Metabolic Network Analysis

Transcription

1 Metabolic Network Analysis Christoph Flamm Institute for Theoretical Chemistry University Vienna Harvest Network Course, Brno, April 1-6, 2011

2 The Systems Biology Modelling Space Network Models Level of Abstraction Process Algebras Stoichiometric Models Rule Based Models Petrinets Kinetic Models Required Knowledge 1/31

3 Genome-scale Reconstruction of Metabolic Networks Figure from Lee KY et al (2010), Microbial Cell Factories 9:94 doi: / /31

4 Phylogenetic tree of metabolic reconstractions Figure from Oberhardt MA et al (2009) Mol Syst Biol 5:320 doi: /msb /31

5 Stoichiometric Matrix S Is formed from the stoichiometric coefficients of a reaction network. Reaction Stoichiometric Coefficient Compound Connectivity Stoichiometry The rows are formed by the chemical compounds. The columns are formed by the chemical reactions. 4/31

6 Examples: Stoichiometric Matrix 2 A 3 B 1 A + 1 B 1 A + 1 C S = A + B A + C S = C A + B B 3 A 2 C B A 2 C 5/31

7 What is a Biological Pathway? S = s 11 s s 1n s 21 s s 2n s m1 s m2... s mn 1 Mathematical description of what a pathway is. 2 Formalism to describe mass flows in the network. (Hint: Mathematicians love vector spaces and operators) 6/31

8 S is a linear transformation operator 2A + B C + A C A + B ρ 2 B A 2 ρ 1 C d [X] = S dt J = d [A] [B] = 1 1 ( 1 1 kρ1 [A] 2 [B] dt k [C] 1 1 ρ2 [C] ) 7/31

9 Linear transformations between (vector) spaces frequency Time Domain time Frequency Domain Row(S) n R S mxn m R Col(S) v dyn v dx/dt v ss 0 0 Null(S) Left null(s) 8/31

10 Singular Value Decomposition (SVD) Reactions S = U Σ V T Metabolites stoichiometry connectivity m x n = Column space m x r Left null space m x m r x r m x n r x n Row space Null space S = r σ i (u i vi T ) i=1 n x n 1 Row space contains all dynamic flux distributions. 2 Null space contains all steady-state flux distributions. 3 Column space contaios all possible time derivatives of the concentration vector. 4 Left null space contains all conservation relationships. 9/31

11 The Kernel Matrix K In steady-state the differential change in species concentrations vanishes d [X] = 0 = S dt J Non-trivial solutions for the flux vector J exists only if there are linear dependencies between columns of S Rang(S) < number of reactions These vectors which span the null space of S are most conveniently organized in a kernal matrix K. Each column vector k i of K solves S k = 0 Any admissible flux in steady-state can be written as linearkombination of vectors k i J = α i k i i 10/31

12 Requirements for null space basis B B A C A C D D S = K = K = K is not interpretable in chemically meaningful terms. Flux through elementary reactions must be positive. Non-negative basis vectors are required. A convex basis has proven useful for this goal. 11/31

13 Hybrid orbitals: are a basis change! sp 3 -Hybrid Orbital tetrahedral Geometry Linear combinations (hybrid orbitals) of H 1s atomic orbitals that match nodal properties of C 2p atomic orbitals to understand the tetrahedral geometry of methane. Figure by MIT OpenCourseWare 12/31

14 Double Description Method A pair (A,R) of real matrices A and R is said to be a double description pair if the following relationship holds Ax 0 if and only if x = Rλ for some λ 0 A is the representation matrix, R is the generating matrix of a polyhedral cone P. Geometrically, the columns of a minimal generating matrix R are in a 1-to-1 correspondence with the extreme rays of P. An iterative procedure inizializing R with K is used to transform the ortho-normal to a convex basis respecting reaction directions. 13/31

15 Properties of linear and convex bases Linear Space Convex Space A x = 0. Set of linearly independent basis vectors ( e i ). v = w i e i with w i [, + ] Unique representation of every point. e i = dim(null(s)). Infinit number of spanning bases. A x = 0 with x 0. Set of conically independent generating vectors ( p i ). v = α i p i with α i [0, + inf] Nonunique representation of every point. p i dim(null(s)). Unique set of generating vectors. 14/31

16 Enzyme Mechanism: Unordered Substrate Binding R 1 : A R 2 : B R 3 : C R 4 : E + A EA R 5 : E + B EB R 6 : EA + B EAB R 7 : EB + A EAB R 8 : EAB E + C EA R4 R6 A E EAB R8 C B R5 R7 EB S = Engl HW et al (2009), Inverse Problems 25: doi: / /25/12/ /31

17 Enzyme Mechanism: Elementary Modes EA EA R4 R6 R4 R6 A A E EAB R8 C E EAB R8 C B B R5 R7 R5 R7 EB EB EA R4 R6 A E EAB R8 C B R5 R7 EB K = Engl HW et al (2009), Inverse Problems 25: doi: / /25/12/ /31

18 PG Asp PG Asp PG Asp PG Asp OAA Mal Fum OAA Mal Fum OAA Mal Fum OAA Mal Fum Succ Succ Succ Succ Gly Gly Gly Gly Cit SucCoA Cit SucCoA Cit SucCoA Cit SucCoA IsoCit IsoCit IsoCit IsoCit Ala Ala Ala Ala Pyr Pyr Pyr Pyr PG Asp PG Asp PG Asp PG Asp OAA Mal Fum OAA Mal Fum OAA Mal Fum OAA Mal Fum Succ Succ Succ Succ Gly Gly Gly Gly Cit SucCoA Cit SucCoA Cit SucCoA Cit SucCoA IsoCit IsoCit IsoCit IsoCit Ala Ala Ala Ala Pyr Pyr Pyr Pyr PG Asp PG Asp PG Asp PG Asp OAA Mal Fum OAA Mal Fum OAA Mal Fum OAA Mal Fum Succ Succ Succ Succ Gly Gly Gly Gly Cit SucCoA Cit SucCoA Cit SucCoA Cit SucCoA IsoCit IsoCit IsoCit IsoCit Ala Ala Ala Ala Pyr Pyr Pyr Pyr PG Asp PG Asp PG Asp PG Asp OAA Mal Fum OAA Mal Fum OAA Mal Fum OAA Mal Fum Succ Succ Succ Succ Gly Gly Gly Gly Cit SucCoA Cit SucCoA Cit SucCoA Cit SucCoA IsoCit IsoCit IsoCit IsoCit Ala Ala Ala Ala Pyr Pyr Pyr Pyr Elementary modes of TCA in E. coli PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA PEP Pyr AcCoA Schuster S et al (1999), Trends Biotechnol 17:53-60 DOI: /S (98) /31

19 Narrow down the Flux Cone Definition ( elementary flux mode ) is a minimal set of enzymes that can operate at steady state with all irreversible reactions used in the right direction. Figure from [Price et al. 2004] 18/31

20 Metabolic Flux Analysis (MFA) normal growth condition. Lys-production condition. The goal of MFA is the quantitative description of cellular fluxes. Figure adapted from Marx A, Bestimmung des Kohlenstoffflusses im Zentralstoffwechsel von Corynebacterium glutamicum mittels 13 C-Isotopenanalyse, PhD-thesis Uni Düsseldorf (1997). 19/31

21 Stoichiometric MFA Relies on balancing of fluxes around metabolites and fixed stoichiometry. Set of linear flux relations: u = q + v v = w w = p p + q = r + v 4 relations for 6 flux variables. Measuring the 2 extracellular fluxes u and r allows to calculate all other fluxes from the network stoichiometry: v = u r, w = u r, p = u r, q = r In many cases however the stoichiometric constraints and the external fluxes do not provide enough information to estimate all fluxes of interest. Figure adapted from [Wiechert, 2001] 20/31

22 Fail cases for stoichiometric MFA parallel pathways without any related flux measurement. certain metabolic cycles. bi-directional reaction steps. split pathways when cofactors are not balanced. Figure adapted from [Wiechert, 2001] 21/31

23 Flux Balance Analysis Formulate as an linear programing problem with additional constraints (capacity of enzymes, external fluxes,...) and an appropriate optimization function. Obj Function max v Biomass v cose max v ATP v cose min P vnadh v cose min δ i max v Biomass P v 2 i Explanation biomass yield (same as groth rate) ATP yield redox potential reaction steps biomass yield per flux unit Schuetz R et al (2007), Systematic evaluation of objective functions for predicting intracellular fluxes in E. coli, Mol Sys Biol 3:119 doi: /msb /31

24 Primer: Linear Programming (LP) Linear programming is an optimization method requiring 2 inputs: 1 A linear objective function. 2 A set of linear constraints. Example: Production planning problem Product machine 1 machine 2 machine 3 A B Total machine running time is 8 hours/day. Profit: 10e/A and 40e/B. Question: How many units of product A and B need to be manifactured in order to maximize profit? 23/31

25 1 maximize profit: Expressed as LP Problem z = F(x 1, x 2 ) = 10 x x 2 2 subject to the linear constraints: 40 x x x x x x 1, x 2 0 Admissible solutions: x 1 = 0 x 2 = 0 = z = 0 x 2 = 0 x 1 = 12 = z = 120 x 1 = 0 x 2 = 8 = z = /31

26 20 LP Problem: Graphical Solution 15 X X 1 40 x x x x x x x x x x 2 8 z = 10 x x 2 x 2 = 1 4 x z 25/31

27 LP Problem: Formal Formulation The linear objective function is generally a sum of terms that contain weighted measurable elements from a metabolic model. Maximize: Z = c 1 x 1 + c 2 x 2 + = c T x Subject to: a 11 x 1 + a 12 x 2 + a 1n x n b 1 a 12 x 1 + a 22 x 2 + a 2n x n b 2. a m1 x 1 + a m2 x 2 +. a mn x n b m A x b 26/31

28 Isotope Labeling Experiments 1 Introduce a isotope labeled substrate into the cell culture at metabolic steady state. 2 Allow the system to reach an isotopic steady state. 3 Measure (e.g. NMR, MS) relative labeling in metabolic intermediates and by products. 4 Estimate fluxes from these measurements. Isotope measurements provide many additional independent constraints for MFA. The cell s flux state and the administered isotope label fully and uniquely determine the isotopomere patterns of metabolites at steady state. Quantitative interpretation requires a mathematical model, which relates metabolic flux to isotopomere abundance. Figure adapted from [Wiechert, 2001] 27/31

29 Isotopomers Are defined as isomeres of a metabolit that differ only in the labeling state of their individual atoms (e.g. carbon [ 12 C, 13 C], hydrogen [ 1 H, 2 H] or oxygen [ 16 O, 17 O, 18 O]. 2 N isotopomeres are possible for a metabolite with N atoms that may be in one of two states (unlabeled or labeled). Example (glucose C 6 H 12 O 6 ) Atoms # of Isotopomeres C (2 6 = 64) O (3 6 = 726) H (2 12 = 4096) C, H ( = ) C, H, O ( = ) 28/31

30 Measuring Isotopomere Distribution Any experimental technique capable of detecting differences between isotopomeres can be used to measure the labeling state. The two dominating technologies are: 1 Nuclear magnetic resonance spectroskopy (NMR). 2 Mass spectrometry (MS). Figure adapted from [Wiechert, 2001] 29/31

31 Atom Transition Network Estimating fluxes from isotopomere patterns is an inverse problem. The atom-atom mapping between reaction educts and products must be known. Getting this information is an NP-hard problem. Therefore most approaches use the heuristic principle of minimal chemical distance. 6 C 9 C 8 1 C 8 C C1 C 2 C C 5 <=> C 7 C 7 C 5 C 9 C 2 C 3 C 4 C 6 3 C C 4 30/31

32 Further Reading Gagneur J, Klamt S. Computation of elementary modes: a unifying framework and the new binary approach. BMC Bioinf 5: doi: / Wagner C, Urbanczik R. The Geometry of the Flux Cone of a Metabolic Network. Biophys J 89: , 2005 doi: /biophysj Price ND, Reed JL, Palsson BO. Genome-scale models of microbial cells: Evaluating the consequences of constraints. Nature Rev Nicrobiol 2: , 2004 doi: /nrmicro1023 Orth JD, Ines Thiele I, Palsson BO What is flux balance analysis? Nature Biotech 28: , 2010 doi: /nbt.1614 Antoniewicz MR, Kelleher JK, Stephanopoulos G. Elementary metabolic units (EMU): A novel framework for modeling isotopic distributions. Metab Eng 9:68-86, 2007 doi: /j.ymben Sauer U. Metabolic networks in motion: 13 C flux analysis. Mol Sys Biol 2:62, 2006 doi: /msb Wiechert W. 13 C Metabolic Flux Analysis. Metab Eng 3(3): , 2001 doi: /mben /31