fba (1) author: tõnis aaviksaar tallinn 2006 ccfft
TRANSCRIPT
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FBA (1)
Author: Tõnis Aaviksaar
TALLINN 2006
CCFFT
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2
Contents
• Intro• Systems of linear equations• Solution by row operations• Steady state mass balance• Linear Programming
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Metabolic Networks• Metabolic networks consist of
reactions between metabolites
• Flux Balance Analysis (FBA) calculates flux patterns from a system of linear equations– Flux value = rate of reaction– Flux pattern is a collection of
flux valuesO2KG
OOXA
OAce-CoAOPYR
OPEP
CO2
CO2
PEPAce-CoA
Mal
Fum
Suc-CoAIcit
CO2
3PG O3PG
GAP DHAP
FBP
E4P
S7P
X5P
R5P
Ru5P
HOO4C 3CH22CH2
1COOH
1C
2CH3
O OH
1C
2CH3
O OH
2C
3CH3
O
1CO OH
HOO4C 3CH22C
O
1COOH
2CH23CH
HO 6COOH
HOO1C 4C
H H
5COOH
HOO5C 4CH23CH2
2C
O
1COOH
mdh
fumA
D
sucD
Eicd
acn
gltA
pckAacs
QACE I ppd
gpmeno
BPG
B
pgk
gapA
tpiA
fbaA1
F6P
fbp1pfk21
G6P
pgi
ppcoadA2
oadA1
GF
zwf
H1
rpe
rpiB
H2
SBP
fbp2
fbaA2
tal
rbcL
CO2
CO2
CO2
pfk22
CO2
OR5P
OE4P
ODHAP
OG6P
OF6P
ORu5P
OS7P
OGAP
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4
v3
v4
v2
v1
A
B
C
....
........
v5
Flux Patterns
v3
v4
v2
v1
A
B
C
....
........
v5
v1
v2
v3
v4
v5
v =v3
v4
v2
v1
A
B
C
....
........
v5
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5
Metabolic Networks• Metabolic networks consist of
reactions between metabolites– Thousands of metabolites– More reactions than metabolites
• Flux Balance Analysis (FBA) calculates flux patterns from a system of linear equations– Flux value = rate of reaction– Flux pattern is a collection of
flux valuesO2KG
OOXA
OAce-CoAOPYR
OPEP
CO2
CO2
PEPAce-CoA
Mal
Fum
Suc-CoAIcit
CO2
3PG O3PG
GAP DHAP
FBP
E4P
S7P
X5P
R5P
Ru5P
HOO4C 3CH22CH2
1COOH
1C
2CH3
O OH
1C
2CH3
O OH
2C
3CH3
O
1CO OH
HOO4C 3CH22C
O
1COOH
2CH23CH
HO 6COOH
HOO1C 4C
H H
5COOH
HOO5C 4CH23CH2
2C
O
1COOH
mdh
fumA
D
sucD
Eicd
acn
gltA
pckAacs
QACE I ppd
gpmeno
BPG
B
pgk
gapA
tpiA
fbaA1
F6P
fbp1pfk21
G6P
pgi
ppcoadA2
oadA1
GF
zwf
H1
rpe
rpiB
H2
SBP
fbp2
fbaA2
tal
rbcL
CO2
CO2
CO2
pfk22
CO2
OR5P
OE4P
ODHAP
OG6P
OF6P
ORu5P
OS7P
OGAP
![Page 6: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/6.jpg)
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FBA
• Steady-state mass balance equations• Weighted sums (linear combinations) of
– Reaction stoichiometries– Flux patterns
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Contents
• Intro• Systems of linear equations• Solution by row operations• Steady state mass balance• Linear Programming
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Linear equation
2x1 + 3x2 + 4x3 = 11
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Linear equation
Linear equationa1x1 + a2x2 + a3x3 = b
in matrix form
x1
x2
x3
a1 a2 a3 b× =
2x1 + 3x2 + 4x3 = 11
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Linear equation
Linear equationa1x1 + a2x2 + a3x3 = b
in matrix form
x1
x2
x3
a1 a2 a3 b× = a1x1 + a2x2 + a3x3 =
2x1 + 3x2 + 4x3 = 11
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System of linear equations
System of linear equationsa11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3
in matrix form
x1
x2
x3
× =
a11 a12 a13
a21 a22 a23
a31 a32 a33
a11x1 + a12x2 + a13x3
a21x1 + a22x2 + a23x3
a31x1 + a32x2 + a33x3
=
b1
b2
b3
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System of linear equations
System of linear equationsa11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3
in matrix form
x1
x2
x3
× =
a11 a12 a13
a21 a22 a23
a31 a32 a33
a11x1 + a12x2 + a13x3
a21x1 + a22x2 + a23x3
a31x1 + a32x2 + a33x3
=
b1
b2
b3
![Page 13: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/13.jpg)
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System of linear equations
System of linear equationsa11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3
in matrix form
x1
x2
x3
× =
a11 a12 a13
a21 a22 a23
a31 a32 a33
a11x1 + a12x2 + a13x3
a21x1 + a22x2 + a23x3
a31x1 + a32x2 + a33x3
=
b1
b2
b3
![Page 14: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/14.jpg)
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System of linear equations
System of linear equationsa11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3
in matrix form
x1
x2
x3
× =
a11 a12 a13
a21 a22 a23
a31 a32 a33
a11x1 + a12x2 + a13x3
a21x1 + a22x2 + a23x3
a31x1 + a32x2 + a33x3
=
b1
b2
b3
![Page 15: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/15.jpg)
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System of linear equations
System of linear equationsa11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3
in matrix form
x1
x2
x3
× =
b1
b2
b3
a11 a12 a13
a21 a22 a23
a31 a32 a33
![Page 16: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/16.jpg)
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Linear Combination of Columns
System of linear equationsa11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3
in matrix form
x1
x2
x3
× =
b1
b2
b3
a11 a12 a13
a21 a22 a23
a31 a32 a33
![Page 17: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/17.jpg)
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Linear Combination of Columns
System of linear equationsa11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3
in matrix form
=
a11
a21
a31
a12
a22
a32
a13
a23
a33
x1 + x2 + x3
x1
x2
x3
×
a11 a12 a13
a21 a22 a23
a31 a32 a33
![Page 18: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/18.jpg)
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Linear Combination of Columns
System of linear equationsa11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3
in matrix form
=
x1
x2
x3
×
a11 a12 a13
a21 a22 a23
a31 a32 a33
a11x1 + a12x2 + a13x3
a21x1 + a22x2 + a23x3
a31x1 + a32x2 + a33x3
![Page 19: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/19.jpg)
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Linear Combination of Columns
System of linear equationsa11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3
in matrix form
x1
x2
x3
× =
b1
b2
b3
a11 a12 a13
a21 a22 a23
a31 a32 a33
![Page 20: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/20.jpg)
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Contents
• Intro• Systems of linear equations
– Rows correspond to equations– Linear combination of columns
• Solution by row operations• Steady state mass balance• Linear Programming
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Matrices
• Identity matrixI
• Inverse of a matrixAA-1 = Iif AB = I and BA = I then B = A-1 and A = B-1
• Solution to a system of linear equationsAx = bA-1Ax = A-1b
I =
1
1
1
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Matrices
• Identity matrixI
• Inverse of a matrixAA-1 = Iif AB = I and BA = I then B = A-1 and A = B-1
• Solution to a system of linear equationsAx = bA-1Ax = A-1bIx = A-1bx = A-1b
I =
1
1
1
x1
x2
x3
×Ix =
1
1
1
=
x1
x2
x3
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Matrices
• Identity matrixI
• Inverse of a matrixAA-1 = Iif AB = I and BA = I then B = A-1 and A = B-1
• Solution to a system of linear equationsAx = bA-1Ax = A-1bIx = A-1bx = A-1b
I =
1
1
1
x1
x2
x3
×Ix =
1
1
1
=
x1
x2
x3
![Page 24: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/24.jpg)
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Matrix Row Operations
–7 –6 –12 –33
5 5 7 24
1 4 5
1 6/7 12/7 33/7
5 5 7 24
1 4 5
1 6/7 12/7 33/7
5/7 –11/7 3/7
–6/7 16/7 2/7
· (–1 / 7)
– 5 · R1
– 1 · R1
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Matrix Row Operations
–7 –6 –12 –33
5 5 7 24
1 4 5
1 6/7 12/7 33/7
5 5 7 24
1 4 5
· (–1 / 7)
– 5 · R1
– 1 · R1
1 −3
1 5
1 2
.….
1 6/7 12/7 33/7
5/7 –11/7 3/7
–6/7 16/7 2/7
· 7 / 5
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Matrix Row Operations
• Equivalent systems of equations– Two systems of equations are equivalent if they have
same solution sets
• Row operations produce equivalent systems of equations– Changing the order of rows– Multiplication of a row by a constant
2x = 4 is equivalent to 4x = 8
– Addition of a row to another row2x1 + 3x2 = 5-x1 + 2x2 = 1
2x1 + 3x2 = 5x1 + 5x2 = 6
is equivalent to
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Matrix Row Operations
• Equivalent systems of equations– Two systems of equations are equivalent if they have
same solution sets
• Row operations produce equivalent systems of equations– Changing the order of rows– Multiplication of a row by a constant
2x = 4 is equivalent to 4x = 8
– Addition of a row to another row2x1 + 3x2 = 5-x1 + 2x2 = 1
2x1 + 3x2 = 5x1 + 5x2 = 6
is equivalent to
![Page 28: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/28.jpg)
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Gaussian Elimination
• Given a system of linear equationsAx = b
• Matrix A is augmented by b[A | b]
• Which is then simplified by row operations to produce[I | c]
• Which corresponds to system of equationsIx = c
• Which is equivalent to the original systemAx = b
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• System of equations−7x1 − 6x2 − 12x3 = −33
5x1 + 5x2 + 7x3 = 24
x1 + 4x3 = 5
• Row-Reduced [A | b] ~ [I | c] =
• Simplified equivalent system of equations, only one solutionx1 = −3
x2 = 5
x3 = 2
Row-Reduced [A | b] Examples
1 −3
1 5
1 2
−7 −6 −12 −33
5 5 7 24
1 4 5
[A | b] =
![Page 30: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/30.jpg)
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Row-Reduced [A | b] Examples
• System of equationsx1 − x2 + 2x3 = 1
2x1 + x2 + x3 = 8
x1 + x2 = 5
• Row-Reduced [A | b]
• Simplified equivalent system of equations, infinite number of solutions (solution space)x1 + x3 = 3
x2 − x3 = 2
1 1 3
1 −1 2
![Page 31: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/31.jpg)
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Row-Reduced [A | b] Examples
• System of equations2x1 + x2 + 7x3 − 7x4 = 2
−3x1 + 4x2 − 5x3 − 6x4 = 3
x1 + x2 + 4x3 − 5x4 = 2
• Row-Reduced [A | b]
• Inconsistent system, no solutions0x1 + 0x2 + 0x3 − 0x4 ≠ 1
1 3 −2
1 1 −3
1
![Page 32: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/32.jpg)
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Gaussian Elimination
• Carl Friedrich Gauss (1777–1855)
• Chiu-chang suan-shu or The Nine Chapters on the Mathematical Art, written around 250 BCE
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Contents
• Intro• Systems of linear equations• Solution by row operations
– Equivalent linear systems– Reduced row echelon form
• Steady state mass balance• Linear programming
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Steady State Approximation
• Steady state conditionFluxes ≠ 0
Concentrations = const
• Steady state mass balanceCompound production = consumption
Production – consumption = 0
v2
v1
A
....
....
![Page 35: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/35.jpg)
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Mass Balance Equations
v1 – v2 = 0
v2 – v3 – v5 = 0
v3 – v4 = 0
1 -1
1 -1 -1
1 -1
A
B
C
v1 v2 v3 v4 v5 v1
v2
v3
v4
v5
× =
Nv = 0
v3
v4
v2
v1
A
B
C
....
........
v5
![Page 36: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/36.jpg)
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Stoichiometry Matrix
…
…
…
…
…
H2O
O2
H2
N2
NH3
v1 v2 …
v1
v2
…
×v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
![Page 37: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/37.jpg)
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Stoichiometry Matrix
…
…
…
…
…
H2O
O2
H2
N2
NH3
v1 v2 …
v1
v2
…
×v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
![Page 38: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/38.jpg)
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Stoichiometry Matrix
-2 …
…
…
…
…
H2O
O2
H2
N2
NH3
v1 v2 …
v1
v2
…
×v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
![Page 39: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/39.jpg)
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Stoichiometry Matrix
-2 …
…
2 …
…
…
H2O
O2
H2
N2
NH3
v1 v2 …
v1
v2
…
×v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
![Page 40: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/40.jpg)
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Stoichiometry Matrix
-2 …
1 …
2 …
…
…
H2O
O2
H2
N2
NH3
v1 v2 …
v1
v2
…
×v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
![Page 41: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/41.jpg)
41
Stoichiometry Matrix
-2 …
1 …
2 …
…
…
H2O
O2
H2
N2
NH3
v1 v2 …
v1
v2
…
×v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
![Page 42: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/42.jpg)
42
Stoichiometry Matrix
-2 …
1 …
2 …
…
…
H2O
O2
H2
N2
NH3
v1 v2 …
v1
v2
…
×v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
![Page 43: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/43.jpg)
43
Stoichiometry Matrix
-2 …
1 …
2 -3 …
-1 …
2 …
H2O
O2
H2
N2
NH3
v1 v2 …
v1
v2
…
×v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
![Page 44: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/44.jpg)
44
Stoichiometry Matrix
-2 …
1 …
2 -3 …
-1 …
2 …
H2O
O2
H2
N2
NH3
v1 v2 …
v1
v2
…
×v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
![Page 45: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/45.jpg)
45
Stoichiometry Matrix
-2
1
2
H2O
O2
H2
N2
NH3
v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
-3
-1
2
v1 + v2 + ….
v1 v2
![Page 46: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/46.jpg)
46
Stoichiometry Matrix
-2
1
2
H2O
O2
H2
N2
NH3
v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
-3
-1
2
v1 + v2 + …. =
v1 = 3
v2 = 2
v1 v2
![Page 47: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/47.jpg)
47
Stoichiometry Matrix
-6
3
6
H2O
O2
H2
N2
NH3
v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
-6
-3
6
v1 + v2 + …. =
v1 = 3
v2 = 2
v1 v2
![Page 48: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/48.jpg)
48
Stoichiometry Matrix
-6
3
6
H2O
O2
H2
N2
NH3
v2
H2N2
1N2 + 3H2 = 2NH3
2H2O = 2H2 + 1O2
O2
NH3
H2Ov1
-6
-3
6
v1 + v2 + …. =
v1 = 3
v2 = 2
v1 v2
![Page 49: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/49.jpg)
49
Calculable Fluxes
v1 – v2 = 0
v2 – v3 – v5 = 0
v3 – v4 = 0
v3
v4
v2
v1
A
B
C
....
........
v5
1 -1
1 -1 -1
1 -1
A
B
C
v1 v2 v3 v4 v5 v1
v2
v3
v4
v5
× =A
B
C
Nv = 0
![Page 50: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/50.jpg)
50
Calculable Fluxes
v1 – v2 = 0
v2 – v3 – v5 = 0
v3 – v4 = 0
v3
v4
v2
v1
A
B
C
....
........
v5
1 -1
1 -1 -1
1 -1
A
B
C
v1 v2 v3 v4 v5 v1
v2
v3
v4
v5
× =A
B
C
Nv = 0
![Page 51: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/51.jpg)
51
Calculable Fluxes
v3
v4
v2
v1
A
B
C
....
........
v5
1 -1
1 -1 -1
1 -1
A
B
C
v1 v2 v3 v4 v5 v1
v2
v3
v4
v5
× =
Nv = 0
A
B
C
-1
1 -1 -1
1
A
B
C
v2 v3 v5
v1
v4
+ =1
-1
v1 v4
v2
v3
v5
× ×
Nclcvclc + Nexpvexp = 0 v1 = 1.0
v4 = .4
A
B
C
![Page 52: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/52.jpg)
52
Calculable Fluxes
v3
v4
v2
v1
A
B
C
....
........
v5
1 -1
1 -1 -1
1 -1
A
B
C
v1 v2 v3 v4 v5 v1
v2
v3
v4
v5
× =
Nv = 0
A
B
C
-1
1 -1 -1
1
A
B
C
v2 v3 v5
1.0
.4+ =
1
-1
v1 v4
v2
v3
v5
× ×
Nclcvclc + Nexpvexp = 0 v1 = 1.0
v4 = .4
A
B
C
![Page 53: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/53.jpg)
53
Calculable Fluxes
v3
v4
v2
v1
A
B
C
....
........
v5
1 -1
1 -1 -1
1 -1
A
B
C
v1 v2 v3 v4 v5 v1
v2
v3
v4
v5
× =
Nv = 0
A
B
C
-1
1 -1 -1
1
A
B
C
v2 v3 v5
+ =
v2
v3
v5
×
Nclcvclc + Nexpvexp = 0
1.0
-.4
bexp
A
B
C
v1 = 1.0
v4 = .4
![Page 54: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/54.jpg)
54
Calculable Fluxes
v3
v4
v2
v1
A
B
C
....
........
v5
1 -1
1 -1 -1
1 -1
A
B
C
v1 v2 v3 v4 v5 v1
v2
v3
v4
v5
× =
Nv = 0
A
B
C
Nclcvclc + Nexpvexp = 0
-1
1 -1 -1
1
A
B
C
v2 v3 v5
=-1.0
.4
A
B
C
bclc
v2
v3
v5
×
![Page 55: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/55.jpg)
55
Row-Reduced [A | b] Examples
• System of equationsx1 − x2 + 2x3 = 1
2x1 + x2 + x3 = 8
x1 + x2 = 5
• Row-Reduced [A | b]
• Simplified equivalent system of equations, infinite number of solutions (solution space)x1 + x3 = 3
x2 − x3 = 2
1 1 3
1 −1 2
![Page 56: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/56.jpg)
56
Dependent and Free Fluxes
-1 1
1 -1 -1
1 -1
A
B
C
v2 v3 v5 v1 v4 v2
v3
v5
v1
v4
× =A
B
C
1 -1
1 -1
1 -1 1
v2 v3 v5 v1 v4
v3
v4
v2
v1
A
B
C
....
........
v5
Nv = 0Nclcvclc + Nexpvexp = 0
![Page 57: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/57.jpg)
57
Dependent and Free Fluxes
v1 v2 v5 v3 v4
1 -1 -1
1 -1 -1
1 -1
1 -1
1 -1 -1
1 -1
+ R3
+ R2
1 -1 -1
1 -1 -1
1 -1
+ R3v3
v4
v2
v1
A
B
C
....
........
v5
![Page 58: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/58.jpg)
58
Dependent and Free Fluxes
v1 v2 v3 v4 v5
-1 1 1
1 -1 -1
1 -1
1 -1 -1
1 -1 -1
1 -1
+ R1
· (-1)
-1 1 1
-1 1
1 -1
v3
v4
v2
v1
A
B
C
....
........
v5
![Page 59: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/59.jpg)
59
Dependent and Free Fluxes
v1 v2 v3 v4 v5
1 -1 -1
-1 1 1
1 -1
1 -1 -1
1 -1 -1
1 -1
+ R2
· (-1)
1 -1
-1 1 1
1 -1
v3
v4
v2
v1
A
B
C
....
........
v5
![Page 60: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/60.jpg)
60
Contents
• Intro• Systems of linear equations• Solution by row operations• Steady state mass balance
– Steady state mass balance– Stoichiometry matrix– Dependent and free fluxes
• Linear Programming
![Page 61: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/61.jpg)
61
Underdetermined Systems
• Be content with infinite solution space• Make more measurements• Assign “experimental” values• Assume that the microorganism “tries” to optimize
an objective– Maximize biomass production– Maximize ATP production– ….
![Page 62: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/62.jpg)
62
The Simplex Method
• Objective function– A linear function
• Constraints– Linear inequalities
• Assumption that all variables are nonnegative– xi ≥ 0
• Solution space is a convex polytope– An optimal solution is a vertex– Move to neighboring vertex with highest objective
function value
![Page 63: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/63.jpg)
63
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z −v3 =0C → max
![Page 64: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/64.jpg)
64
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z −v3 =0
+v1 −v2 =0
+v2 −v3 −v4 =0
+v1 ≤5
C
A
B
→ max
![Page 65: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/65.jpg)
65
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z −v3 =0
+v1 −v2 =0
+v2 −v3 −v4 =0
+v1 +x5 =5
C
A
B
→ max
![Page 66: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/66.jpg)
66
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z −v3 =0
+v1 −v2 ≤0
+v1 −v2 ≥0
+v2 −v3 −v4 ≤0
+v2 −v3 −v4 ≥0
+v1 +x5 =5
C
A
A
B
B
→ max
![Page 67: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/67.jpg)
67
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z −v3 =0
+v1 −v2 ≤0
−v1 +v2 ≤0
+v2 −v3 −v4 ≤0
−v2 +v3 +v4 ≤0
+v1 +x5 =5
C
A
A
B
B
→ max
![Page 68: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/68.jpg)
68
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z −v3 =0
+v1 −v2 +x1 =0
−v1 +v2 +x2 =0
+v2 −v3 −v4 +x3 =0
−v2 +v3 +v4 +x4 =0
+v1 +x5 =5
C
A
A
B
B
→ max
![Page 69: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/69.jpg)
69
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z −v3 =0
+v1 −v2 +x1 =0
−v1 +v2 +x2 =0
+v2 −v3 −v4 +x3 =0
−v2 +v3 +v4 +x4 =0
+v1 +x5 =5
C
A
A
B
B
→ max
![Page 70: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/70.jpg)
70
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z −v2 +v4 +x4 =0
+v1 −v2 +x1 =0
−v1 +v2 +x2 =0
+x3 +x4 =0
−v2 +v3 +v4 +x4 =0
+v1 +x5 =5
C
A
A
B
B
→ max
![Page 71: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/71.jpg)
71
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z −v2 +v4 +x4 =0
+v1 −v2 +x1 =0
−v1 +v2 +x2 =0
+x3 +x4 =0
−v2 +v3 +v4 +x4 =0
+v1 +x5 =5
C
A
A
B
B
→ max
![Page 72: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/72.jpg)
72
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z −v1 +v4 +x2 +x4 =0
+x1 +x2 =0
−v1 +v2 +x2 =0
+x3 +x4 =0
−v1 +v3 +v4 +x2 +x4 =0
+v1 +x5 =5
C
A
A
B
B
→ max
![Page 73: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/73.jpg)
73
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z −v1 +v4 +x2 +x4 =0
+x1 +x2 =0
−v1 +v2 +x2 =0
+x3 +x4 =0
−v1 +v3 +v4 +x2 +x4 =0
+v1 +x5 =5
C
A
A
B
B
→ max
![Page 74: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/74.jpg)
74
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z +v4 +x2 +x4 +x5 =5
+x1 +x2 =0
+v2 +x2 +x5 =5
+x3 +x4 =0
+v3 +v4 +x2 +x4 +x5 =5
+v1 +x5 =5
C
A
A
B
B
→ max
![Page 75: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/75.jpg)
75
Simplex Example (1)
v3
z
v2
v1
A
B
C
....
........
v4
+z +v4 +x2 +x4 +x5 =5
+x1 +x2 =0
+v2 +x2 +x5 =5
+x3 +x4 =0
+v3 +v4 +x2 +x4 +x5 =5
+v1 +x5 =5
C
A
A
B
B
→ max
![Page 76: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/76.jpg)
76
Simplex Example (2)
z
v3
v1
A B
C
....
........
v4
v2
....
2A + 3B = 1C
+z −v3 =0
+v1 −2v3 −v4 +x1 =0
−v1 +2v3 +v4 +x2 =0
+v2 −3v3 +x3 =0
−v2 +3v3 +x4 =0
+v1 +x5 =1
+v2 +x6 =1
![Page 77: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/77.jpg)
77
Simplex Example (2)
z
v3
v1
A B
C
....
........
v4
v2
....
2A + 3B = 1C
+z −v3 =0
+v1 −2v3 −v4 +x1 =0
−v1 +2v3 +v4 +x2 =0
+v2 −3v3 +x3 =0
−v2 +3v3 +x4 =0
+v1 +x5 =1
+v2 +x6 =1
![Page 78: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/78.jpg)
78
Simplex Example (2)
+z −1/3v2 +1/3x4 =0
+v1 −2/3v2 −v4 +x1 +2/3x4 =0
−v1 +2/3v2 +v4 +x2 +2/3x4 =0
+x3 +x4 =0
−1/3v2 +v3 +1/3x4 =0
+v1 +x5 =1
+v2 +x6 =1
![Page 79: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/79.jpg)
79
Simplex Example (2)
+z −1/3v2 +1/3x4 =0
+v1 −2/3v2 −v4 +x1 +2/3x4 =0
−v1 +2/3v2 +v4 +x2 +2/3x4 =0
+x3 +x4 =0
−1/3v2 +v3 +1/3x4 =0
+v1 +x5 =1
+v2 +x6 =1
![Page 80: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/80.jpg)
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Simplex Example (2)
+z −1/2v1 +1/2v4 +1/2x2 =0
+x1 +x2 =0
−3/2v1 +2/3v2 +3/2v4 +3/2x2 −x4 =0
+x3 +x4 =0
−1/2v1 +v3 +1/2v4 +1/2x2 =0
+v1 +x5 =1
+3/2v1 −3/2v4 −3/2x2 +x4 +x6 =1
![Page 81: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/81.jpg)
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Simplex Example (2)
+z −1/2v1 +1/2v4 +1/2x2 =0
+x1 +x2 =0
−3/2v1 +2/3v2 +3/2v4 +3/2x2 −x4 =0
+x3 +x4 =0
−1/2v1 +v3 +1/2v4 +1/2x2 =0
+v1 +x5 =1
+3/2v1 −3/2v4 −3/2x2 +x4 +x6 =1
![Page 82: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/82.jpg)
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Simplex Example (2)
+z +1/3x4 +1/3x6 =1/3
+x1 +x2 =0
+v2 +x6 =1
+x3 +x4 =0
+v3 +1/3x4 +1/3x6 =1/3
+v4 +x2 −2/3x4 +x5 −2/3x6 =1/3
+v1 −v4 −x2 +2/3x4 +2/3x6 =2/3
![Page 83: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/83.jpg)
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Simplex Example (2)
+z +1/3x4 +1/3x6 =1/3
+x1 +x2 =0
+v2+x6 =1
+x3 +x4 =0
+v3+1/3x4 +1/3x6 =1/3
+v4+x2 −2/3x4 +x5 −2/3x6 =1/3
+v1 −v4−x2 +2/3x4 +2/3x6 =2/3
z
v3
v1
A B
C
....
........
v4
v2
....
2A + 3B = 1C
![Page 84: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/84.jpg)
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Simplex Example (2)
+z +1/3x4 +1/3x6 =1/3
+x1 +x2 =0
+v2+x6 =1
+x3 +x4 =0
+v3+1/3x4 +1/3x6 =1/3
+v4+x2 −2/3x4 +x5 −2/3x6 =1/3
+v1 −v4−x2 +2/3x4 +2/3x6 =2/3
z
v3
v1
A B
C
....
........
v4
v2
....
2A + 3B = 1C
![Page 85: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/85.jpg)
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Simplex Example (2)
+z +1/3x4 +1/3x6 =1/3
+x1 +x2 =0
+v2+x6 =1
+x3 +x4 =0
+v3+1/3x4 +1/3x6 =1/3
+v4+x2 −2/3x4 +x5 −2/3x6 =1/3
+v1+x5 =1
z
v3
v1
A B
C
....
........
v4
v2
....
2A + 3B = 1C
![Page 86: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/86.jpg)
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Simplex Example (2)
+z +1/3x4 +1/3x6 =1/3
+x1 +x2 =0
+v2+x6 =1
+x3 +x4 =0
+v3+1/3x4 +1/3x6 =1/3
+v4+x2 −2/3x4 +x5 −2/3x6 =1/3
+v1+x5 =1
z
v3
v1
A B
C
....
........
v4
v2
....
2A + 3B = 1C
![Page 87: FBA (1) Author: Tõnis Aaviksaar TALLINN 2006 CCFFT](https://reader033.vdocument.in/reader033/viewer/2022051216/5697bfa21a28abf838c961ab/html5/thumbnails/87.jpg)
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Contents
• Intro• Systems of linear equations• Solution by row operations• Steady state mass balance• Linear Programming
– Objective Function– Convex Solution Space– The Simplex Method– Multiple Optimal Solutions