The ability of flux balance analysis to predict evolution of central metabolism scales with the initial distance to the optimum

Abstract

The most powerful genome-scale framework to model metabolism, flux balance analysis (FBA), is an evolutionary optimality model. It hypothesizes selection upon a proposed optimality criterion in order to predict the set of internal fluxes that would maximize fitness. Here we present a direct test of the optimality assumption underlying FBA by comparing the central metabolic fluxes predicted by multiple criteria to changes measurable by a (13)C-labeling method for experimentally-evolved strains. We considered datasets for three Escherichia coli evolution experiments that varied in their length, consistency of environment, and initial optimality. For ten populations that were evolved for 50,000 generations in glucose minimal medium, we observed modest changes in relative fluxes that led to small, but significant decreases in optimality and increased the distance to the predicted optimal flux distribution. In contrast, seven populations evolved on the poor substrate lactate for 900 generations collectively became more optimal and had flux distributions that moved toward predictions. For three pairs of central metabolic knockouts evolved on glucose for 600-800 generations, there was a balance between cases where optimality and flux patterns moved toward or away from FBA predictions. Despite this variation in predictability of changes in central metabolism, two generalities emerged. First, improved growth largely derived from evolved increases in the rate of substrate use. Second, FBA predictions bore out well for the two experiments initiated with ancestors with relatively sub-optimal yield, whereas those begun already quite optimal tended to move somewhat away from predictions. These findings suggest that the tradeoff between rate and yield is surprisingly modest. The observed positive correlation between rate and yield when adaptation initiated further from the optimum resulted in the ability of FBA to use stoichiometric constraints to predict the evolution of metabolism despite selection for rate.

Figure 1. Evolution of metabolic fluxes and measures of optimality and predictability.

We consider three ways to analyze changes in metabolism that relate an ancestor (Anc, blue) to an evolved isolate (E_i, green) in regard to an FBA-predicted optimum (Opt, red). A) Evolution of metabolic fluxes can be evaluated from the perspective of changes in proximity to the theoretical maximum for a given optimality criterion (Δ% Optimality). B) A vector of flux ratios defines a position in multi-dimensional flux space. One can then consider the relative Euclidian distance of a given evolved population in this space from its optimum (D_EO) compared to that of an ancestor from its optimum (D_AO; plotted as log(D_EO/D_AO)). C) At the most detailed level, one can compare the FBA-predicted value for a given flux ratio versus that observed via ¹³C labeling.

Figure 2. Evolved changes in central carbon metabolism for the LTEE populations after 50,000 generations of adaptation on glucose.

A) The flux pathways measured for the LTEE lines are denoted with numbers and red arrows. The genes knocked out in the knockout data set and the entry point of lactate into the network are both indicated. B) A heat map of the difference between evolved and ancestral flux ratios from the LTEE populations. The right side indicates flux ratios predicted for the ancestral line according to each optimality criterion. The number of the flux ratio corresponds to the numbered pathways in A. Single asterisks denote significant changes as calculated by ANOVA, double asterisks are also significant by Tukey-HD.

Figure 3. Measures of optimality and predictability after adaptation of LTEE populations to glucose for 50,000 generations.

A,D) The % optimality of the ancestor (black) and evolved isolates (grey, same order as Fig. 2); B,E) distance to optimal flux distribution (plotted as log(D_EO/D_AO)); and C, F) comparison of predicted to observed flux ratios for FBA-predictions based upon BM/S (A–C) or ATP/S (D–F). Error bars represent standard errors of three biological replicates.

Figure 4. Measures of optimality and predictability after adaptation to lactate for ∼900 generations.

A,D) The % optimality of the ancestor (black) and evolved isolates (grey); B,E) distance to optimal flux distribution (plotted as log(D_EO/D_AO)); and C, F) comparison of predicted to observed flux ratios for FBA-predictions based upon BM/S (A–C) or ATP/S (D–F).

Figure 5. Measures of optimality and predictability after adaptation of gene knockouts on glucose for ∼600–800 generations.

A,B) The % optimality of the ancestor (black) and evolved isolates (grey); C,D) distance to optimal flux distribution for FBA-predictions based upon BM/S (A,C) or ATP/S (B,D).

Figure 6. Evolutionary change in % optimality versus initial % optimality of the ancestor across data sets for BM/S.

Error bars represent standard errors between evolved populations.