Bioinformatics: Network Analysis Flux Balance Analysis and Metabolic Control Analysis COMP 572 (BIOS 572 / BIOE 564) - Fall 2013 Luay Nakhleh, Rice University 1 Flux Balance Analysis (FBA) ✤ Flux balance analysis (FBA), an optimality-base method for flux prediction, is one of the most popular modeling approaches for metabolic systems. ✤ Flux optimization methods do not describe how a certain flux distribution is realized (by kinetics or enzyme regulation), but which flux distribution is optimal for the cell; e.g., providing the highest rate of biomass production at a limited inflow of external nutrients. ✤ This allows us to predict flux distributions without the need for a kinetic description. 2 Flux Balance Analysis (FBA) ✤ FBA investigates the theoretical capabilities and modes of metabolism by imposing a number of constraints on the metabolic flux distributions: ✤ The assumption of a steady state: S×v=0. ✤ Thermodynamics constraints: ai≤vi≤bi. ✤ An optimality assumption: the flux distribution has to maximize (or, minimize) an objective function f(v) r f(v)= civi i=1 X 3 P R IMER Geometric Interpretation of FBA Box 1 Mathematical representation of metabolism Metabolic reactions are represented as a v3 v3 v3 stoichiometric matrix (S) of size m s n. Constraints Optimization Every row of this matrix represents one 1) Sv = 0 maximize Z unique compound (for a system with m 2) ai < vi < bi compounds) and every column represents one reaction (n reactions). The entries v in each column are the stoichiometric v1 1 v1 coefficients of the metabolites participating Unconstrained Allowable in a reaction. There is a negative coefficient solution space solution space Optimal solution for every metabolite consumed and a v v v positive coefficient for every metabolite 2 2 2 that is produced. A stoichiometric coefficient of zero is used for every Figure 1 The conceptual basis of constraint-based modeling. With no constraints, the flux metabolite that does not participate in a distribution of a biological network may lie at any point in a solution space. When mass balance constraints imposed by the stoichiometric matrix S (labeled 1) and capacity constraints imposed particular reaction. S is a sparse matrix by the lower and upper bounds (ai and bi) (labeled 2) are applied to a network, it defines an because most biochemical reactions allowable solution space. The network may acquire any flux distribution within this space, but involve only a few different metabolites. points outside this space are denied by the constraints. Through optimization of an objective The flux through all of the reactions in a function, FBA can identify a single optimal flux distribution that lies on the edge of the network is represented by the vector v, allowable solution space. which has a length of n. The concentrations [Source: “What is flux balance analysis?”, Nat Biotech.] of all metabolites are represented by the analyze single points within the solution function. In practice, when only one 4 vector x, with length m. The system of mass space. For example, we may be interested reaction is desired for maximization or balance equations at steady state (dx/dt = in identifying which point corresponds to minimization, c is a vector of zeros with a 0) is given in Fig. 2c26: the maximum growth rate or to maximum value of 1 at the position of the reaction Sv = 0 ATP production of an organism, given its of interest (Fig. 2d). Any v that satisfies this equation is particular set of constraints. FBA is one Optimization of such a system is said to be in the null space of S. In any method for identifying such optimal points accomplished by linear programming realistic large-scale metabolic model, within a constrained space (Fig. 1). (Fig. 2e). FBA can thus be defined as there are more reactions than there are FBA seeks to maximize or minimize an the use of linear programming to solve compounds (n > m). In other words, objective function Z = cTv, which can be the equation Sv = 0, given a set of upper there are more unknown variables than any linear combination of fluxes, where and lower bounds on v and a linear equations, so there is no unique solution c is a vector of weights indicating how combination of fluxes as an objective to this system of equations. much each reaction (such as the biomass function. The output of FBA is a particular Although constraints define a range of reaction when simulating maximum flux distribution, v, which maximizes or solutions, it is still possible to identify and growth) contributes to the objective minimizes the objective function. glucose gDW–1 h–1; DW, dry weight) and set- constrained to an uptake rate of 0 mmol growth of deleting every pairwise combina- ting the maximum rate of oxygen uptake to gDW–1 h–1. Constraints can also be tailored tion of 136 E. coli genes to find double gene an arbitrarily high level, so that it does not to the organism being studied, with lower knockouts that are essential for survival of limit growth. Then, linear programming is bounds of 0 mmol gDW–1 h–1 used to simu- the bacteria. used to determine the maximum possible late reactions that are irreversible in some FBA has limitations, however. Because it flux through the biomass reaction, result- organisms. Nonzero lower bounds can also does not use kinetic parameters, it cannot ing in a predicted exponential growth rate force a minimal flux through artificial reac- predict metabolite concentrations. It is also of 1.65 h–1. Anerobic growth of E. coli can be tions (like the biomass reaction) such as the only suitable for determining fluxes at steady calculated by constraining the maximum rate ‘ATP maintenance reaction’, which is a bal- state. Except in some modified forms, FBA of uptake of oxygen to zero and solving the anced ATP hydrolysis reaction used to sim- does not account for regulatory effects such system of equations, resulting in a predicted ulate energy demands not associated with as activation of enzymes by protein kinases growth rate of 0.47 h–1 (see Supplementary growth13. Constraints can even be used to or regulation of gene expression. Therefore, Tutorial for computer code). simulate gene knockouts by limiting reac- its predictions may not always be accurate. As these two examples show, FBA can be tions to zero flux. used to perform simulations under differ- FBA does not require kinetic parameters The many uses of flux balance analysis ent conditions by altering the constraints and can be computed very quickly even for Because the fundamentals of flux balance analy- on a model. To change the environmental large networks. This makes it well suited sis are simple, the method has found diverse conditions (such as substrate availabil- to studies that characterize many different uses in physiological studies, gap-filling efforts ity), we change the bounds on exchange perturbations such as different substrates or and genome-scale synthetic biology3. By alter- reactions (that is, reactions representing genetic manipulations. An example of such a ing the bounds on certain reactions, growth on metabolites flowing into and out of the sys- case is given in example 6 in Supplementary different media (example 1 in Supplementary tem). Substrates that are not available are Tutorial, which explores the effects on Tutorial) or of bacteria with multiple gene 246 VOLUME 28 NUMBER 3 MARCH 2010 NATURE BIOTECHNOLOGY Formulation of an FBA Problem P R IMER Figure 2 Formulation of an FBA problem. (a) A metabolic network reconstruction consists of a A B + C Reaction 1 list of stoichiometrically balanced biochemical a Genome-scale B + 2C D Reaction 2 reactions. (b) This reconstruction is converted into a metabolic reconstruction ... mathematical model by forming a matrix (labeled S), Reaction n in which each row represents a metabolite and each column represents a reaction. Growth is incorporated into the reconstruction with a biomass reaction (yellow column), which simulates metabolites Reactions ... consumed during biomass production. Exchange 1 2 n BiomassGlucoseOxygen v1 reactions (green columns) are used to represent the A –1 v b Mathematically represent 2 s e t i l o b a t e M e t a b o l i t e s B 1 –1 flow of metabolites, such as glucose and oxygen, metabolic reactions C 1 –2 ... = 0 in and out of the cell. (c) At steady state, the flux and constraints D 1 * v through each reaction is given by Sv = 0, which n ... –1 vbiomass v defines a system of linear equations. As large –1 glucose v m oxygen models contain more reactions than metabolites, there is more than one possible solution to these Stoichiometric matrix, S Fluxes, v equations. (d) Solving the equations to predict the maximum growth rate requires defining an objective –v1 + ... = 0 function Z = cTv (c is a vector of weights indicating c Mass balance defines a v1 – v2 + ... = 0 how much each reaction (v) contributes to the system of linear equations objective). In practice, when only one reaction, such v1 – 2v2 + ... = 0 as biomass production, is desired for maximization v2 + ... = 0 or minimization, c is a vector of zeros with a value etc. of 1 at the position of the reaction of interest. In the growth example, the objective function is Z = vbiomass (that is, c has a value of 1 at the position of the d Define objective function To predict growth, Z = v biomass reaction). (e) Linear programming is used biomass (Z = c * v + c * v ... ) to identify a flux distribution that maximizes or 1 1 2 2 minimizes the objective function within the space of allowable fluxes (blue region) defined by the v constraints imposed by the mass balance equations 2 Z and reaction bounds.