Relationship between fitness and heterogeneity in exponentially growing microbial populations

Abstract

Despite major environmental and genetic differences, microbial metabolic networks are known to generate consistent physiological outcomes across vastly different organisms. This remarkable robustness suggests that, at least in bacteria, metabolic activity may be guided by universal principles. The constrained optimization of evolutionarily motivated objective functions, such as the growth rate, has emerged as the key theoretical assumption for the study of bacterial metabolism. While conceptually and practically useful in many situations, the idea that certain functions are optimized is hard to validate in data. Moreover, it is not always clear how optimality can be reconciled with the high degree of single-cell variability observed in experiments within microbial populations. To shed light on these issues, we develop an inverse modeling framework that connects the fitness of a population of cells (represented by the mean single-cell growth rate) to the underlying metabolic variability through the maximum entropy inference of the distribution of metabolic phenotypes from data. While no clear objective function emerges, we find that, as the medium gets richer, the fitness and inferred variability for Escherichia coli populations follow and slowly approach the theoretically optimal bound defined by minimal reduction of variability at given fitness. These results suggest that bacterial metabolism may be crucially shaped by a population-level trade-off between growth and heterogeneity.

Figure 1

(A) Constraint-based models use large-scale reconstructions of cellular metabolic networks, encoded in a stoichiometric matrix $\mathbf{S}$, together with biochemical or regulatory constraints on fluxes. Nonequilibrium steady states (NESSs) of the network satisfy the conditions $\mathbf{S}\mathbf{v}=0$. The (high-dimensional, as $N\gg M$) polytope of solutions is the feasible space $\mathcal{F}$ of the system. The biomass output associated with each feasible flux vector (phenotype) represents its fitness. (B) Empirical mean biomass rate $v_{\text{bm}}^{\text{xp}}$ (markers, from (15,32)) and maximum biomass output $v_{\text{bm}}^{\text{max}}$ (continuous line) predicted by flux balance analysis for the metabolic network and glucose uptakes from (15,32). (C) Experiment-derived averages for a small set of fluxes and the bounds defined by the feasible space can be used to inform a maximum entropy inference procedure that allows us to determine the most likely distribution of phenotypes for the entire network. The distribution inferred for each experiment yields a point in the (fitness, information) plane. (D) Inferred fitness-information values for a set of 33 experiments probing E. coli growth in glucose-minimal medium. Green line: fitness-information (F-I) bounds numerically computed by EP for the 33 experiments (all curves perfectly overlap). In each condition, fitness values $⟨v{⟩}_{\text{bm}}=v_{\text{bm}}^{\text{optm}}$ have been rescaled by the corresponding value of $v_{\text{bm}}^{\text{max}}$. Markers: fitness is computed according to the inferred distribution, i.e., $⟨v{⟩}_{\text{bm}}=v_{\text{bm}}^{\text{inf}}$. The black marker denotes the rescaled fitness $⟨v_{\text{bm}}{⟩}_{q(\mathbf{v};\beta =0)}/v_{\text{bm}}^{\text{max}}$ corresponding to a uniform distribution on $\mathcal{F}$ ($I=0$, $\beta =0$), which separates the upper branch of the F-I bound (growth faster than $⟨v_{\text{bm}}{⟩}_{q\left(\mathbf{v};0\right)}$, $\beta >0$) from the lower one (growth slower than $⟨v_{\text{bm}}{⟩}_{q\left(\mathbf{v};0\right)}$, $\beta <0$). The gray area represents the infeasible region. In color bars in (B) and (D), orange and blue shades are used for data coming from (15) and (32), respectively.

Figure 2

(A) Vertical distance between the F-I bound and inferred F-I pairs versus empirical growth rate. The dashed line is a guide for the eye. (B) Norms of the projections of the fields $\mathbf{c}$ onto the feasible space $\mathcal{F}$ (supporting material, “projections of the coefficients along individual flux directions”) as a function of the experimentally measured biomass. In color bars, orange and blue shades are used for data coming from (15) and (32), respectively.

Figure 3

Mean variance of fluxes (Eq. 15) through six metabolic pathways as a function of the growth rate (empirical values) in optimal (purple) and inferred (cyan) distributions. (Note that markers appear darker at points where many of them overlap. This is mainly due to replicated experiments in (32).)

Figure 4

Explained variance of the coefficients (in percentage, red line) and reconstruction error of the original fields (continuous blue line) and phenotypes (dashed blue line) as a function of the number of PCA components employed to compute their low-rank counterpart.

Figure 5

Mean values of the acetate excretion rate (top) and biomass production rate (bottom) measured (purple) and predicted from the inferred distribution (cyan) for the different experiments from (15,32). For comparison, bare flux balance analysis predictions correspond to $v_{\text{ex ace}}=0$ and $v_{\text{bm}}/v_{\text{bm}}^{\text{max}}=1$, respectively. Error bars represent experimental errors as reported in ([15, 32])