Topological Properties of the Stoichiometric Matrix By: Linelle T Fontenelle

Topology Some geometric problems do not depend on the exact shape of the object, but rather on the way that points are connected in space Topology is a mathematical term which has the following motivating insight: some geometric problems depend not on the exact shape of the object involved but rather the way they are put together. To determine what properties these problems rely on, we must define the notion of topological equivalence. Two spaces are topologically equivalent if one can be deformed into the other without tearing apart space or sticking distinct parts together. Two spaces are topologically equivalent if one can be deformed into another without tearing space apart or sticking distinct parts together

Stoichiometric Matrix Consider the following system: X1 X2 x3 v1 v2 v3

Stoichiometric Matrix In matrix notation: S is a linear transformation of the flux vector, v, to a vector of time derivatives of the concentration vector Where: S is a linear transformation of the flux vector v to a vector of time derivatives of the concentration vector. A linear transformation is a special kind of mathematical function which has to meet certain conditions (eg if and only if). The input and output are both vectors. (The output variables are linear functions of the input variables. {A function T from Rm to Rn is called a linear transformation if there is an mxn matrix A such that T(x) = A(x) for all x in Rn

S Matrix as a Linear Map Concentration Solution Space -Column Space -Left null space Flux Solution Space -Row space -Null space S matrix consists of 4 fundamental subspaces. S matrix acts as the linear transformation between the space of rxn activities and time derivatives of concentration space. http://www.biophysj.org

Row Space of S Spanned by all the independent rows of S Space in which changes in the concentration values contribute to the flux rates The row space is the space in which the changes in the concentration values contribute to the flux rates. The row space defines the space of thermodynamic transduction which derives biochemical rxns to proceed http://gcrg.ucsd.edu/classes/4_slides_Smatrix.pdf

Null Space of S Consists of all vectors that satisfy Svss = 0 i.e. dx/dt = 0 Spans the steady state pathway space of a biochemical network Holds true for steady state solutions. The null space spans the steady state pathway space of a biochemical network http://gcrg.ucsd.edu/classes/4_slides_Smatrix.pdf

Left Null Space of S Consists of all vectors that define the dependencies of the rows in S Constrains the conserved relationships The left null space constrains all the conserved relationships such as total electric charge and elemental balances http://gcrg.ucsd.edu/classes/4_slides_Smatrix.pdf

Column Space of S Spanned by all the independent columns of S Defines the dynamic concentration space in which metabolites are formed and consumed The column space defines the dynamic concentration space in which metabolites are formed and consumed http://gcrg.ucsd.edu/classes/4_slides_Smatrix.pdf

Connectivity Properties of the S Matrix 6 2 3 Reactions Metabolites sij Every column of S is a chemical rxn with a defined set of values. The fluxes however are the values that represent the activity of the rxns and indicates how much is going thru them. S connects all the metabolites in a defined metabolic system. Metabolic networks must make all the biomass components of the cell in order for it to grow. Relationship between node connectivity and lethality fraction – lethality fraction of less connected metabolites is higher than that of highly connected metabolites Every column of S is a chemical rxn with a defined set of values. The fluxes however are the values that represent the activity of the rxns and indicates how much is going thru them. S connects all the metabolites in a defined metabolic system. Metabolic networks must make all the biomass components of the cell in order for it to grow.

Metabolite Connectivity Metabolite Connectivity of E.coli, H. influenzae, H. pylori and S. cerevisiae Some metabolites participate in a large no. of rxns. For eg ATP participates in more than 160 rxns in E.coli and S. cerevisiae. The conc of these metabolites r important since any changes in them affects many rxns. There r also metabolites that participate in only 2 rxns. The connectivity of metabolites on a log-log graph. Power law distribution of probabilities said to be scale free (biological significance is not clear) http://gcrg.ucsd.edu/classes/4_slides_Smatrix.pdf

Elementary Topological Properties Based on nonzero elements of S The Binary form of S Elementary topological properties are determined based on the nonzero elements of S S

Elementary Topological Properties Participation Number Connectivity Number Nonzero entries in the rows and columns of S give the elementary topological properties. Participation number is the no. of compounds that participate in rxn j (most likely 3). The connectivity no. gives the no. of rxns in which compound I participates. All compounds have to be present for a rxn to take place (not so in functional state which implies fundamental difference between flux and concentration maps). Internal metabolites must have a participation no. of 2 or more or else there will be a dead end in the network. External metabolites typically have 1 single rxn associated with them

Expanded Elementary Topological Properties Compound Adjacency Matrix: Ax = Diagonal elements summation gives the no. of rxns in which compound xi participates. The off diagonal elements gives the number of rxns in which both compounds xi and xj participate and shows how extensively the 2 compounds are topologically connected in the network Ax x = Leads to symmetrical matrix. Diagonal elements summation gives the no. of rxns in which compound xi participates. The off diagonal elements gives the number of rxns in which both compounds xi and xj participate and shows how extensively the 2 compounds r topologically connected in the network n x m = 3 x 4 m x m = 4 x 4 m x n = 4x 3

Expanded Elementary Topological Properties Reaction Adjacency Matrix: Av = Av x = n x m = 3 x 4 Diagonal elements give participation no. Off diagonal elements count how many compounds 2 rxns have in common m x n = 4x 3 n x n = 3 x 3 Diagonal elements give participation no. Off diagonal elements count how many compounds 2 rxns have in common

Example of Adjacency Matrix A cofactor-coupled reaction: C + AP CP + A S

Example of Adjacency Matrix Ax = Av = = = = =

Conclusion/Future Direction Elementary topological properties give network connectivity and component participation information Leads to combined and simultaneous characterization of metabolites and reactions: SVD (decouples and decorrelates systemic properties)