870 likes | 900 Views
Dive into the world of metabolic networks and linear equations in this comprehensive guide, covering Flux Balance Analysis, matrix operations, and system solutions. Explore the key concepts with detailed examples and explanations.
E N D
CCFFT FBA (1) Author: Tõnis Aaviksaar TALLINN 2006
Contents • Intro • Systems of linear equations • Solution by row operations • Steady state mass balance • Linear Programming
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 values
.... .... .... v1 v1 v1 A A A v2 v2 v2 B B B v3 v3 v3 v5 v5 v5 C C C v4 v4 v4 .... .... .... .... .... .... Flux Patterns v =
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 values
FBA • Steady-state mass balance equations • Weighted sums (linear combinations) of • Reaction stoichiometries • Flux patterns
Contents • Intro • Systems of linear equations • Solution by row operations • Steady state mass balance • Linear Programming
Linear equation 2x1 + 3x2 + 4x3 = 11
Linear equation Linear equation a1x1 + a2x2 + a3x3 = b in matrix form 2x1 + 3x2 + 4x3 = 11 × =
Linear equation Linear equation a1x1 + a2x2 + a3x3 = b in matrix form 2x1 + 3x2 + 4x3 = 11 × = =
System of linear equations System of linear equations a11x1 + a12x2 + a13x3 = b1 a21x1 + a22x2 + a23x3 = b2 a31x1 + a32x2 + a33x3 = b3 in matrix form × = =
System of linear equations System of linear equations a11x1 + a12x2 + a13x3 = b1 a21x1 + a22x2 + a23x3 = b2 a31x1 + a32x2 + a33x3 = b3 in matrix form × = =
System of linear equations System of linear equations a11x1 + a12x2 + a13x3 = b1 a21x1 + a22x2 + a23x3 = b2 a31x1 + a32x2 + a33x3 = b3 in matrix form × = =
System of linear equations System of linear equations a11x1 + a12x2 + a13x3 = b1 a21x1 + a22x2 + a23x3 = b2 a31x1 + a32x2 + a33x3 = b3 in matrix form × = =
System of linear equations System of linear equations a11x1 + a12x2 + a13x3 = b1 a21x1 + a22x2 + a23x3 = b2 a31x1 + a32x2 + a33x3 = b3 in matrix form × =
Linear Combination of Columns System of linear equations a11x1 + a12x2 + a13x3 = b1 a21x1 + a22x2 + a23x3 = b2 a31x1 + a32x2 + a33x3 = b3 in matrix form × =
Linear Combination of Columns System of linear equations a11x1 + a12x2 + a13x3 = b1 a21x1 + a22x2 + a23x3 = b2 a31x1 + a32x2 + a33x3 = b3 in matrix form × = x1+ x2+ x3
Linear Combination of Columns System of linear equations a11x1 + a12x2 + a13x3 = b1 a21x1 + a22x2 + a23x3 = b2 a31x1 + a32x2 + a33x3 = b3 in matrix form × =
Linear Combination of Columns System of linear equations a11x1 + a12x2 + a13x3 = b1 a21x1 + a22x2 + a23x3 = b2 a31x1 + a32x2 + a33x3 = b3 in matrix form × =
Contents • Intro • Systems of linear equations • Rows correspond to equations • Linear combination of columns • Solution by row operations • Steady state mass balance • Linear Programming
Matrices • Identity matrix I • Inverse of a matrix AA-1=I ifAB=IandBA=IthenB=A-1andA=B-1 • Solution to a system of linear equations Ax=b A-1Ax=A-1b I =
Matrices • Identity matrix I • Inverse of a matrix AA-1=I ifAB=IandBA=IthenB=A-1andA=B-1 • Solution to a system of linear equations Ax=b A-1Ax=A-1b Ix=A-1b x=A-1b I = Ix = × =
Matrices • Identity matrix I • Inverse of a matrix AA-1=I ifAB=IandBA=IthenB=A-1andA=B-1 • Solution to a system of linear equations Ax=b A-1Ax=A-1b Ix=A-1b x=A-1b I = Ix = × =
Matrix Row Operations · (–1 / 7) – 5 · R1 – 1 · R1
Matrix Row Operations · (–1 / 7) – 5 · R1 – 1 · R1 · 7 / 5 .….
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 to4x = 8 • Addition of a row to another row 2x1 + 3x2 = 5 -x1 + 2x2 = 1 2x1 + 3x2 = 5 x1 + 5x2 = 6 is equivalent to
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 to4x = 8 • Addition of a row to another row 2x1 + 3x2 = 5 -x1 + 2x2 = 1 2x1 + 3x2 = 5 x1 + 5x2 = 6 is equivalent to
Gaussian Elimination • Given a system of linear equations Ax=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 equations Ix=c • Which is equivalent to the original system Ax=b
Row-Reduced [A | b] Examples • 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 solution x1 = −3 x2 = 5 x3 = 2 [A | b] =
Row-Reduced [A | b] Examples • System of equations x1 − 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
Row-Reduced [A | b] Examples • System of equations 2x1 + x2 + 7x3 − 7x4 = 2 −3x1 + 4x2 − 5x3 − 6x4 = 3 x1 + x2 + 4x3 − 5x4 = 2 • Row-Reduced [A | b] • Inconsistent system, no solutions 0x1 + 0x2 + 0x3 − 0x4≠1
Gaussian Elimination • Carl Friedrich Gauss (1777–1855) • Chiu-chang suan-shu or The Nine Chapters on the Mathematical Art, written around 250 BCE
Contents • Intro • Systems of linear equations • Solution by row operations • Equivalent linear systems • Reduced row echelon form • Steady state mass balance • Linear programming
.... v1 A v2 .... Steady State Approximation • Steady state condition Fluxes ≠ 0 Concentrations = const • Steady state mass balance Compound production = consumption Production – consumption = 0
.... v1 A v2 B v3 v5 C v4 .... .... Mass Balance Equations v1–v2 = 0 v2– v3 –v5 = 0 v3 –v4 = 0 × = Nv = 0
Stoichiometry Matrix H2O v1 2H2O = 2H2 + 1O2 O2 H2 × N2 v2 1N2 + 3H2 = 2NH3 NH3
Stoichiometry Matrix H2O v1 2H2O = 2H2 + 1O2 O2 H2 × N2 v2 1N2 + 3H2 = 2NH3 NH3
Stoichiometry Matrix H2O v1 2H2O = 2H2 + 1O2 O2 H2 × N2 v2 1N2 + 3H2 = 2NH3 NH3
Stoichiometry Matrix H2O v1 2H2O = 2H2 + 1O2 O2 H2 × N2 v2 1N2 + 3H2 = 2NH3 NH3
Stoichiometry Matrix H2O v1 2H2O = 2H2 + 1O2 O2 H2 × N2 v2 1N2 + 3H2 = 2NH3 NH3
Stoichiometry Matrix H2O v1 2H2O = 2H2 + 1O2 O2 H2 × N2 v2 1N2 + 3H2 = 2NH3 NH3
Stoichiometry Matrix H2O v1 2H2O = 2H2 + 1O2 O2 H2 × N2 v2 1N2 + 3H2 = 2NH3 NH3
Stoichiometry Matrix H2O v1 2H2O = 2H2 + 1O2 O2 H2 × N2 v2 1N2 + 3H2 = 2NH3 NH3
Stoichiometry Matrix H2O v1 2H2O = 2H2 + 1O2 O2 H2 × N2 v2 1N2 + 3H2 = 2NH3 NH3
Stoichiometry Matrix H2O v1 2H2O = 2H2 + 1O2 O2 H2 v1+ v2+ …. N2 v2 1N2 + 3H2 = 2NH3 NH3
Stoichiometry Matrix H2O v1= 3 v1 v2= 2 2H2O = 2H2 + 1O2 O2 H2 v1+ v2+ …. = N2 v2 1N2 + 3H2 = 2NH3 NH3
Stoichiometry Matrix H2O v1= 3 v1 v2= 2 2H2O = 2H2 + 1O2 O2 H2 v1+ v2+ …. = N2 v2 1N2 + 3H2 = 2NH3 NH3
Stoichiometry Matrix H2O v1= 3 v1 v2= 2 2H2O = 2H2 + 1O2 O2 H2 v1+ v2+ …. = N2 v2 1N2 + 3H2 = 2NH3 NH3
.... v1 A v2 B v3 v5 C v4 .... .... Calculable Fluxes v1–v2 = 0 v2– v3 –v5 = 0 v3 –v4 = 0 × = Nv = 0
.... v1 A v2 B v3 v5 C v4 .... .... Calculable Fluxes v1–v2 = 0 v2– v3 –v5 = 0 v3 –v4 = 0 × = Nv = 0