MIT OpenCourseWare http://ocw.mit.edu 6.047 / 6.878 Computational Biology: Genomes, Networks, EvolutionFall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.6.047/6.878 Lecture 22: Metabolic Modeling 2 November 20, 2008 1 Review In the last lecture, we discussed how to use linear algebra to model metabolic networks with flux-balance analysis (FBA), which depends only on the stoichiometry of the reactions, not the kinetics. The metabolic network is represented as an n × m matrix M whose columns are the m reactions occurring in the network and whose rows are the n products and reactants of these reactions. The entry M(i, j) represents the relative amount of metabolite i consumed or produced by reaction j. A positive value indicates production; a negative value indicates consumption. The nullspace of this matrix consists of all the m × 1 reaction flux vectors that are possible given that the metabolic system is in steady state; i.e. all the sets of fluxes that do not change the metabolite concentrations. The nullspace ensures that the system is in steady state, but it there are also additional constraints in biological systems: fluxes cannot be infinite, and each reaction can only travel in the forward direction (in cases where the backward reaction is also biologically possible, it is included as a separate column in the matrix). After bounding the fluxes and constraining the directions of the reactions, the resulting space of possible flux vectors is called the constrained flux-balance cone. The edges of the cone are known as its extreme pathways. By choosing an objective function—some linear combination of the fluxes—to maximize, we can calculate the optimal values for the fluxes using linear programming and the simplex algorithm. For example, the objective function may be a weighted sum of all the metabolite fluxes that represents the overall growth rate of the cell (growth objective). Alternatively, we can maximize use of one particular product by find the m × 1 flux vector that is in the flux-balance cone and has the largest negative value of that product. At the end of the last lecture, we also discussed knockout phenotype predictions, a classic application of metabolic modeling. Experimental biologists often gather information about the function of a protein by generating transgenic organisms in which the gene encoding that protein has been disrupted, or knocked out. We can simulate in silico the effect of knocking out a particular enzyme on metabolism by assuming that the reaction catalyzed by that enzyme does not occur at all in the knockout: zeroing out the jth column of M effectively removes the jth reaction from the network. What used to be an optimal solution may now lie outside of the constrained flux-balance cone and thus no longer be feasible in the absence of the jth reaction. We can determine the new constrained flux-balance cone and calculate the new optimum set of fluxes to maximize our objective function, predicting the effect of the knockout on metabolism. 2 Knockout Phenotype Prediction in Eukaryotes (Forester et al, 2003; Famili et al, 2003) The last lecture gave an example of using knockout phenotype prediction to predict metabolic changes in response to knocking out enzymes in E. coli, a prokaryote. Eukaryotic cells, including animal cells, have more complex organization and more complex gene regulation and gene expression pathways. In this lecture, a recent attempt to perform similar knockout phenotype prediction in yeast, a eukaryote, was presented (Forster et al, 2003 and Famili et al, 2003). The authors predicted whether a variety of enzyme knockouts would be able to grow under a few different environmental conditions and compared the predictions to experimental results. They achieved 81.5% agreement between their predictions and experiment, but looking at the data more closely reveals that all of the discrepancies were false positives, in which the yeast were predicted to grow but did not. In fact, the model predicted that almost every knockout would grow. While not bad, these results were not as compelling as those for the prokaryotic case, highlighting the need to incorporate the ability of cells to regulate gene expression in response to environmental and metabolic changes into flux-balance analysis of eukaryotes.3 Overview of Today’s Material: Extensions to FBA Today we discuss a number of extensions to flux-balance analysis that provide additional predictive power. First, we demonstrate the ability of FBA to give quantitative predictions about growth rate and reaction fluxes given different environmental conditions. We then describe how to use FBA to predict time-dependent changes in growth rates and metabolite concentrations using quasi steady state modeling. Next, we discuss two approaches for taking gene expression changes into account in the FBA model: by building the rules of gene regulation into a Boolean network or by using available expression data from microarray experiments to constrain the flux-balance cone. Finally, we provide an example of using expression data to predict the state of the environment from the metabolic state, rather than the other way around. 4 Quantitative Flux Prediction (Edwards, Ibarra, & Palsson, 2001) Since FBA maximizes an objective function, resulting in a specific value for this function, we should in theory be able to extract quantitative information from the model. An early example of this was done by Edwards, Ibarra, and Palsson (2001), who predicted the growth rate of E. coli in culture given a range of fixed uptake rates of oxygen and two carbon sources (acetate and succinate), which they could control in a batch reactor. They assumed that E. coli cells adjust their metabolism to maximize growth (using a growth objective function) under given environmental conditions and used FBA to model the metabolic pathways in the bacterium. The controlled uptake rates fixed the values of the oxygen and acetate/succinate input fluxes into the network, but the other fluxes were calculated to maximize the value of the growth objective. The authors’ quantitative growth rate