Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Edda Klipp Systembiologie 4 – Flux Balance Analysis Sommersemester 2010 Humboldt-Universität zu Berlin Institut für Biologie Theoretische Biophysik

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Reminder: Stoichiometry Stoichiometric matrix Vector of metabolite concentrations Vector of reaction rates Parameter vector Systems equations in matrix form In steady state: K represents the basis vector for all possible steady state fluxes.

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Elementary Modes/Extreme Pathways

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Flux Balance Analysis in Metabolic Networks

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin InSilico Network Reconstruction

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Stoichiometric Matrix for the Hypothetical Network

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Defining Mass Balance Constraints The mass balance constraints are defined by summing the rates of production and degradation for each metabolite in the network

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Re-Formulation Additional constraints: Capacity:  j ≤ v j ≤  j

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Additional constraints Capacity and Thermodynamic Constraints:  j ≤ v j ≤  j Constraints on the metabolic network consist of mass balance constraints and flux constraints (reversibility constraints). Linear programming can be used to determine the optimal use of the metabolic network subject to the imposed constraints.

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Phenotypic Phase Plane Each phase is denoted by Pn x,y, where “n” denotes the number of the demarcated phase, and “x,y” denotes the two uptake rates on the axis of the PhPP. The PhPP can also be generated for a mutant genotype; represented as P gene n x,y. One demarcation line in the PhPP is defined as the line of optimality (LO). This line represents the optimal relation between respective metabolic fluxes. The LO is identified by varying the x-axis flux and calculating the optimal y-axis flux with the objective function defined as the growth flux

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Linear Programming  Optimization of linear objective functions subject to linear equations and linear unequalities.  Special case of convex optimization. Linear programs are problems that can be expressed in canonical form: Maximize: c T x Subject to: Ax ≤ b. where x represents the vector of variables (to be determined), c and b are vectors of (known) coefficients and A is a (known) matrix of coefficients. There are different solution algorithms. Most well-known one: SIMPLEX algorithm

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Additional Information An E. coli (ECK12) system definition is available on the website of Palsson’s group. website of Palsson’s group Free LP package (GLPK, simplex) is freely available on the Internet.GLPK LP solvers are available in Matlab ® optimization toolbox.Matlab ® optimization toolbox FBA course at Palsson’s group website

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Example for Objective Function Optimization of the system with different objective functions (Z). Case I gives a single optimal point, whereas case II gives multiple optimal points lying along an edge.

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Predicting the E.coli Optimal Growth Ibarra et al. Escherichia coli k-12 undergoes adaptive evolution to achieve in silico predicted optimal growth. Nature Daniel Segre, Dennis Vitkup, and George M. Church. Analysis of optimality in natural and perturbed metabolic networks. PNAS, vol. 99, Edwards et al. Characterizing the metabolic phenotype. A phenotype phase plan. Biotechnology and bioengineering Kenneth et al. Advances in flux balance analysis. Current Opinion in Biotechnology Schillling et. Al Combining pathway analysis with flux balance analysis for the comprehensive study of metabolic systems. Biotechnology and bioengineering, 2001.

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin What is the biological interpretation of any point in the flux cone ?

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin (I) Narrowing the Steady State Flux Cone The steady state flux cone contains infinite flux distributions! Only a small portion of them is physiologically feasible. More constraints on the external fluxes. These depend on factors as: –Organism –Environment and accessibility substrates –maximum rates of diffusion mediated transport –Etc…

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin (II) Calculating Optimal Flux Distribution The constrained flux cone in E.coli contains ~10^6 (Schilling 2001) How can we identify a “biologically meaningful” flux? Assumption … the metabolic network is optimized with respect to a certain objective function Z. Z will be a linear function. Later, we will deal with how exactly to choose Z

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Minimize/Maximize S.T + inequality constraints What we want to do is find the vector v in the flux cone which maximizes Z. This optimization problem is a classical linear programming (LP) problem that can be solved using the simplex algorithm. W. Wiechert. Journal of Biotechnology(2002) (II) Calculating Optimal Flux Distribution …this can be can formulated as an optimization problem:

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin (III) How to Choose the Objective Function Z We want to choose a Z that is biologically meaningful. Reasonable options could be: 1.Z: Cellular growth (maximization) 2.Z: Particular metabolite engineering (maximization) 3.Z: Energy consumption (minimization) We want a v that: (A) Resides in side the cone. (B) maximizes Z=B+D+2E. Example: cellular growth is correlated with the production of B,D and 2E.

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin 1. “It has been shown that under rich growth conditions (i.e. no lack of phosphate and nitrogen), E. Coli grows in a stoichiometrically optimal manner.” (Schilling 2001, Edwards 1994) We shall use Z which reflects: Cellular Growth (III) How to Choose the Objective Function Z 2. “It is reasonable to hypothesize that unicellular organisms have evolved toward maximal growth performance.” (Segre, 2002.)

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin (IV) Phenotype phase planes - PPP Predicting Cellular Growth X axis – Succinate uptake rate Observations: Schilling 2001 Y axis – Oxygene uptake rate Z axis - Growth rate (maximal value of the objective function as function of succinate and oxygen uptake) Metabolic network is unable to utilize succinate as sole carbon source in anaerobic conditinos. Region 1: oxygen excess – this region is wasteful – (less carbon is available for biomass production since it is oxidized to eliminate the excess oxygen.) Line of optimality Succinate Oxygene Growth rate

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin (IV) Phenotype phase planes - PPP Predicting Cellular Growth X axis – Succinate uptake rate Observations: Schilling 2001 Y axis – Oxygene uptake rate Z axis – Growth rate (maximal value of the objective function as function of succinate and oxygen uptake) Region 2 – limitation on both oxygen and succinate Region 3 – the uptake of additional succinate has a negative effect. Cellular resources are required to eliminate excessive succinate. Succinate Oxygene Growth rate

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin (IV) Phenotype phase planes - PPP Predicting Cellular Growth The EPs can be projected onto the plane. EPs are used to explain the different regions from a pathway perspective PPPs were also constucted for Malate/oxygen and Glucose/oxygen

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Robustness Flux dependencies. Centre: phenotypic phase plane (PhPP). It shows the maximum biomass production that is achievable at every possible combination of O2 and succinate uptake rates. The line of optimality corresponds to the conditions that are necessary for maximal biomass yield (g DW cell mmol–1 carbon source, where DW is dry weight). Robustness analysis of the two uptake rates is shown in the two side panels. Left: effect on growth rate of varying O2 uptake at a fixed succinate uptake rate. Right: the effect on biomass generation of varying the succinate uptake rate at a fixed O2 uptake rate.

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Model vs. biological experiments

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Does E. coli behave according to optimal behavior predictions? E. coli was grown with malate as sole carbon source. A range of substrate concentrations and temperatures were used in order to vary the malate uptake rate (MUR). Oxygen uptake rate (OUR) and growth rate were measured.

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Does E. coli behave according to optimal behavior predictions? Malate/oxygen PPP Ibarra et al., Nature The experimentally determined growth rate were on the line of optimality of the PPP !

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Does E. coli behave according to optimal behavior predictions? Malate/oxygen PPP Ibarra et al., Nature 2002 Is the optimal performance on malate stable over prolonged periods of time? Evolution of E. coli on malate was studied for 500 generations in a single condition… 2- An adaptive evolution was observed with an increase of 19% in growth rate! 3- Same adaptive evolution was observed for succinate and Malate!

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Does E. coli behave according to optimal behavior predictions? Why does this adaptive evolution occur? In other words why is the starting point at the bottom of the hill?

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Does E. coli behave according to optimal behavior predictions? Same experiments were made using glycerol as sole carbon source Day 0 – Sub optimal growth Day 1-40 – evolution toward optimal growth Day 40 –optimal growth Day 60 –optimal growth (no change) Why?

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Exploitation of Constraints A growing toolbox for constraint-based analysis. The two steps that are used to form a solution space — reconstruction and the imposition of governing constraints — are illustrated in the centre of the figure. As indicated, several methods are being developed at various laboratories to analyze the solution space. Ci, concentration of compound i; EP, extreme pathway; vj, flux through reaction j; vnet, net flux through loop. Price, 2004

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Bibliography [1] Daniel Segre, Dennis Vitkup, and George M. Church. Analysis of optimality in natural and perturbed metabolic networks. PNAS, vol. 99, [2] C. H. Schilling, D. Letscher and Bernhard Palsson. Theory for the Systemic Definition of Metabolic Pathways and their use in Interpreting Metabolic Function from a Pathway-Oriented Perspective. J. theor. Biol. (2000) [3] Schillling et. Al Combining pathway analysis with flux balance analysis for the comprehensive study of metabolic systems. Biotechnology and bioengineering, [4] Edwards et al Characterizing the metabolic phenotype” A phenotype phase plan. Biotechnology and bioengineering [5] Kenethh et al. Advances in flux balance analysis. Current Opinion in Biotechnology. [6] Ibarra et al. Escherichia coli k-12 undergoes adaptive evolution to achiev in silico predicted optimal growth. Nature [7] W. Wiechert. Modeling and simulation: tools for metabolic engineering. Journal of biotechnology(2002) [8] Cornish-Bowden. From genome to cellular phenotype- a role for meatbolic flux analysis? Nature biotechnology, vol 18, [9] Schuster et al. Detection of elelmtary flux modes in biochemical networks: a promising tool for pathway analysis and metabolic engineering. TIBTECH 1999 [10] J. Papin, Nathan D Price, B. Palsson. Extreme pathway lengths and reaction participation in genome scale metabolic networks. Genome research, [11] Stelling eta l. Metabolic netwrok structure determines key aspects of functionality and regulation. Nature [12] A general definition of metabolic pathways useful for systematic organization and analysis of complex metabolic networks.

Humboldt- Universität Zu Berlin Edda Klipp, Humboldt-Universität zu Berlin Growth Phenotypes of insilico deletion strains The biomass yields are normalized with respect to the results for the full metabolic genotype. The α and β value for the constraints on the external fluxes for glucose and oxygen uptake are defined as follows (units- mmole g-1 hr-1): Phase 1 - vglc = 10, voxy = 23; LO - vgic = 10, voxy = 20.3; Phase 2 - vglc = 10, voxy = 17; Phase 3 - vglc = 10, voxy = 12; Phase 4 - vglc = 10, voxy = 8; Phase 5 - vglc = 10, voxy = 3; Phase 6 - vglc = 10, voxy = 0. Maximal biomass yields on glucose for all possible single gene deletions in central intermediary metabolism. The environmental variables (uptake rate/external metabolic fluxes) are set to correspond to a point within each of the phases of the wild-type PhPP (figure inset). The maximal yields were calculated using flux-balance analysis with the objective of maximizing the growth flux.