Example Problems:Chapters 6 & 7

Systems Biology Study Group

Sarah Munro

11-19-2007

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

E

Axt

Ext

bypxt

A B C

byp

v1 v22b1

D

cof

byp

2cof

b3

b2v3v5

v4

v6

Reaction Network Map

E

Axt

Ext

bypxt

A B C

byp

2

D

cof

byp

2

cof

Metabolite Connectivity Map

b1 v1 v2

v3

v4

v6

v5

b3

b2

100000000bypxt

010000000Ext

001-000000Axt

1-00000110byp

00001-11-00cof

01-0101000E

000011-100D

0001-1-0010C

0000002-2-1B

001000001-A

bbbvvvvvv

S

321654321

1000

1100

0100

0110

0010

0011

1001

0001

1353123454363312

Urea

Arginine

Fumarate

ccinateArgininosu

Aspartate

Citrulline

Ornithine

P-Carbamoyl

............

S1

Create E1, the elemental matrix for S1:

Ornithine C5H13N2O2

Carbamoyl Phosphate CH2NO5P

Citrulline C6H13N3O3

Aspartate C4H6NO4

Argininosuccinate C10H17N4O6 Fumarate C4H2O4

Arginine C6H15N4O2 Urea CH4N2O

00000001P

24041321N

12464325O

415217613132H

164104651C

UreaArgFumsucc-ArgAspCitrulOrnithP-Carb

E1

Multiply the elemental and stoichiometric matrices in MATLAB:

0001P

0000N

1014O

2022H

0000C

1.3.5.31.2.3.45.4.3.63.3.1.2

S1E1

E1·S1 ≠ 0 Something is missing!

0001H

1-010OH

0001HPO

1000Urea

1100Arginine

0100Fumarate

0110ccinateArgininosu

0010Aspartate

0011Citrulline

1001Ornithine

0001P-Carbamoyl

1.3.5.31.2.3.45.4.3.63.3.1.2

2

4

S2

00000001P

24041321N

12464325O

415217613132H

164104651C

UreaArgFumsucc-ArgAspCitrulOrnithP-Carb

E2

001

000

014

121

000

HOHHPO 24

Multiply the new elemental and stoichiometric matrices in MATLAB:

0000P

0000N

0000O

0000H

0000C

1.3.5.31.2.3.45.4.3.63.3.1.2

S2E2

E2·S2 = 0 The S matrix is now correct !

H2O

HPO4

H+

1

1-001-10000E

00011-1-100D

01-0000010C

0000011-1-1B

001-000001-A

vvvvvvvvv

S

eca654321

100110000E

000111100D

010000010C

000001111B

001000001A

vvvvvvvvv

S

eca654321

1-010C

001-1B

01-01-A

vvvv

S

ca21

2

1010C

0011B

0101A

vvvv

S

ca21

2

100v

001v

110v

011v

CBA

S

c

a

2

1T2

1010v

0101v

1021v

0112v

vvvv

SS A

c

a

2

1

ca21

2T2v

Reaction Adjacency Matrix, Av:

How many compounds participate in va? In v1?

How many compounds do v2 and vc have in common?

210C

121B

012A

CBA

SS A T22x

Compound Adjacency Matrix, Ax:

How many reactions does compound A participate in?

How many reactions do A and B participate in together? What about compounds A and C?

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

Teusink_Glycolysis

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

Teusink_Glycolysis_core

Rxn Name Rxn Abbrev Rxn #'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

Metab Names Metab Abbrev Metab #'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' 'NAD' 13'NADH' 'NADH' 14'CO2' 'CO2' 15'Extracellular Glucose' 'GLCo' 16'Ethanol' 'ETOH' 17

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 endendend

%calculate transpose of SbinSbinT = transpose(Sbin);

%calculate Ax, the compound adjacency matrixAx = Sbin*SbinT; %calculate Av, the reaction adjacency matrixAv = SbinT*Sbin; % bar plot of the number of metabolites y, that participate in x reactions[m,n] = size(Ax);y = [];for i = 1:my = [y Ax(i,i)];endmaxreactions = 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')

1 2 3 4 50

2

4

6

8

10

12

14

number of reactions

num

ber

of c

ompo

unds

Participation of Compounds in Reactions for Glycolysis Core

0 5 10 15 20 25 300

5

10

15

20

25

30

number of reactions

num

ber

of c

ompo

unds

Participation of Compounds in Reactions in E. coli Core

What’s Next?Singular Value Decomposition?

Calculating Extreme Pathways?

Running Simulations using ODE solvers?