A. Papin, Nathan D. Price and Bernhard Ø. Palsson

3 Introduction Reaction Network Stoichiometric Matrix Duplication is only for easy drawing

4 Background For every metabolite in the system we get the following equation:

5 Background Lets look at B for example: Since the time constants associated with growth are much larger than those associated with each individual reaction we assume:

6 Background We get: Every solution of this set of equation is a steady state that the system can be in.

7 Background Reminder: Such a system is called homogenous. Such a system always has a solution (the zero solution). If it has more than one solution it has an infinite number of solutions. The set of all the solutions is a vector space. This vector space is called the null space. From the rank theorem of linear algebra we know: ( is the number of reactions)

8 Background Defining the null space In order to define the null space we need to find a spanning set. Reminder: A spanning set for a vector space of dimension is a set of vectors, such that every other vector in can be written as a linear combination of the vectors in. Mathematically: The minimal possible size of a spanning set is. If the spanning set satisfies this then it is called a base and all the vectors in it are linearly independent.

9 Background Defining the null space Since we have for every. This implies that every member of the base is a possible steady state. A Problem Mathematically, can take negative values. Biologically this creates a problem since each vector defines a flux which can not be “reversed”.

10 Background The solution Notice that we are only interested in solutions where for every (since the reactions must take place in the “right” direction). We find a spanning set such that every such solution can be written as a linear combination of the vectors in where all the coefficients take non-negative values. Notice that the vectors in such a set can be linearly independent.

11 Background The solution These vectors will be called genetically independent. Genetically independent vectors are a group of vectors in which no vector can be expressed as a linear combination of the other vectors such that all the coefficients are non negative. An algorithm to find a genetically independent minumum spanning set is described in Clarke’s paper “Complete set of steady states for the general stoichiometric dynamical systems” and will not be shown in framework of this presentation.

12 Background The resulting solution space takes the space of a convex polyhedral cone.

13 Extreme Pathways The genetically independent spanning set in the above example is the following:

14 Extreme Pathways Notice that each such vector defines a pathway in the Reaction network.

15 Extreme Pathways These pathways are called extreme pathways. From the way they were calculated we know that every possible steady state flux can be expressed as a non negative linear combination of these extreme pathways. The extreme pathways define the topological structure of the network.

16 Extreme Pathways We now define there extreme pathway matrix: equals the relative flux value through the reaction in the extreme pathway.

17 Extreme Pathway Length A property of the extreme pathways which we are interested in is the length of the extreme pathways. These lengths can be calculated from the extreme pathway matrix. First we transform to a binary matrix by changing all the non zero values to 1.

18 Extreme Pathway Length We then simply multiply with. The numbers in the position represent the length of the extreme pathway. The numbers in the represent the shared length of the and extreme pathways. The numbers in the position represent the length of the extreme pathway. The numbers in the represent the shared length of the and extreme pathways.

19 Extreme Pathway Length Why is this true? Lets look at the 1,3 entry for example: 1 st row of : 3 rd column of :

20 Extreme Pathway Length Using this method we can calculate the extreme pathway lengths for various organisms. In this article the lengths of the extreme pathways responsible for producing amino acids were calculated for: 1.Haemophilus influenzae – AKA Pfeiffer's bacillus or Bacillus influenzae. 2.Helicobacter pylori – A bacteria that infects the lining of the human stomach.

21 Extreme Pathway Length Extreme Pathway Length Haemophilus influenzae These distributions have more than one peak. This implies that there are often multiple common extreme pathway lengths around which deviations are made

22 Extreme Pathway Length Extreme Pathway Length Helicobacter pylori valine and alanine are almost identical except that the histogram is shifted. It takes five extra reaction steps to make valine for shorter extreme pathways and only three extra reactions for the longer ones. Conclusion: The number of extra reaction steps need to create valine instead of alanine depends on the length of the pathway. Conclusion: The number of extra reaction steps need to create valine instead of alanine depends on the length of the pathway.

23 Extreme Pathway Reaction Participation Another property of the extreme pathways which we are interested in is the reaction participation in the extreme pathways. The reaction participation of a reaction is the number of extreme pathways that the reaction takes place in. ‘s reaction participation is 3 for example.

24 Extreme Pathway Reaction Participation We want to calculate the reaction participation value for each of the reactions. Recall that is the matrix obtained from by changing all the non zero values to 1.

25 Extreme Pathway Reaction Participation This can be achieved by multiplying with. The numbers in the position represent in how many extreme pathways the reactions participates in. The numbers in the position represent in how many extreme pathways both reaction and reaction participates in.

26 Extreme Pathway Reaction Participation Why is this true? Lets look at the 2,4 entry for example: 2 nd row of : 4 th column of :

27 Extreme Pathway Reaction Participation What can we learn from the extreme pathway reaction participation matrix? Lets look the at for example: participates in 3 extreme pathways. Since there are only 3 extreme pathways we know that participates in all the extreme pathways.

28 Extreme Pathway Reaction Participation What else can we learn from the extreme pathway reaction participation matrix? Lets look the at and for example: participates in 3 extreme pathways. participates in 3 extreme pathways. Since there are 3 extreme pathways in which they both appear in we know that one takes place iff the other takes place.

29 Extreme Pathway Reaction Participation The reactions in region 1 participate in all of the extreme pathways. Conclusion: all the reactions in region 1 either participate or not together.

30 Extreme Pathway Reaction Participation This information if of value. If we know all the reactions that must occur together we can control (or completely prevent) a reaction by affecting a different reaction. As we just saw, in some cases this information is easily seen in the matrix. We will now describe an algorithm which based on the reaction participation matrix will find all the reactions that must occur (or not occur) together.

31 Extreme Pathway Reaction Participation The algorithm: 1.Each reaction is in a group of its own. 2.While changes: I.Check if there exist such that: II.For every such couple merge the two groups. Naïve implementation: iterations. The implementation can be improved to work in iterations by choosing the pairs on which we perform the test more carefully. is the reaction participation matrix. Can we make the algorithm work faster? Answer: Who cares?

32 Extreme Pathway Reaction Participation Example: We could make a small optimization. When we reach a reaction that was already joined with another reaction we need not check it. This however doesn’t change the worst case complexity. We could make a small optimization. When we reach a reaction that was already joined with another reaction we need not check it. This however doesn’t change the worst case complexity.