E.coli MG1655, NCM3722 and ML308 have been selected as the model strains in this work and are referred to as MG, NCM and ML respectively from this point on. In this section, the simulated acetate excretion pattern is presented against experimental data to demonstrate the accuracy of the model prediction. Subsequently we elucidate the linear interdependency of the proteomic cost parameters. In particular, we reveal the similarities and differences between the three E. coli strains. With respect to the PP pathway ratio (PPP%), previous studies on the slow-growth strain MG show that the portion of carbon that goes into the PP pathway can be approximately 20% of the total carbon intake [40, 41]. In this work, we set the upper bound of the carbon flowing into PP pathway (uPPP%) to 25, 35 and 40% to investigate the potential effect of the change in PPP% on proteomic cost parameters and model prediction (more justification is provided in Additional file 2, section 3). On cellular energy demand, we refer to the original energy demand specified in the core model [34] as nominal energy demand, and present first the set of results which were generated on this basis. Subsequently, we show how an adjusted energy demand (particularly applied to ML, referred to as ML-new) affects the patterns of the estimated proteomic cost parameters and the accuracy of biomass yield prediction.
Model prediction of overflow metabolism with nominal energy demand
Here, the accuracy of the predicted acetate excretion rate is compared with experimental data. Variation in the proteomic cost parameters with the changed carbon level diverted into the PP pathway is also presented.
Model prediction of acetate production
Figure 1 shows that model prediction of the pattern of acetate excretion is in good agreement with the experimental observations for three different E. coli strains. The onset of the production of acetate is concomitant with the drop in the respiratory flux, indicating a switch between fully-respiration to respiration-fermentation mode. As the growth rate further increases, the acetate flux becomes dominant while the extent of respiration is gradually diminishing. It is worth noting that zero acetate production was commonly observed at low growth rates of different strains [3, 16, 41, 42]; To emphasise the (strain-specific) acetate production pattern, we only collect the data with non-zero acetate production. For all the strains, data involving growth rates lower than those presented in Fig. 1 are associated with non-detectable acetate excretion, hence are not shown here.
Linear relationships between proteomic cost parameters
When the nominal energy demand is adopted (indicated by “-nom”), the change of uPPP% leads to insignificant changes to the \( {w}_r^{\ast }-{w}_f^{\ast } \) line for each strain (Fig. 2). Between different strains, MG and NCM share nearly identical lines. The lines of ML-nom deviate from those of the former two, but not significantly (although this closeness will be altered with the adjusted energy demand, see Fig. 2 ML-new and the section below). In any case, \( {w}_r^{\ast } \) is clearly higher than the corresponding \( {w}_f^{\ast } \), implying that respiration has a higher (lower) proteomic cost (efficiency) than fermentation for energy production, which is consistent with what was derived from protein abundances data for comparable parameters in [16].
To inspect the insignificant disparity in the \( {w}_r^{\ast }-{w}_f^{\ast } \) lines when all strains use the nominal energy demand, Eq. (12) is re-arranged to
$$ {w}_r^{\ast }=\frac{v_{f,0}}{v_{r,0}}{w}_f^{\ast }+\frac{1}{v_{r,0}} $$
(19)
The slope and intercept of the \( {w}_r^{\ast }-{w}_f^{\ast } \) line are dictated by \( \frac{v_{f,0}}{v_{r,0}} \) and \( \frac{1}{v_{r,0}} \), respectively. vf, 0 can be determined directly by the experimental measurement of acetate production. vr, 0, on the other hand, is a result of the combination of (measured) rates of acetate production and the mass and energy balance structure of the metabolic model.
For a specific strain, vf, 0 only depends on the pattern of acetate excretion, not affected by assumed level of uPPP%. Therefore, the impact of uPPP% on the \( {w}_r^{\ast }-{w}_f^{\ast } \) line is through affecting the value of vr, 0, which turns out to be rather moderate. Between different strains, the ratio of vf, 0 and vr, 0 and the value of vr, 0 are nearly identical between MG and NCM, regardless of the level of uPPP% adopted, resulting in the very much overlapped pattern of the \( {w}_r^{\ast }-{w}_f^{\ast } \) relationship between MG and NCM. For ML, the value of the slope is slightly smaller than MG and NCM, while the intercept is about 25% larger (as shown in Additional file 1: Table S2). Figure 2 also suggests that the proteomic cost (efficiency) of respiration pathways for ML is higher (lower) than that for MG and NCM, regardless the modification in the energy demand.
Compared to the \( {w}_r^{\ast }-{w}_f^{\ast } \) relationship, that of \( {b}^{\ast }-{w}_f^{\ast } \) appears to be affected by the level of uPPP% more visibly (Fig. 3). Between different species, the difference is also more pronounced, and closeness is present between the two rapid-growth strains NCM and ML (as presented in Additional file 2: Figure S1).
For a specific strain, the increase of uPPP% gradually moves the \( {b}^{\ast }-{w}_f^{\ast } \) line to the right (yellow arrow, Fig. 3), corresponding to an increase in b∗ (blue arrow, Fig. 3). This trend can be explained by inspecting a re-arrangement of Eq. (11):
$$ {b}^{\ast }=\left({k}_r\frac{v_{f,0}}{v_{r,0}}-{k}_f\right){w}_f^{\ast }+\frac{k_r}{v_{r,0}} $$
(20)
Equation (20) suggests that the shift of the \( {b}^{\ast }-{w}_f^{\ast } \) line results from the change in the respiratory flux (note the intercept, \( \frac{k_r}{v_{r,0}} \)). In E. coli, the PP pathway and the TCA cycle are two major sources for the production of NADPH [40]. At a given growth rate, the amount of NADPH needed for cell growth is fixed based on the mass and energy balance. As uPPP% increases, more carbon is predicted to enter the PP pathway. In the model simulation, an increase in the amount of NADPH produced via PP pathway would force a drop of the flux into the TCA cycle in order to maintain the constant total production rate of NADPH. The reduction in the TCA flux in turn manifests in a lower vr. In the overflow region, Eq. (6) is bounded by the equal sign. As shown earlier, the level of uPPP% has a negligible impact (when nominal energy demand is adopted) on the \( {w}_r^{\ast }-{w}_f^{\ast } \) line. Also recall that the relation between the rate of acetate excretion and steady state growth rate is fixed by the experimentally measured growth data. With all the other quantities (\( {w}_f^{\ast } \), vf, \( {w}_r^{\ast } \) and λ) fixed in Eq. (6), the drop in vr due to the increase in uPPP% will necessarily be accompanied by an increase in b∗.
Between different strains, b∗ varies significantly. In particular, b∗ for MG is remarkably larger than that of NCM and ML (as presented in Additional file 2: Figure S1). This disparity can again be explained by Eq. (6). For a certain value of \( {w}_f^{\ast } \), \( {w}_r^{\ast } \) is rather similar among different strains (with nominal energy demand) as shown by Fig. 2. In the overflow region, the respiration flux vr of MG is much smaller than the others (see Fig. 1), which thus leads to a lower value of the \( {w}_r^{\ast }{v}_r \) term for MG than NCM and ML. As the value of the \( {w}_f^{\ast }{v}_f \) term (for any selected value of \( {w}_r^{\ast } \)) is similar between these strains, due to their similarity in the relationship between \( {w}_r^{\ast } \) and \( {w}_f^{\ast } \), the value of the remaining term on the left-hand side of Eq. (6), b∗λ, must be higher for MG than for the other two strains. On the other hand, in the overflow region and at a same acetate excretion rate vf, the growth rate of MG has been shown to be much lower than that of NCM and ML. Now, a higher value of b∗λ coupled with a lower value of λ will undoubtedly lead to a higher value of b∗for MG, compared to the other two strains.
The above mathematical explanation in fact coincides with the known biological fact that the inverse of b∗ is proportional to the rate of protein synthesis [31]: the slower the rate of protein synthesis, the higher the value of b∗. Thus for the slow-growing strain MG, it is expected to have a higher value of b∗compared to the fast-growing strains NCM and ML.
Predicted evolution of PP pathway flux
The results presented above show rather moderate impact of the upper limit of PP pathway ratio (uPPP%) on the linear interdependency of the proteomic cost parameters. With an interest in the FBA solution of the flux distribution in PP pathway (at different growth rates), simulation results were recorded for three strains with uPPP%set to 35%; other uPPP% levels displayed a similar trend (as presented in Additional file 2: Figures S4 and S5). Flux variability analysis (FVA) [43] was performed to confirm that the trend of PPP% presented here was unique.
In general, PPP% gradually increases with the growth rate. Two turning points can be observed, which divide the whole curve into three distinct phases (Fig. 4a). A close inspection of the model simulation revealed that the variation of the predicted PP pathway ratio was co-related particularly with three fluxes, namely NAD transhydrogenase (NADTRHD), transketolase (TKT2) and NADP transhydrogenase (THD2).
In phase I, only NADTRHD is active, with zero fluxes for both TKT2 and THD2. The enzymatic reaction NADTRHD functions to convert NADPH into NADH. Thus in phase I, it is likely that the amount of NADPH produced exceeds the required amount for biosynthesis; NADTRHD is thus activated to consume the surplus NADPH.
In phase II, an on/off swap occurs between NADTRHD and TKT2 while THD2 still remains silent. We infer that in this phase, NADPH produced satisfies the demand, but the amount of carbon flowing into the PP pathway surpasses the rate of the carbon withdrawal (for the synthesis of biomass precursors). Therefore, TKT2 is activated to direct the extra amount of four-carbon and five-carbon compounds back to the glycolysis.
In phase III, THD2 is finally switched on and becomes significantly active in the high-growth-rates region. TKT2 increases progressively while NADTRHD remains silent. It is presumed that in this phase, as the growth rate becomes higher, more NADPH is required for biomass synthesis. NADP transhydrogenases (THD2) is activated to produce NADPH needed in rapid growth. The surplus carbon flux in the PP pathway, which might result from the high glucose uptake rate at a high growth rate, is directed back to glycolysis via TKT2.
It would be desirable to verify the theoretical prediction of the evolution of PPP% with experimental measurements, which unfortunately have not been widely reported in the literature. Nevertheless, Fig. 4b shows a comparison with one set of experimental observations available [42], which suggests a good degree of qualitative similarity.
Adjusting cellular energy demand improves the prediction of biomass yield
Although combining the PAT constraint with the core model succeeded in predicting the rates of acetate production, the accuracy in biomass yield varied and was especially unsatisfactory for ML strain (Fig. 5). A similar deficiency in yield prediction was also reported in [28]. Focusing on the yield, two features can be observed: (i) in the overflow region, for a fixed growth rate (associated with an acetate excretion rate) the biomass yield for ML is higher than MG and NCM; and (ii) the rate of the drop in yield (i.e. the slope) of ML is sharper than the other two strains.
Intuitively, feature (i) suggests that in ML, the amount of energy required per unit mass of biomass formation should be less than NCM or MG. Therefore we collected the growth data of ML and remodelled the cellular energy demand (see Methods).
It is worth noting that for ML, the negative value of M (Table 1) clearly indicates a constrained applicable range of the maintenance parameters, i.e. valid only within the overflow region. As growth rate decreases, if M stays unchanged, the overall energy consumption (Eq. (16)) will drop to a negative value, which is clearly not biologically feasible. This then implies a certain degree of nonlinearity in the global relationship between (total or maintenance) energy requirement and growth rate. Such proposition was previously referred to as “varied non-growth-associated maintenance” [3]. Non-linearity in energy consumption manifesting before and after the onset of the overflow metabolism has also been observed and discussed in a recent work [44].
Model prediction of biomass yield with adjusted energy demand
We first re-estimated the set of proteomic cost parameters for ML with the adjusted energy demand (see Table 1 and Methods). Applying updated values of \( {w}_f^{\ast } \), \( {w}_r^{\ast } \) and b∗ together with the adjusted maintenance energy, our model is now able to effectively capture the unique trend of biomass yield for ML, without any compromise in the accuracy of predicting acetate excretion (Fig. 6). Simulation results for ML with adjusted energy demand are referred to as “ML-new”.
It is worth noting that, our model also succeeds in matching the elevated reduction in the yield of ML in the overflow region as the growth rate increases. This captured trend appears to originate from the low energy demand of ML. Approximately, the yield reduction rate can be considered as being proportional to the ratio of the increase in the acetate excretion (ac2 − ac1) and the increase in glucose uptake rate (glc2 − glc1), while the growth rate rises from λ1 to λ2:
$$ yield\ reduction\ rate\propto \frac{ac_2-{ac}_1}{glc_2-{glc}_1}, for\ {\lambda}_1\to {\lambda}_2\ \left({\lambda}_2>{\lambda}_1\right) $$
(21)
NCM and ML exhibit similar acetate excretion rates, hence a similar value in “ac2 − ac1”. However, the energy demand per unit growth of ML is much lower than NCM, which means that with a similar increase in acetate production, the increase in substrate intake (i.e. glc2 − glc1) for ML will be lower than NCM to achieve a given increment in the growth rate. According to Eq. (21), the yield reduction rate of ML will thus be higher than NCM.
Impact of the adjusted energy demand on \( {w}_r^{\ast }-{w}_f^{\ast } \) and \( {b}^{\ast }-{w}_f^{\ast } \) relationships
To investigate the impact of the change in cellular energy demand on the linear relationships of \( {w}_f^{\ast } \), \( {w}_r^{\ast } \) and b∗ of ML-new, we recalculated constants kf, vf, 0, kr and vr, 0 at different uPPP% values (25, 35 and 40%) to update the linear equations describing \( {w}_r^{\ast }-{w}_f^{\ast } \) line and \( {b}^{\ast }-{w}_f^{\ast } \) (Eqs. (11) and (12)). The resulting \( {w}_r^{\ast }-{w}_f^{\ast } \) lines for ML-new are plotted in Fig. 2, together with the results obtained earlier for MG/NCM/ML with nominal energy demand.
The switch to the adjusted energy demand makes the \( {w}_r^{\ast } \) value for ML-new much higher than that of ML-nom, the latter being rather close to those of the MG-nom and NCM-nom. This implies that the adjustment of the energy demand of ML leads to an enlarged gap in the proteomic efficiency between respiration and fermentation.
The similarity among MG-/NCM−/ML-nom has already been discussed in the previous section. Here we mainly focus on the discrepancy with ML-new. We found that both the slope and intercept of \( {w}_r^{\ast }-{w}_f^{\ast } \) line for ML-new are about three times larger than ML-nom (as shown in Additional file 1: Table S4). The dramatic changes in the slope and intercept of ML-new predominantly result from the reduction in the respiratory flux vr when applying the adjusted energy demand (see Fig. 7 and Additional file 2: Figure S3).
The link between the drop in vr and the increase in \( {w}_r^{\ast } \) has been discussed in the previous section. The results presented herein indicate that it is the energy demand that plays a major role in distinguishing the \( {w}_r^{\ast }-{w}_f^{\ast } \) relationship between different strains, not the uPPP% or the acetate excretion pattern.
Applying the adjusted energy demand also has an impact on the relationship between b∗ and \( {w}_f^{\ast } \). As shown in Fig. 8, the \( {b}^{\ast }-{w}_f^{\ast } \) lines are significantly right-shifted when the model is changed from ML-nom to ML-new (i.e. red lines are located in a much right area than yellow lines). Given the identical pattern of acetate excretion between ML-nom and ML-new (as both predicted the same set of experimental data), the amount of energy produced through fermentation remains unchanged. For ML-new, as the energy demand per unit of growth is much lower than that of the nominal strain, the respiratory flux vr must decrease significantly to avoid energy overproduction, as confirmed in Fig. 7. Although the value of \( {w}_r^{\ast } \) for ML-new is higher than that for ML-nom (for a given value of \( {w}_f^{\ast } \), Fig. 2), the value of the product \( {w}_r^{\ast }{v}_r \) for ML-new still becomes lower (as the increase in \( {w}_r^{\ast } \) is not able to compensate for the sharp drop in vr). With no change in \( {w}_f^{\ast }{v}_f \) between ML-new and ML-nom (for a given \( {w}_f^{\ast } \)), Eq. (6) again dictates b∗ to become higher for ML-new than ML-nom, hence the right-shifting of the \( {b}^{\ast }-{w}_f^{\ast } \) lines.
In the case of ML-nom, it was shown earlier in Fig. 3 that the increase in uPPP% would lead to a reduction of vr, which in turn would lead to an increase in b∗ or right-shifting of the \( {b}^{\ast }-{w}_f^{\ast } \) line. Now for ML-new, an enlarged gap between the \( {b}^{\ast }-{w}_f^{\ast } \) lines at different uPPP% levels is observed compared to the case of ML-nom. This implies that the effect of vr reduction due to the increase in uPPP% is more pronounced with the adjusted energy demand.