The number of active metabolic pathways is bounded by the number of cellular constraints at maximal metabolic rates

Growth rate is a near-universal selective pressure across microbial species. High growth rates require hundreds of metabolic enzymes, each with different nonlinear kinetics, to be precisely tuned within the bounds set by physicochemical constraints. Yet, the metabolic behaviour of many species is characterized by simple relations between growth rate, enzyme expression levels and metabolic rates. We asked if this simplicity could be the outcome of optimisation by evolution. Indeed, when the growth rate is maximized—in a static environment under mass-conservation and enzyme expression constraints—we prove mathematically that the resulting optimal metabolic flux distribution is described by a limited number of subnetworks, known as Elementary Flux Modes (EFMs). We show that, because EFMs are the minimal subnetworks leading to growth, a small active number automatically leads to the simple relations that are measured. We find that the maximal number of flux-carrying EFMs is determined only by the number of imposed constraints on enzyme expression, not by the size, kinetics or topology of the network. This minimal-EFM extremum principle is illustrated in a graphical framework, which explains qualitative changes in microbial behaviours, such as overflow metabolism and co-consumption, and provides a method for identification of the enzyme expression constraints that limit growth under the prevalent conditions. The extremum principle applies to all microorganisms that are selected for maximal growth rates under protein concentration constraints, for example the solvent capacities of cytosol, membrane or periplasmic space.


Source code
Code for finding coconsumption EFMs The Python and Matlab-code used for finding coconsuming EFMs are attached in a compressed folder as a supplement. In the compressed folder, we have also added a text-file with instructions.

Strain information
All experiments were performed with E. coli strain MG1655.
Cells were seeded from a frozen glycerol stock into 5 ml liquid N-C-minimal medium +glucose and cultured in 30x115 mm conical tubes, shaking with 220 rpm at 37 o C. During the subsequent 12 hours they were sequentially diluted into tubes with the desired medium, to achieve exponential growth and removal of undesired carbon. Then, depending on the specific growth rate in each condition, cells of the various cultures were diluted to different densities and 200 µl of each was transferred to a Greiner 96-well, flat bottom plate. The plate was kept shaking at 37 o C and densities were measured at 600 nm using a Spectramax 384 plus (Molecular Devices). Cell densities were chosen such that 8 doublings could take place before growth in the plate could be detected. Every condition was represented by 10 micro-wells (i.e. technical replicates) during each experiment, to make sure enough volume was available for sampling. Samples were taken during growth, filtrated and stored at -20 o C for further analysis. This experiment was performed in triplo, meaning that three biological replicates were done on separate days. Carbon substrate uptake measurements 50µl of undiluted samples were analysed on their mannose, maltose, succinate and xylose content using HPLC (Shimadzo, LC-20AT) at a flow rate of 0.5 mL/min. Calibration samples were made for individual-and triple carbon sources in N-C-minimal medium, to determine the concentrations and validate good separation. Compounds were separated on an ROA-Organic Acid H+ column (Phenomenex, Rezex) and detected using refractive index (Shimadzu, RID-10A) and UV-Vis (Shimadzu, SPD-20A).
Acetate concentrations were measured using an enzyme essay described by Smith et al. [4]. Samples were diluted either 10 or 100 times to stay in the linear range of NADH detection. The essay was conducted in a 96-well, flat bottom plate at 37 o C and NADH oxidation was measured at 340 nm using a Spectramax 384 plus (Molecular Devices).

Data analysis
The three plate-reader experiments resulted in two types of data: OD measurements and HPLC (High Performance Liquid Chromatography) analyses of growth medium samples. The OD measurements were taken every 5 minutes during the full growth experiment and in total 104 growth medium samples were taken at different ODs for all conditions.
The OD measurements were analysed using Matlab. Background OD was subtracted and time windows of at least two hours were selected in which the natural logarithm of the measurements was sufficiently linear: (R 2 > 0.95). For these windows the specific growth rate (µ = 1 OD dOD dt ) was calculated and the maximum is reported below.
The HPLC analyses were normalized using a peak in the chromatogram that corresponded to a constant compound (phosphate) in the medium. Compound concentrations were calculated using a linear calibration curve that was made for all compounds. Since we were interested in the decrease of substrate concentration, rather than in the absolute value of these concentrations, the concentrations were normalized such that the t 0 -concentration is equal to the intended initial concentration for that compound, thereby correcting for small pipetting errors.

Results
The calculated growth rates for all 12 conditions are summarized in 1. Data can be found in file: SI_growth_rates.txt. In almost all cases the growth rate increases or remains equal when an extra compound is added: only the combination of mannose and maltose leads to a lower growth rate than on maltose alone. The addition of succinate to the medium always leads to an increase in growth rate.  We determined for all conditions the mean specific uptake rate for all the compounds. The resulting data is shown in Table 2 and the standard errors of the means are shown in Table 3. This data can be found in SI_q_S_comp_cond.xlsx.  In Figure 2 we plot, for all conditions except for glucose, the relation between the optical density of cells in the sample and the concentrations of the compounds succinate, maltose, mannose and xylose. The dataset can be found in the file SI_OD_conc_per_cond.xlsx.  Figure 2: During the growth experiments, the concentration of carbon sources was measured. The letters that indicate the conditions denote the available carbon sources in the medium: S=Succinate, L=maLtose, M=Mannose, X=Xylose, G=Glucose. We here show the decrease in these concentrations as the OD (Optical Density) of the culture increases. Data from three biological replicates was normalized for initial concentration and then shown together. The best linear approximation was calculated and shown by a dashed line.