Influence of protein biosynthetic rates on pathway regulation
In a previous work we showed that the protein biosynthetic rate of an organism has a strong influence on activation strategies of metabolic pathways [12]. The protein biosynthetic rate of an organism corresponds to the rate at which proteins can be produced. As we’ve shown in a previous work, the genomic copy number of ribosomal RNAs is strongly associated to the protein biosynthetic capacity of an organism [12]. Therefore, we analyzed optimal targets of pathway control with an additional constraint on the rate of change of enzyme concentrations for different protein biosynthetic rates (see Additional file 1: Text S2 for the problem formulation). We analyzed the regulatory effort in a linear pathway (without feedback inhibition) for three different protein biosynthetic rates across 150 models with random kinetic parameters and dilution values. We observed that a high protein biosynthetic rate leads to a decrease of the regulatory effort at the first reaction of the pathway (Wilcoxon test p-value =1.27·10−5 between regulatory effort for m=0.06 and m=0.15) and a concomitant increase at the terminal step of the pathway (Wilcoxon test p-value <10−16, between regulatory effort for m=0.06 and m=0.15, Figure 1A). This observation also holds when considering differences in the time-scales between metabolite, transcriptional and growth dynamics (Additional file 1: Figure S7).
Thus, the protein biosynthetic rate has an influence on the optimal regulatory effort at each pathway position. By decreasing protein biosynthetic rates, we limit the amount of change in the individual enzymes while there is no constraint on the amount of change in metabolite concentrations. In consequence, in relative terms the effects of changes in protein concentrations on metabolite concentrations propagate faster through the network. Hence, also the control of the first enzyme on the flux through the entire pathway becomes more immediate. In contrast, if protein concentrations can be adapted more rapidly, also a control of the terminal enzymes of pathways is of advantage since adjusting the concentration of the terminal enzyme of a pathway allows for an immediate change of pathway output while for slow changes in protein concentrations a part of this function can be taken over by the initial enzyme.
We investigated whether we could observe this pattern of regulation in vivo. To this end, we analysed the regulatory effort at different pathway positions across several hundred prokaryotes with different protein biosynthetic capacities. As a proxy for the regulatory effort that is exerted on each enzyme, we used the length of non-coding regions upstream of each gene (see Methods).
We ordered organisms according to the copy number of ribosomal RNAs in their genome and grouped them into organisms with low and high protein biosynthetic rates correspondingly (lower and upper 50% of organisms). Subsequently, we determined the average relative length of promoter regions for the initial as well as terminal enzymes in sparsely regulated metabolic pathways (Figure 1B). In a direct comparison of relative promoter lengths between organisms with low and high protein biosynthetic rates we found a significant decrease at the beginning of pathways (Wilcoxon test p-value =1.2·10−2) and a significant increase at the end of pathways (Wilcoxon test p-value =3.56·10−5). We performed this analysis in more detail for two pathways amongst the transcriptionally sparsely regulated pathways in the largest number of organisms: tryptophan biosynthesis and protoheme IX biosynthesis, an intermediate of heme biosynthesis. In both cases we observed a decrease in the regulatory effort targeting the first reaction and an increase in the last reaction of the corresponding pathways for organisms with faster protein biosynthetic rates (Figure 1C). Thus, as predicted by the optimization, there is a shift in the regulatory effort from the first to the terminal enzyme with an increasing protein synthesis rate.
These results also provide an explanation why we observed that the increase in the frequency of transcriptional regulatory interactions in pathways in E. coli is stronger at the end of metabolic pathways than at the beginning [13], since E. coli has a high copy-number of rRNAs in comparison to other prokaryotes.
Feedback inhibition in linear pathways
A frequently encountered mechanism in the control of metabolic pathways is a feedback inhibition of the initial enzyme by the product [34-36]. To study this mechanism, the problem was modified by introducing a competitive inhibition of the first enzyme e
1 of the pathway by the product P (Figure 2):
$$ \upsilon_{1}(t)=\frac{k_{cat,1}\cdot e_{1}(t)\cdot s(t)}{s(t)+K_{m}\left(1+\frac{p(t)}{k_{r,1}}\right)} $$
((5))
with k
r,1 corresponding to the strength of the feedback inhibition. For increasing values of k
r,1 inhibition is weaker and for decreasing values it is stronger.
In a first step, we analyzed how the introduction of a feedback inhibition influences the individual components of the objective function, the regulatory effort as well as the initial enzyme concentrations. We performed these comparisons for two different values of the weighting factor σ using unit inhibitory constants (Figure 3).
We did not observe a significant change in the objective function values after introduction of the inhibition across both weights for initial enzyme concentrations (Figure 3A). The contribution of the regulatory component J
reg
(deviation from the initial enzyme concentrations) was significantly decreased across all cases. This was mostly marked for a low weight of initial enzyme concentrations (Wilcoxon test p-value =8.92·10−11) while it was not as strong for a high weight of initial enzyme concentrations (Wilcoxon test p-value =2.9·10−2). Though we observed a tendency of the initial concentrations of enzymes to increase with introduction of the feedback inhibition, this increase was not significant (Wilcoxon test p-value >0.1 for each case).
Analysing changes in the regulatory effort targeting individual enzymes (Figure 2), we found a strong decrease in the first enzyme. This change was strongest (Wilcoxon test p-value <10−16) for a low weight of initial enzyme concentrations. In consequence, particularly for pathways with lowly abundant proteins (low σ-values) the introduction of a feedback inhibition appears to relieve the requirement of a control of the first enzyme. Thus, due to the presence of the feedback inhibition, the flux through the entire pathway can be controlled through transcriptionally regulating the terminal step of the pathway. These results substantiate the observation that flux through a pathway can be controlled much more precisely through a regulation of the terminal enzyme. This is in contrast to the classical biochemical picture of pathway control in which the first enzyme has been considered the most relevant [35]. However, please note that since our optimization approach focuses on optimal responses to changes in product consumption while assuming a constant supply of the substrate of the pathway, the relevance of the individual enzymes might also differ if we consider changes in substrate concentrations.
For a high weight of initial enzyme concentrations the reduction in the regulatory effort was reduced but still significant (Wilcoxon test p-value =2.8·10−4). Thus, while protein abundance has an influence on the strength of the reduction, it is significant across all the considered cases. This is in line with previous observations that the utilization of post-translational regulation is not influenced by the abundance of proteins [13].
As discussed above, the introduction of a feedback inhibition reduces the efficiency of the first enzyme. Thus, higher concentrations of e
1 are required to achieve the same flux (cf. Additional file 1: Figure S10). In consequence, also initial enzyme concentrations are increased. To further investigate this effect, we repeated the optimization with different values of the inhibition constant k
r,1 between [0, 100]. We observed that an increase of k
r,1 had a strong effect on the initial concentration of e
1 (Figure 4A). However, with an increasing value of k
r,1 also the strength of the inhibition is reduced and thus also its control over the flux through the first enzyme. In consequence, if k
r,1 is larger than a specific threshold value, the first enzyme is again under transcriptional control (Figure 4B). These two opposing trends result in the optimality of a specific value of k
r,1 at which the inhibition is still strong enough to exert a regulatory effect on the first enzyme, while the costly increase of the concentration of e
1 to maintain pathway flux is minimized (Figure 4A).
Optimal regulatory programs for complex pathway topologies
Since metabolism often involves more complex topologies than the simple linear pathway considered above, we investigated optimal regulatory programs in two more complex pathway topologies: a converging pathway leading from two substrates to a product and a diverging pathway producing two distinct products from a single substrate (See Additional file 1: Text S3 for the problems formulations).
Optimal regulatory programs in pathways with a converging reaction
In a first setup, we considered a pathway in which two substrates are converted into a common product which is drained through ν
growth
(Figure 5A). For the individual steps, we assumed irreversible Michaelis-Menten-kinetics as above.
The regulatory effort in different pathway positions is displayed in Figure 5A. While most of the regulation was observed in the initial and terminal steps of the pathway, there was almost no regulation at intermediate positions (e
2, e
4, e
5).
These results show that in the case of a metabolic pathway with two convergent branches, it is still sufficient to control both the initial and the terminal steps of the pathway and there is no regulation around the converging step. Thus, the flux through the entire pathway, also after the branch, can be controlled by the initial enzymes of the individual pathway branches.
To verify the prediction that there are no differences in the regulatory effort around a convergent branch, we analyzed the average length of promoter regions of proteins catalyzing the corresponding reactions across our collection of prokaryotes. In confirmation of the optimization results, we did not observe a change in the length of promoter regions between reactions prior to a converging reaction and the converging reaction itself (Figure 5C, Wilcoxon test p-value =0.29).
Optimal regulatory programs in pathways with a divergent branch
In a second step, we analyzed regulatory programs to control a metabolic pathway that diverges into two distinct branches. We assumed that the products of the individual branches are drained with two different rates, ν
g1 and ν
g2, to account for potential differences in their production, for instance, when concentrations need to be adapted independently. Initially, we considered a pathway with eight reactions with irreversible Michaelis-Menten kinetics (Figure 5B).
We found that apart from a regulation in the initial and terminal reactions, also frequently the enzymes after the pathway branch (e
3, e
4) were regulated (Figure 5B). In contrast, upstream of the branch, we did not observe much regulation. Hence, a regulation after the branch appears to provide a better control of flux through the entire pathway than before the branch. This has important consequences on pathways in which the product of the pathway is substrate to further pathways since, in principle, no transcriptional control of the terminal step of the pathway would be required in such a case.
To test the predictions of the optimization, we analyzed the relative length of promoter regions of enzymes catalyzing reactions before and after branching reactions in the metabolic networks of our prokaryote collection. We found that in sparsely regulated metabolic pathways there was a significant increase in the length of promoter regions of enzymes after the branching reaction compared to the preceding reactions (Figure 5C, Wilcox test p-value =6.96·10−11). A specific example for the regulatory effort before and in a set of branching reactions in adenosine nucleotide biosynthesis is shown in Figure 5D.
Feedback inhibition over different positions of the pathway
Since we did not observe any regulation in the intermediate enzymes, we considered the impact of feedback inhibition in a reduced network in which e
7 and e
8 were removed (Figure 6A). Three different cases of feedback inhibition were considered for unit inhibitory constants (Figure 6A): 1) an inhibition of the initial step of a pathway by the two products (panel 2 in Figure 6A), 2) an inhibition of the branching enzymes e
3 and e
4 by the products P
1 and P
2, respectively, (panel 3 in Figure 6A) and 3) a combination of the two previous cases (panel 4 in Figure 6A) which corresponds to a nested feedback inhibition [37].
In general, we observed a decrease in the regulatory effort for enzymes targeted by feedback inhibition (Figure 6B). In the case of a feedback inhibition from the products of a pathway on the first step of the pathway, we observed a drastic decrease in the regulatory effort at this position. This decrease was not as strong when the products of the pathway additionally influenced the reactions after the branchpoint (e
3 and e
4).
The introduction of different types of feedback inhibition reduced the overall regulatory effort that is required to control the flux through the pathway across all different cases (Figure 6C). We observed the strongest decrease in the regulatory effort required for the case in which the products inhibited the initial reaction as well as the reaction after the branching point. In consequence, in principle a transcriptional control of the terminal steps of the individual pathway branches would be sufficient for a full control of the flux through the pathway. Thus, the introduction of feedback inhibition allows the reduction of the required number of transcriptional regulatory control points from five to two. The optimality of this pattern of feedback inhibition is exemplified by its implementation in several pathways in E. coli metabolism including aromatic amino acid biosynthesis, the combined metabolism of lysine, methionine as well as threonine, branched chain amino acid biosynthesis and purine biosynthesis (Figure 6D).
Post-translational regulation reduces the transcriptional regulatory effort targeted at enzymes
As our optimization results have shown, post-translational regulation in general reduces the regulatory effort that is required to control the flux through a metabolic pathway. To test this assumption, we investigated the association between the occurrence of post-translational regulation and the length of promoter regions. As reference for post-translational regulation across our organism set, we used the dbPTM data base [33] that contains a large number of experimentally verified and predicted sites of post-translational modifications across all domains of life. We used these protein modifications as a reference for post-translational regulation since large-scale information about feedback inhibition in metabolism is only available for very few organisms.
For transcriptionally sparsely regulated metabolic pathways, we compared how the presence of post-translational regulation influenced promoter lengths at different pathway positions (Figure 7). We observed for all pathway positions that promoter lengths were significantly shorter if an enzyme is post-translationally regulated (Wilcoxon test for initial reactions: p-val =8.6·10−8, intermediate reactions: p-val ≤2.2·10−16, terminal reactions: p-val =1.9·10−3). The relative difference in promoter lengths was largest for initial reactions. Thus, the in vivo validation confirmed that post-translational regulation reduced the amount of regulatory effort targeted at an enzyme. Beyond the consideration of optimal programs for pathway regulation this result also represents a very interesting aspect considering that it is usually assumed that post-translational and transcriptional regulation act on completely different time-scales [38]. Our optimization predicts and the validation shows that despite this separation of time-scales, both types of regulation appear to be interchangeable to some extent.