Linear Equations and Metabolic Networks: A Comprehensive Guide

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 × =

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 × =

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 = × =

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

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 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

Linear Equations and Metabolic Networks: A Comprehensive Guide