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Example Problems: Chapters 6 & 7 Systems Biology Study Group Sarah Munro

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Examples Drawing networks Creating the S Matrix Verifying the S Matrix Topological Properties of the network S for E. coli core metabolism S for Glycolysis

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E Axt Ext bypxt AB C byp v1v1 v2v2 2 b1b1 D cof byp 2 cof b3b3 b2b2 v3v3 v5v5 v4v4 v6v6 Reaction Network Map

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E Axt Ext bypxt ABC byp 2 D cof byp 2 cof Metabolite Connectivity Map b1b1 v1v1 v2v2 v3v3 v4v4 v6v6 v5v5 b3b3 b2b2

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Create E1, the elemental matrix for S1: Ornithine C 5 H 13 N 2 O 2 Carbamoyl Phosphate CH 2 NO 5 P Citrulline C 6 H 13 N 3 O 3 Aspartate C 4 H 6 NO 4 Argininosuccinate C 10 H 17 N 4 O 6 Fumarate C 4 H 2 O 4 Arginine C 6 H 15 N 4 O 2 Urea CH 4 N 2 O

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Multiply the elemental and stoichiometric matrices in MATLAB: E1·S1 ≠ 0 Something is missing!

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Multiply the new elemental and stoichiometric matrices in MATLAB: E2·S2 = 0 The S matrix is now correct !

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H2OH2O HPO 4 H+H+

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Reaction Adjacency Matrix, A v : How many compounds participate in v a ? In v 1 ? How many compounds do v 2 and v c have in common?

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Compound Adjacency Matrix, A x : How many reactions does compound A participate in? How many reactions do A and B participate in together? What about compounds A and C?

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1 Teusink et al. Eur. J. Biochem. (267) 2000 Teusink_Glycolysis

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1 Teusink et al. Eur. J. Biochem. (267) 2000 Teusink_Glycolysis_core

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Rxn NameRxn AbbrevRxn # 'Hexokinase''vGLK'1 'Glucose-6-phosphate isomerase''vPGI'2 'Phosphofructokinase''vPFK'3 'Aldolase''vALD'4 'Glyceraldehyde 3-phosphate dehydrogenase''vGAPDH'5 'Phosphoglycerate kinase''vPGK'6 'Phosphoglycerate mutase''vPGM'7 'Enolase''vENO'8 'Pyruvate kinase''vPYK'9 'Pyruvate decarboxylase''vPDC'10 'Glucose transport''vGLT'11 'Alcohol dehydrogenase''vADH'12 'ATPase activity''vATP'13

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Metab NamesMetab AbbrevMetab # 'Glucose in Cytosol‘'GLCi'1 'Glucose 6 Phosphate''G6P'2 'Fructose 6 Phosphate''F6P'3 'Fructose-1,6 bisphosphate''F16P'4 'Triose-phosphate''TRIO'5 '1,3-bisphosphoglycerate''BPG'6 '3-phosphoglycerate''P3G'7 '2-phosphoglycerate''P2G'8 'Phosphoenolpyruvate''PEP'9 'Pyruvate''PYR'10 'Acetaldehyde''ACE'11 'High energy phosphates''P'12 'NAD' 13 'NADH' 14 'CO2' 15 'Extracellular Glucose''GLCo'16 'Ethanol''ETOH'17

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function [Ax, Av, Sbin] = topo_properties(S) %Plots the number of metabolites y that participate in x reactions %Function file input is a mxn matrix that defines the stoichiometry of a %reaction network %Function file outputs include: Ax = compound adjacency matrix, %Av = reactions adjacency matrix, Sbin = binary form of Smatrix %Generate binary form of S matrix [m,n] = size(S); Sbin = zeros(m,n); for i= 1:m for j= 1:n if S(i,j)~=0; Sbin(i,j) = 1; if S(i,j) == 0; Sbin(i,j) = 0; end %calculate transpose of Sbin SbinT = transpose(Sbin);

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%calculate Ax, the compound adjacency matrix Ax = Sbin*SbinT; %calculate Av, the reaction adjacency matrix Av = SbinT*Sbin; % bar plot of the number of metabolites y, that participate in x reactions [m,n] = size(Ax); y = []; for i = 1:m y = [y Ax(i,i)]; end maxreactions = max(y); minreaction = min(y); reactions = [minreactions:1:maxreactions]; compounds = zeros(1,length(reactions)); for j = 1:length(reactions); I = find(y == reactions(j)); compounds(j) = [length(I)]; end bar(reactions,compounds) xlabel('number of reactions') ylabel('number of compounds')

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What’s Next? Singular Value Decomposition? Calculating Extreme Pathways? Running Simulations using ODE solvers?

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