Fig 1.
The cost vector formalism shows what determines the number of EFMs in the optimal solution.
We here consider a simplified model with 2 EFMs (blue and orange), and 2 constraints. In reality, the costs of many more EFMs have to be compared, and potentially also of more constraints. The cost vector of the ith EFM denotes the fractions of the first and second constrained enzyme pool that this EFM uses when producing one unit of objective flux. The cell-synthesis flux produced by EFM i is denoted by λi, and the corresponding enzyme costs are λidi(x). The cost of mixing EFMs 1 and 2 corresponds to the weighted sum of the cost vectors: λ1 d1(x) + λ2 d2(x). The mixture is feasible as long as none of the constraints is exceeded: λ1 d1(x) + λ2 d2(x) ≤ 1. The objective value, λ1 + λ2, is maximized by fitting a vector sum of as many vectors as possible in the constraint box. This solution is shown by the dashed vectors. The pure usage of one EFM with off-diagonal cost vector leads to underuse of one constraint, while diagonal cost vectors can exhaust both constrained pools. A mixture of EFMs will always be a combination of an above-diagonal and a below-diagonal vector. All EFMs and mixtures thereof, can be ranked by a dot on the diagonal that denotes the average cost per unit cell-synthesis flux (see Lemma 4 in S1 Appendix for a proof). Pure usage of above-diagonal cost vectors is ranked by projecting the cost vector horizontally to the diagonal, while pure usage of below-diagonal vectors is ranked by vertical projection. Mixtures are ranked by placing a dot at the intersection of the diagonal with the line between the two cost vectors. The (mixture of) EFM(s) with the lowest average cost (i.e., with the dot closest to the origin) leads to the highest growth rate (the mathematical proof is included in S1 Appendix). The enzymatic costs of an EFM depend on the intracellular metabolite concentrations, i.e., the saturation of enzymes. The shaded regions indicate alternative positions for the cost vectors at different intracellular metabolite concentrations, two of them are shown. The blue and orange cost vectors lead to the highest growth rate when using only that EFM. We see that in the left figure the orange EFM gives rise to a higher growth rate. Upon a change of environmental conditions, the cost vectors can change, and the mixture of EFMs can become better than either single EFM (right figure). A change like this would lead to a change in metabolic behaviour.
Fig 2.
Proportionality of reaction rates and growth rates, shown by many microorganisms, is an indication of low metabolic complexity.
Measured uptake rates [28–33] were gathered from experiments in which growth rate was varied in carbon-limited chemostats. For each species we normalized the measured growth rate to the so-called critical growth rate: the growth rate at which the production of overflow products starts. Uptake rates were normalized relative to the uptake rate of the species at the critical growth rate. Up to the critical growth rate, all microorganisms show a simple proportional relation between the growth rate and uptake rates of glucose and oxygen. In Section 8 we explain why this proportionality is an indication of the usage of only one EFM. After the critical growth rate, the reaction rates are no longer proportional, a phenomenon called overflow metabolism.
Fig 3.
Illustration of the extremum principle.
The extremum principle states that the dimensionality of the solution space is determined by the number of enzyme-expression constraints, rather than by the dimensionality of the metabolic network. The constraints result from biophysical limits, e.g., limited solvent capacities within cellular compartments. Our cost vector formalism, explained in Fig 1, enables us to analyze metabolism in the low-dimensional constraint space, instead of in the high-dimensional flux space that is normally used.
Fig 4.
The cost vector formalism provides insight in how growth rate maximization leads to overflow metabolism.
a) A core model with two EFMs that individually lead to cell synthesis (orange: respiration and blue: acetate overflow). All considered reactions have an associated enzyme, whose activity depends on kinetic parameters and the metabolite concentrations. We varied growth rate by changing the external substrate concentration. Given this external condition, the growth rate was optimized under two enzymatic constraints (limited cytosolic enzyme Σ ei,cyto ≤ 1 and limited membrane area etransport ≤ 0.3). b) The predicted substrate uptake fluxes directed towards respiration and overflow are in qualitative agreement with the experimental data (shown before in Fig 2) of several microorganisms scaled with respect to the growth rate (μcrit) and uptake rate (qcrit) at the onset of overflow [4, 38, 39]. c) The cost vectors (solid arrows) of the two EFMs before (left) and after (right) the respirofermentative switch. The x-coordinate of the cost vectors denote the fraction of the cytosolic volume that is needed to produce one unit objective flux with the corresponding EFM. The y-coordinate shows the necessary fraction of the available mebrane area. The position of the cost vectors are shown for the optimized metabolite concentrations; the shaded regions show alternative positions of the cost vectors at different enzyme and metabolite concentrations. The dashed vectors show the usage of the EFMs in the optimal solution.
Fig 5.
Predictions and experimental results of the perturbation of the size of limited enzyme pools during growth using a mixture of EFMs.
In the cost vector plots, panels a) and b), the red vector denotes the optimal solution in the unperturbed organism. Upon experimental perturbation, the available area in constraint space can change, indicated by the shaded grey areas. The green, blue, and grey vectors show the new optimal solutions under increasingly strong perturbations. The predicted effect on the flux through the acetate branch is shown in panels c and d). a,c) Analysis of perturbations that tighten both protein pools with the same amount shows that flux and growth rate will decrease proportionally, as observed experimentally (e)) for the overexpression of LacZ on different carbon sources (data from Basan et al. [4]). b,d) Perturbations that tighten an enzyme pool that is mostly used by one EFM (here denoted by CO2) initially cause an increase in flux through the other EFM in the mixture(Ac). Eventually, at stronger limitations, this flux also decreases. f) This behaviour is observed, a.o., for translation inhibitor experiments using chloramphenicol (S1 Appendix Section 7).
Fig 6.
Under-utilization of enzymes and co-consumption can be understood with our kinetic, constrained-based approach.
a) Model simulations of the metabolic switch of L. lactis are shown (dashed lines), along with experimental data from [44]. The flux predictions for both pathways are expressed as a fraction of the total flux through both pathways. Enzyme concentrations are normalized to the concentrations at a growth rate of 0.15 and then log-scaled. The model reproduces the switch from mixed-acid to homolactic fermentation at constant enzyme concentrations, because of its consideration of enzyme kinetics. Details of this model are described in S3 Appendix. To obtain a perfect fit with the data, a larger model should be invoked, but this is beyond the scope of this paper. We emphasize that protein concentrations can remain constant while pathway usage changes. b) An example is shown of a metabolic network with EFMs that use either succinate or xylose (orange and blue circles respectively), and an EFM (green circles) that uses two carbon sources. Grey squares denote products that are essential for cell growth. The co-consumption EFM can synthesize one cell component with succinate, and the other with xylose. The reaction that connects the upper and lower parts of the network therefore becomes inessential. This leads to a possible reduction in protein costs and therefore to a growth rate advantage. We indeed measured a growth rate increase by the co-consumption of succinate and xylose, as shown in the inset in which different biological replicates are indicated with different points. Results of the other combinations that were tested can be found in S4 Appendix.