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Humboldt-Universität
zu Berlin
Edda Klipp
Systembiologie 3 - Stoichiometry
Sommersemester 2010
Humboldt-Universität zu BerlinInstitut für BiologieTheoretische Biophysik
Humboldt-Universität
zu Berlin
Stoichiometric Analysis of Cellular Reaction Systems
2A B + C 2D
E F
G
v1 v3
v2
- Analysis of a system of biochemical reactions- Network properties- Enzyme kinetics not considered
http://www.genome.ad.jp/kegg/pathway/map/map01100.html
Humboldt-Universität
zu Berlin
Stoichiometric Coefficients
Stoichiometric coefficients denote the proportions, with which the molecules of substrates and products enter the biochemical reactions.
Example Catalase2222 OOH2OH2
Stoichiometric coefficients for Hydrogenperoxid, water, oxygen -2 2 1
Stoichiometric coefficients can be chosen such that they agree with molecularity, but not necessarily.
2222 OOHOH2
1
-1 1 1/2
2222 OOH2OH2
2 -2 -1
Their signs depend on the chosen reaction direction. Since reactions are usually reversible,one cannot distinguish between „substrate“ and „product“. v - v
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Time Course of Concentrations
Usually described by ordinary differential equations (ODE)
d
d
H O
tv2 2 2 d
d
H O
tv2 2
d
d
O
tv2 Example catalase
for this choice of stoichiometric coefficienten:
2222 OOH2OH2
-2 2 1
Humboldt-Universität
zu Berlin
Time Course of Concentrations
Several reactions at the same time all rate equations must beconsidered at the same time.
S 1 S 2
S 3
1 2 3
4
43
322
4211
vS
vvS
vvvS
Usually described by ordinary differential equations (ODE)
One can summarize the stoichiometric coefficients in matrix N. The rows refer to the substances, the columns refer to the reactions.
4321 SSS
100001101011
3
2
1
N
Column: reaction
Row: Substance
External metabolites are not included in N.
Humboldt-Universität
zu BerlinBalance equations/Systems equations
In general: We consider the substances Si and their
stoichiometric coefficients nij in the respective reaction j.
If the biochemical reactions are the only reason for the change of concentration of metabolites, i.e. if there is no mass flow by convection, diffusion or similarThen one can express the temporal behavior of concentrations by the balance equations.
d
d
S
tn vi
ij jj
r
1
r – number of reactionSi – metabolite concentrationvj – reaction ratenij – stoichiometric coefficient
Humboldt-Universität
zu BerlinSummary
Stoichiometric matrix
Vector of metabolite concentrations
Vector of reaction rates
Parameter vector
ijnN ni ...1 rj ...1
T1,..., nSSS
T1 rvv ,...,v
T1,..., mppp
With N can one write systems equations clearly.
d
d
Sv
tN
Metabolite concentrations and reaction rates are dependent on kinetic parameters.
pSS pSvv ,
Humboldt-Universität
zu BerlinThe Steady State
Reaction systems are frequently considered in steady state,Where metabolite concentrations change do not change with time.This describes an implicite dependency of concentrationsand fluxes on the parameters.
0pS,vN 0dt
dSb.z.w.
The flux in steady state is p,pSvJ
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zu Berlin
Example Unbranched pathway
.constSconstS n .,0
S S1 2, S0 S1 S2 Sn
v1 v2 v3
variabel
3
12
2
11
23122
1211
,
0
0
k
vS
k
vS
SkSkdt
dS
Skvdt
dS
Assumption: Linear kinetics
System equations
Matrix formalism
dS1 / dt = v1-v2
dS2 / dt = v2-v3
d S1 1 -1 0 dt S2 0 1 -1
v1
v2
v3
=
S N v.
.=
Steady state 321 vvv
Nv = 0 is usually a non-linear equation system, which cannotbe solved analytically(necessitates knowledge of kinetic().
dSi /dt = 0
Humboldt-Universität
zu Berlin
The Stoichiometric Matrix N
- Characterizes the network of all reactions in the system
- Contains information about possible pathways
vS
Ntd
d
Humboldt-Universität
zu BerlinThe Kernel Matrix K
In steady state holds0
d
d v
SN
t
Non-trivial solutions exist only if the columns of N are linearly dependent. rNRang
Mathematically, the linear dependencies can be expressed by a matrix K withthe columns k which each solve
NK 00Nk
K – null space (Kernel) of N
The number of basis vectors of the kernel of N is NRangr-
Humboldt-Universität
zu BerlinCalculation of the Kernel Matrix
The Kernel matrix K can be calculated with the Gaussian EliminationAlgorith for the solution of homogeneous linear equation systems.
0
0
110
111NK
31
21
11
k
k
k
0
0
3121
312111
kk
kkk
0
1
11
3121
k
kk
Example
Alternative: calculate with computer programmesSuch as „NullSpace[matrix]“ in Mathematica.
Humboldt-Universität
zu BerlinRepresentation of Kernel Matrix
The Kernel matrix K is not uniquely determined. Every linear combination of columns is also a Possible solution. Matrix multiplication with a regular Matrix Q „from right“ gives another Kernel matrix.
For some applications one needs a simple ("kanonical") representation of the Kernel matrix.
A possible and appropriate choice is
K contains many zeros.
KQK ˆ
I
KK 0
I – Identity matrix
Humboldt-Universität
zu BerlinInformations from Kernel Matrix K
-Admissible fluxes in steady state
-Equilibrium reactions
-Unbranched reaction sequences
-Elementary modes
Humboldt-Universität
zu BerlinAdmissible Fluxes in Steady State
110
011N
111
1k 1kK S0 S1 S2 S3
v1 v2 v3
S0 S1 S2 S3
v1
v2
v4
v3
1110
0111N
0110
1k
1011
2k
Sv2
v1v3 111 N
0
1
1
1k
1
0
1
2k
21 kkK
21 kkK
Examples
Humboldt-Universität
zu BerlinAdmissible Fluxes in Steady State
With the vectors ki (k1, k2,…) is also every linear combination A possible columns of K.
for example: instead and also
All admissible fluxes in steady state can be written as linear combinationsof vectors ki :
The coefficients i have the respective units, eg. or .
101
011
21
3
2
1
vvv
v
Mol s 1 mol l h1 1
0110
1k
1011
2k
11
01
0110
1011
123 kkk
Sv2
v1v3für
In steady state holds 0d
d v
SN
t
Humboldt-Universität
zu BerlinEquilibrium Reactions
Case: all elements of a row in K are 0Then: the respective reaction is in every steady state in equilibrium.
00 j
jj
ijji kv
S1 S2
S3
0 1 2
3
0
1
1
1
K
Example
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Unbranched Reaction Steps
S0 S1 S2 S3
S4
v1 v2 v3
v4
0110
1011N
1
0
0
1
,
0
1
1
1
21 kk
1
0
0
1
0
1
1
1
21
4
3
2
1
v
v
v
v
v
32 vv
The basis vectors of nullspace have the same entries for unbranchedreaction sequences.Unbranched reaction sequences can be lumped for further analysis.
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Kernel Matrix –Dead Ends
S0 S1 S2 S4
S3
v1 v2 v3
v4
1000
0110
1011
N
S1, S2, S3 intern, S0, S4 extern
Necessary and sufficient condition for a „Dead end“:One metabolite has only one entry in the stoichiometric matrix (is only onceSubstrate or product).
Flux in steady state through this reaction must vanish in steady state (J4 = 0).
Model reduction: one can neglect those reactants for steady state analyses.
Humboldt-Universität
zu Berlin
Kernel Matrix – Irreversibility
S0 S1 S2 S3
S4
v1 v2 v3
v4
0110
1011N
1
0
0
1
,
0
1
1
1
21 kk
2211 kk v02 v
1
1
1
0
,
0
1
1
1
kk02 v Mathematically possible,
biologically not feasible
Other choice of basis vectors
The basis vectors of a null space are not unique.The direction of fluxes (signs) do not necessarily agree with the direction of irreversible reactions.
(Irreversibility limits the space of possible steady state fluxes.)
S1, S2 internal, S0 , S3, S4 external
Humboldt-Universität
zu BerlinConservation Relations: Matrix G
If compounds or groups are not added to or deprived of a Reaction system, then must their total amount remain constant.
0
dt
EESd .constEES Michaelis-Menten kinetics
2 3A BIsolated reaction: 2 3A B const .
Pyruvatkinase, Na/K-ATPase ATP ADP ADP ATP const .
Examples
Humboldt-Universität
zu BerlinConservation Relations - Calculation
If there exist linear dependencies between the rows of the stoichiometric matrix, then one can find a matrix G such as
0GN N – stoichiometric matrix
Due tod
d
Sv
tN holds 0
d
d v
SGNG
t
The integration of this equation yields the conservation relations.
.S constG
Humboldt-Universität
zu BerlinConservation Relations – Properties of G
The number of independent row vectors g (= number of Independent conservation relations) is given by
Nrankn
(n = number of rows of the stoichiometric matrix = number of metabolites)
GT is the Kernel matrix of NT, and can be found in the same way as K. (Gaussian elimination algorithm)
The matrix G is not unique, with P regular quadratic matrix is again conservation matrix.
PGG
IGG 0Separated conservation conditions:
Humboldt-Universität
zu BerlinConservation Relations – Examples
ATP ADPATP
ADP
11
11N
0GN 11G ADP ATP const .
.S constG
Conservation of atoms or atom groups, e.g. Pyruvatdecarboxylase (EC 4.1.1.1)
CH COCOO H CH CHO CO3 3 2
1
1
1
1
N
0
2
1
4
1
2
1
0
0
3
3
3
Gcarbonoxygen
hydrogen
1
0
0
0
1
1
0
1
0
1
0
1
GCH3CO-groupProtons
Carboxyl group
0011 g Elektric charge
Humboldt-Universität
zu BerlinConservation relations – Simplification of the ODE system
If conservation relations hold for a reaction system, then the ODE system can be reduced, since some equations are
linearly dependent. vS N
'N
NN
000 N
L
INLN
'Rearrange N, L – Linkmatrix(independent upper rows, dependent lower rows)
v'S
S
d
dS
d
d 0
b
a NL
I
tt
Rearrange S respectively(indep upper rows, dep lower rows)
d daS t N0 vReduced ODE system
d
d
d
db aS S
t tL' S S const .b a L'For dependent concentrations hold
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Basic Elements of Biochemical Networks
Systems equations
r – number of reactionsSi – metabolite concentrationsvj – reaction ratesnij – stoichiometric coefficients
Network properties Individual reaction properties
r
jjij
i vndt
dS
1
d
d
Sv
tN
ijnN ni ...1rj ...1 T
1,..., nSSS
T1 rvv ,...,v
Matrix representation
S1
S2
S4
S3
v1 v2
v3
v4
v5
Humboldt-Universität
zu Berlin
Nachtrag vom 10. Mai 2010
Humboldt-Universität
zu Berlin
Non-negative Flux Vectors
In many biologically relevant situations have fluxes fixed signs.We can define their direction such that
Sometimes is the value of individual ratesfixed.
Both conditions restrict the freedom for the choice of Basis vectors for K.
0jv
Example: opposite uni-directional rates instead of net rates, - Description of tracer kinetics or dynamics of NMR labels- different isoenzymes for different directions of reactions- for (quasi) irreversible reactions
fixedvk
Humboldt-Universität
zu BerlinElementary Flux Modes
Situation: some fluxes have fixed signes, others can operate in both directions.Which (simple) pathes connect external substrats?
SP1
P2
P3
v1
v2
v3
SP1
P2
P3
v1
v3
111N
v2
1
0
1
,
0
1
1
ik
1
1
0
,
1
0
1
,
0
1
1
,
1
1
0
,
1
0
1
,
0
1
1
ik
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zu Berlin
Elementary Flux Modus
-An elementary flux mode comprises all reaction steps, Leading from a substrate S to a product P.
-Each of these steps in necessary to maintain a steady state.
-The directions of fluxes in elementary modes fulfill the demands for irreversibility
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Number of Elementary Flux Modes
S0 S1 S2 S3
S4
v1 v2 v3
v4
1
1
1
0
,
1
0
0
1
,
0
1
1
1
,
1
1
1
0
,
1
0
0
1
,
0
1
1
1
k
S0 S1 S2 S3
S4
v1 v2 v3
v4
1
1
1
0
,
1
0
0
1
,
1
0
0
1
,
0
1
1
1
k
The number of elementary modes is at least as high as the number of basisvectors of the null space.