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BIOCHEMICAL

REACTION NETWORKSMetabolic Flux Analysis, MFA

Metabolic flux analysis

What is metabolic flux analysis?

What is the point of MFA?

Key concepts

Examples

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Limitation of macroscopic models

Macroscopic models are very

useful, but there is a lot of

available information that is

not used.

No information about

intracellular reactions is

used. (i.e. the Biochemistry is

completely neglected!)

In Out

Cell

What is MFA?

Metabolic flux analysis is the calculation and analysis of

the flux distribution of the entire biochemical reaction

network in the system of study.

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What information can be used?

Many metabolic pathways are well characterized

the glycolysis

the TCA-cycle

the PPP (pentose phosphate pathway)

Synthetic pathways for amino acids and nucleotides

S1S2

..

.

SN

P1P2

..

.

PM

v1

-rs,N

-rs,1 rp,1

rp,M

X1X2

..

.

XKvJ

v2

.

Biomass

Figure 5.1 A general representation of reactions considered in a metabolic network.N substrates enter the

cell and are converted into M metabolic products via a total ofK intracellular metabolites. The conversions

occur via J intracellular reactions for which the rates are given by v1vJ. Rates of substrate formation

(rs,1,,rs,N) and product formation (rp,1,,rp,M) are also shown.

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Example 1

A B

C

D

E

F

NADH

2 NADH

12

34

5

A hypothetical reaction scheme

Example 1

Assume pseudo-steady state for the intermediates B, D

and NADH 3 constraints on the system.

(1) 1 - 2 - 3 = 0

(2) 345 = 0

(3) 2 25 = 0

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Example 1Solving for 2, 3 and 4 in terms of 1and 5

1. 2 = 252. 3 = 1 253. 4 = 1 35

All five fluxes could thus be calculated fromonly two measured values, 1 and 5, byusing the steady-state constraints.

Example 1

In this simple case, the substitutions could be made by

hand calculations, but this is not in general possible or to

be recommended.

Instead a matrix approach should be used, in which all

reactions are treated using linear algebra.

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Key concepts

The matrix, T (the transpose of the stoichiometric

matrix) *

The steady-state reaction rates,

The net rates of formation, r

*IMPORTANT NOTE: The nomenclature is dif ferent in the 2nd edi tionof NV&L. In the 2nd edition, T denotes the stoichiometric matrix

0

X

p

s

Matrix containing all

coefficients in the reactions;

one row per reaction

substrates

products

intracellular biomass

components, metabolites

The reaction stoichiometries are

defined by the stoichiometric matrix

*IMPORTANT NOTE: The nomenclature is dif ferent in the 2nd edit ionof NV&L. In the 2nd edition, T denotes the stoichiometric matrix

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Example 2 (from NV&L, 2nd ed)

Substrate

Metabolite 2

Metabolite 1

Metabolite 3

C

B

A

v1

v2v3

v4

v6v5

v1=v2Balance for A

v2=v3 +v4Balance for B

v4=v5 +v6Balance for C

v1=v2Balance for A

v1=v2Balance for A





The stoichiometric matrix;

1001000

1000100

1100000

0100010

01100000010001

S P1 P2 P3 A B C

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The net formation rates, r, can be calculatedfrom and T

vv

0

rT

T

Stoichiometric matrix

Vector of fluxes, i.e. steady-state

reaction rates

Vector of net formation rates

Metabolite steady-state net

formation rates

Example 2

654

432

21

6

5

3

1

6

5

4

3

2

1

3,

2,

1,

111000

001110

000011

100000

010000

000100

000001

0

0

0

vvv

vvv

vv

v

v

v

v

v

v

v

v

v

v

r

r

r

r

p

p

p

s

The last three rows give the steady-state requirements

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Example 3

4 metabolites

(A, B, C, D)

3 reactions

(A - B)

(B - C)

(B - D)

1, 2, 3 reaction rates (mol/m3 s)

12

3

Example 3

1010

01100011

1st reaction

2nd reaction

3rd reaction

A B C D

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Example 3

Tr

3

2

1

100

010111

001

D

C

B

A

r

rr

r

Example 3

3

2

321

1

D

C

B

A

r

rr

r

seems reasonable..

Net rates of formationReaction rates fluxes

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So if we know , we can calculate r. However, normally the situationis the opposite, i.e we want to calculate from a (partially) know r

vT

T

0

r

r

2

1

m

cNet formation rates

to be calculated

Measured rates

Rates of formation of

intracellular metabolites = 0We want to calculate

this vector.

Dimensions of vectors involved

P = total number of compounds involved (i.e. thedimension of the vector r)

J = number of fluxes (i.e. the dimension of the vector)

K = number of metabolites at steady-states F = J K = (probably) equal to the degrees offreedom in the system = number of neededmeasured net formation rates, (i.e. the dimension ofthe vector rm)

The dimension of the matrix T is thus (P x J)

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2

1

T

TT

Corresponding to the

non-measured part of r.

Dimensions (P-J) x J

Corresponding to the

measured part of r and themetabolites at steady-state.Dimensions (J+(K-J) x J, i.e.

J x J.

The matrix equation (1) can thus be divided

into two equations (2 and 3)

Tr

1Trc

20

Trm

(2)

(3)

(1)

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Equation 3 is now used to calculate

00

1

22

1

2

1

2

mm rTTT

rT

Multiply both sides from the left by the inverse of the matrix T2

Example 1

revisited

A B

C

D

E

F

NADH

2 NADH

12

34

5

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Example 1Stoichiometry

0

0

0

0

0

2100010

0100100

0110000

1011000

0010001

NADH

D

B

C

E

F

A

sp

X

Example 1

T

0

m

c

r

r

Flux vector

Stoichiometric

matrix (transposed)

measured netproduction rate

r to be calculated

assumed at steady

state ( r = 0)

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Example 1Assume that rA and rF are measured

5

4

3

2

1

20010

11100

00111

10000

00001

00010

01000

NADH

D

B

F

A

C

E

r

r

r

r

r

r

r

measr

calcr

We get

20

Trm

5

4

3

2

1

20010

11100

00111

10000

00001

0

0

0

F

A

r

r

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The fluxes are obtained from

0

1

2

mrT

To calculate the fluxes we need to invert T2. This is a major task to do

Manually, but trivial if using a suitable mathematical program like MATLAB.

We get:

00010

11131

10121

10020

00001

1

2T

In terms of the measured rates, we thus have

the fluxes:

F

FA

FA

F

A

r

rr

rr

r

r

3

2

2

5

4

3

2

1

Note. RememberrA < 0, rF > 0

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What about the non-measured formationrates?

vTr 1cThe non-measured product formation rates are obtained from

5

4

3

2

1

00010

01000

C

E

r

r

F

FA

C

E

C

E

r

rr

r

rv

r

r

2

3

2

4

We get:

Summary

MFA is a systematic way of analyzing fluxes in a

metabolic network

The degree of freedom (rank of) the matrix T determines

how many fluxes must be known to fully resolve the

metabolic fluxes in the network.

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Recipe

Write down all known reactions and define

the stoichiometry

Write down the T-matrix

Determine the rank of T

Consider the observability of the system.

(What rates can and should be measured?) Carry out the experiments and the analysis!

Fundamental equations

vT

T

0

r

r

2

1

m

c

0

rTv

m1

2

vTr 1c

Net formation rate

to be calculated

Measured rate

Intracellular metabolite, no

net formation

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Simplification of metabolic networks

A

E

D

CB

F1 2

3

5

4Consider

Full model5 fluxes, 3 metabolites, 2 df

0 -1 0 1 0 0

0 0 0 -1 1 0

0 0 0 0 -1 1

0 0 1 0 0 -1

1 0 0 0 -1 0

=

-1 0 0 0 0

-1 0 -1 0 0

0 1 0 0 1

0 1 0 0 0

-1 -1 -1 -1 -1

T2-1 =

FA

F

F

A

A

rr

r

r

r

r

5

4

3

2

1

Obviously, 1=2 and 3=4

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The system can be reduced by only consideringbranchpoints

A

E

C

F1 3

5

Consider the reduced system-1 0 0

0 1 0

-1 -1 -1

T2-1 =

0 -1 0 1

0 0 1 -1

1 0 0 -1

=

FA

F

A

rr

r

r

5

3

1

Back to the favourite example

B

C

D

E

F

NADH

2 NADH

12

34

5

NAD+

2 NAD+

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0 0 -1 0 1 0 0 0

1 0 0 0 -1 0 1 -1

0 0 0 0 -1 1 0 0

0 1 0 0 0 1 0 0

0 0 0 1 0 1 -2 2

=

0 0 0 0 1

1 -1 -1 0 0

0 0 1 1 1

0 1 0 0 -2

0 -1 0 0 2

T2 =But.. T2 is NOT invert ible.

The rank of T2 is only 4 (not 5).

Applying the steady-state requirement also for NAD+ shouldgive another constraint leaving only one rate to be measured.

Only one component in each co-factor pair (NADH/NAD+;

NADPH/NADP+; ATP/ADP etc.) should be included in the

stoichiometric matrix. If the steady state requirement is

met for one component it is automatically satisfied for the

other, since they are not independent.

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Steady state approximationA small complication.. (Note 5.1)

dt

dXxX

dt

dx

dt

xXd ii

i )( chain rule

xTq ii

xdt

dx

net volumetric acc. term ofcompound i

volumetric formationrate of biomass

Steady state approximation

iii

XTdt

dX

Surprise!

Dilution term

Must be small forPSS approx. to bevalid!

Intracellular acc. term

This is what is normallyassumed to be = 0!

Volumetric net formation term/ biomass conc.(= qi/x)

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Intracellular concentrations

Metabolite/co-factor Concentration

(M)Metabolite/co-factor Concentration

(M)Glucose (GLC)Glucose-6-P (G6P)

Fructose-6-P (F6P)Fructose-1,6-bisP (FDP)Dihydroxyacetone P (DHAP)

Glyceraldehyde-3-P (GAP)3-Phosphoglycerate (3PG)

500083

1431138

18.5118

2-Phosphoglycerate (2PG)Phosphoenolpyruvate (PEP)

Pyruvate (PYR)ATPADP

Pi

29.523

511850138

1000

Typically the specific growth rate is in the range 0.1 0.5 h-1. Theglycolytic flux for a dilution rate of 0.1 h-1 is in the range 1 mmol

g-1 h-1.

What if the system is

overdetermined? (i.e. the

dimension of rm is > F)

Xy

Compare with linear regression using least squares

TT 21

22

#

2 TTTT

0

rTv

m#

2where

yXXXb tt 1

Estimate of

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What if the system is underdetermined?

Sometimes the flux distribution can be calculated under

the assumption that metabolism is geared towards

maximizing some cellular objective function (e.g.

growth rate).

Linear programming can be used if the objective

function is a simple linear combination of the fluxes.

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