Dynamic optimisation identifies optimal programs for pathway regulation in prokaryotes. - Supplementary Information -
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1 Dynamic optimisation identifies optimal programs for pathway regulation in prokaryotes - Supplementary Information - Martin Bartl a, Martin Kötzing a,b, Stefan Schuster c, Pu Li a, Christoph Kaleta b a Department of Simulation and Optimal Processes, Ilmenau University of Technology, Ilmenau, Germany; b Research Group Theoretical Systems Biology, Friedrich Schiller University, Jena, Germany; c Department of Bioinformatics, Friedrich Schiller University, Jena, Germany Supplementary Figure S1. Time course of pathway activation with unit kinetic parameters. The individual enzyme synthesis rates are d j,max = in a, d j,max =0.005 in b, d j,max =0.01 in c with the free protein synthesis capacity d max =0.01. For each case, the optimal enzyme profiles, growth rate and the corresponding metabolite profiles are shown for the entire time span. The regulatory strategy can be separated into time phases with an activation of the pathway followed by constant growth (optimal operation for the active state of the pathway). Due to a defined time horizon, (here t f = 1000) which is required for the optimisation, there is an artificial drastic increase in the dilution of the product toward the end of the time-course to drain product as well as intermediates from the pathway. 1
2 Supplementary Figure S2. Influence of protein abundance on the order of activation of enzymes in a pathway. Position of each enzyme in the activation sequence for 100 randomly drawn kinetic parameter sets for different individual enzyme synthesis rates (pathway as in the main document). Results are sorted vertically for different enzyme synthesis rates and horizontally for the different enzymes. The x-axis denotes the position in the activation sequence defined as the rank of the activation time of the particular enzyme in the ordered list of activation times for this parameter set. The y-axis indicates the abundance of each enzyme relative to the average abundance of enzymes for each particular set of kinetic parameters. 2
3 Supplementary Figure S3. Time course of pathway activation with unit kinetic parameters for a converging pathway. a We analyzed activation strategies for a converging pathway with the free protein synthesis capacity d max =0.01. Based on the constraints for the optimal pathway activation problem from the main article, we obtained the optimal regulatory strategy shown in b. We observed a sequential activation of enzymes along the individual branches of the pathway ([e 1,e 3,e 5 ] and [e 2,e 4,e 6 ], respectively) with a subsequent sequential activation of the remaining pathway (e 7 -e 11 ). Thus, even for a converging metabolic pathway we observe a sequential activation of enzymes along the individual sequences of enzymes [e 1,e 3,e 5,e 7,e 8,e 9,e 10,e 11 ] as well as [e 2,e 4,e 6,e 7,e 8,e 9,e 10,e 11 ]. 3
4 Supplementary Figure S4. Protein abundances in the arginine biosynthetic pathway. Normalized protein abundance of E. coli grown on glucose minimal media were taken from 6. Abundance data is given as total abundance which is calculated as copy number of each protein multiplied with the respective mass of the protein. 4
5 Dataset All genes Non-metab. genes ρ p-value ρ p-value Reference CAIs (n=3762) x x10 49 Microarray glucose minimal medium (n=4200) x x (aerobic WT data) Protein abundance glucose x x minimal medium (n=756) Microarray glycerol minimal medium (n=3934) x (WT data) Supplementary Table S1 Spearman correlation between protein abundance and operon sizes. The number behind each data set indicates the number of proteins covered by the data set. Columns two through seven give the Spearman correlation and the p- value of the correlation for different protein sets. The last column indicates the source of the data. 5
6 Supplementary Note 1. General influence of protein abundance on operon size To test the general influence of protein abundance on operon size, we tested across all of the species of the MicroCyc collection whether we would find a correlation between the size of operons and the abundance of proteins contained within them (550 organisms with codon adaptation indices and operon information). For each organism we determined, for each operon, the average codon adaptation index of genes contained within them. Then we computed the Spearman correlation between the size of each operon and the average codon adaptation index of proteins contained within it. After correcting for multiple testing using the Benjamini-Yekutieli procedure 34 with a false-discovery-rate of 5%, we found a negative correlation between protein abundance and operon size in ten organisms and a positive correlation between protein abundance and operon size in 258 organisms. For the individual values for each species and gene set see Supplementary Data 1. As this analysis shows, tested across all operons, there is a trend for proteins with higher abundance to be arranged in larger operons for most species. Thus, this general trend is opposite to the predictions of our optimisation approach for individual metabolic pathways. Hence, the significant negative correlation between protein abundance and operon sizes that we find in many pathways is a strong indicator for the utilization of the different pathway activation strategies that we predict. Supplementary Note 2. Relationship between protein synthesis capacity, rrna copy numbers and number of protein coding genes A genomic feature that is a strong determinant of the protein synthesis capacity of an organism (or concentration of ribosomes) is the number of copies of the rrna genes contained in its genome. To demonstrate that there is a link between the number of rrna genes in the genome and ribosome concentration, we use two observations. First, there is a strong correlation between growth rate and ribosome concentration in E. coli 46. Second, we hypothesized that there is a strong correlation between growth rate and the total copynumber of rrna genes. To confirm this hypothesis, we analyzed the relationship between maximal growth rates reported in 47 with the copy-number of rrna genes representing the 5S, 16S and 23S ribosomal subunits as annotated in MicroCyc. Using data from 130 strains and controlling for growth temperature (also reported in 47 ) we found a Spearman correlation of 0.69 between growth rate and the copy number of rrna operons (pvalue= ). In consequence, the copy number of rrna operons can be used as a proxy for ribosomal concentration and for the free protein synthesis capacity of the cell. 6
7 If free protein synthesis capacity decreases with the number of protein coding genes, we would expect that there is a positive correlation between the number of protein coding genes and the copy-number of rrna operons (i.e. a higher number of ribosomes is required to produce the increased number of genes). This is confirmed by an analysis of the same 130 species for which we tested the relationship between ribosomal copynumber and growth rate: controlling for growth rate we found a Spearman correlation of 0.45 between ribosomal RNA copy-number and the number of protein coding genes of a species (p-value= ). Supplementary Note 3. Codon adaptation indices as proxies for protein abundance Since codon adaptation indices are affected by a large number of factors, we tested their reliability as estimator of protein abundance. To this end, we compared the correlation between measured absolute protein abundances and microarray data as well as codon adaptation indices for E. coli. For protein abundance 6 and microarray data from E. coli K12 48 grown on glucose-minimal medium, we found a Spearman correlation of ρ=0.515 (p-value=0, n=758) between codon adaptation indices 49 as well as protein abundance and a Spearman correlation of ρ=0.58 (p-value=0, n=758) between microarray data as well as protein abundance. Thus, protein abundances show a similar good correlation with microarray data and codon adaptation indices. Moreover, we repeated this analysis of protein-abundance dependent effects on operon sizes using several protein abundance and microarray data sets in E. coli. Based on protein abundance data, we determined for each operon the average abundance of proteins contained within them. For microarray data, we transformed the log-transformed expression signals to the linear scale by exponentiating them and computed the average of the transformed values across replicates. For each operon, we determined the average abundance of proteins contained within them for the protein abundance data and the average expression signal for microarray data. Then we computed the correlation of the size of operons with the average codon adaptation index, the average exponentiated microarray signal (2 data sets) and protein abundance individually. We performed these test for different sets of genes corresponding to 1) all annotated protein coding genes and 2) all protein-coding genes not annotated as metabolic genes. In agreement with the tests of the influence of protein abundances on operon sizes conducted across all species, we found that there is in general a significant positive correlation between protein abundance and operon sizes (Supplementary Table S1). 7
8 Supplementary Note 4. Comparability of codon adaptation indices across organisms To assess whether codon adaptation indices are comparable across organisms, we checked whether the average codon adaptation indices across organisms reflect the abundance of the products of each pathway in a typical bacterial cell. This analysis is based on the hypothesis that with an increasing abundance of the products of a pathway also the abundance of the enzymes catalyzing the corresponding reactions should increase. To this end, we used information about the biomass composition of wild-type E. coli cells provided in 51. In the Supplementary Material of that work, the molar amount of each biomass component per gram dry-weight of cells is given. Even though this biomass composition is specific for E. coli, we expect that it reflects the average molar fraction of the corresponding metabolites in a typical prokaryotic cell. For each of the 99 pathways in our analysis, we determined whether it yielded a product or an intermediate to a product that corresponds to a metabolite whose amount is given in the biomass composition data (29 pathways). If one pathway gave rise to a product that is the intermediate of several biomass metabolites, we summed up the corresponding molar amounts from the biomass composition. Subsequently, we determined, for each pathway for each organism, the average codon adaptation indices for the enzymes belonging to the pathway. Then we calculated the average for all these values across all organisms. In the comparison of the total molar amount of products of each pathway and the average codon adaptation indices of proteins of these pathways, we obtained a Spearman correlation of ρ=0.71 (p-value= , n=29). Thus, there is a very strong correlation between the average abundance of enzymes associated to a pathway across a large number of organisms measured by codon adaptation indices and the molar amount of the product of the pathway in a typical prokaryotic cell. This is a strong indicator that the codon adaptation indices we used can be indeed compared across species. The corresponding data is provided in Supplementary Information Dataset S1. Supplementary Note 5. Control for metabolism-independent effects on operon sizes By controlling for average non-metabolic operon size, we can exclude non-specific global effects on operon size for Hypotheses 1 and 2 in the analysis of the influence of individual enzyme synthesis rate and free protein synthesis capacity on operon size. This is due to the observation that the factors number of protein coding genes and rrna operon copy 8
9 number have a global impact across all the operons in a genome. However, average abundance of enzymes within a pathway has only local effects on the particular pathway. In order to exclude non-specific effects on Hypothesis 3, we can use results from Supplementary Note 1. There we found in an analysis across 550 organisms that operon sizes increase with protein abundance for most organisms (258 organisms with a significant positive correlation, ten organisms with a significant negative correlation). This is opposite to the predictions of Hypothesis 3. Hence, the different pathway activation strategies that we identified oppose the general trend of high-abundant proteins to be arranged in larger operons. In consequence, finding a negative correlation between protein abundance and operon sizes for a particular pathway is a strong indicator that these pathway activation strategies are indeed used. Supplementary Note 6. Control for phylogenetic bias An often encountered problem in analyses of traits across several species is that data points are not independent from each other for each species. Thus, traits in closely related species are similar to each other than traits in more distantly related species. Several approaches to deal with this problem have been proposed These approaches are all based on a phylogenetic tree across all of the species that are part of the analysis (additionally the tree needs to contain information about the evolutionary distance between species). The species contained within MicroCyc cover all prokaryotic species including Archaea and Bacteria. While some trees across all species have been reconstructed based on ribosomal sequences (c.f. 55 ), they lack resolution in some of the strains of the same species in our analysis such as distinct strains of E. coli. To exclude that some of the effects that we observe are due to particular subsets of closely related species, we tested whether the removal of distinct clades from our analysis impacts the conclusions we draw. To determine clades among the species that we considered in our analysis, we used the classification of species provided within NCBI Taxonomy ( Note that NCBI Taxonomy corresponds to a classification tree and thus no phylogenetic tree with proper evolutionary distances. We reconstructed the sub-tree of NCBI Taxonomy that contained all of the species from the MicroCyc-collection by removing all leaves of the tree (corresponding to individual species) that were not contained within the MicroCyc-collection and removing all internal nodes of the tree without a remaining leave corresponding to a species in MicroCyc. 9
10 Using this tree of species from the MicroCyc-collection as basis, we randomly deleted internal nodes from the tree (along with all child-nodes and leaves within the tree) until approximately two thirds of the species from the MicroCyc-collection remained. Based on this subset of species, we repeated our analysis of the three distinct hypotheses concerning genomic influences on operon sizes of pathways as well as the co-expression of high- and low-abundant enzymes. We repeated this procedure 1000 times and counted each time how often each of the hypotheses was accepted or rejected (using the same procedure as in the main manuscript). We only considered pathways occurring in at least 70 species. Then we determined for each of the 1000 validation runs whether any of the hypotheses was more often rejected than accepted. Among the 1000 validation runs, Hypothesis 1 was rejected more often than accepted in 21 cases (2.1% of all runs), Hypothesis 2 was rejected more often than accepted in one case (0.1% of all runs) and Hypothesis 3 was rejected more often than accepted in no case. We repeated the test also for the coexpression of high- and low-abundant enzymes with earlier or later enzymes of the same pathway 1000 times with a reduced organism set. Across the 1000 runs, we did not find a significant coexpression of low-abundant enzymes with later enzymes of the same pathway in 14 runs (1.4% of all runs). In all cases the average positional coexpression bias was larger with the actual operon structure if compared to a randomized operon structure (i.e. enzymes are more often coexpressed with later enzymes of the same pathway, indicated by the black bars in Fig. 5 of the main manuscript). High-abundant enzymes were not significantly more often coexpressed with earlier enzymes of the same pathway if compared with a randomized operon structure in 144 runs (14.4% of all runs). However, in all but seven cases (0.7% of all runs), the average positional coexpression bias with the actual operon structure was smaller than with a randomized operon structure (i.e. enzymes were more often coexpressed with earlier enzymes of the same pathway). Thus, even though the tendency of high-abundant enzymes to be more often coexpressed with earlier enzymes is not significant in all cases, the comparison of actual and randomized operon structure shows the predicted tendency in almost all cases. In summary, these results show that our results are robust against randomly removing clades from the species considered in our analysis. 10
11 Supplementary Note 7. Timing in the activation of the arginine biosynthetic pathway Our results show that well-ordered activation strategies like a sequential synthesis of the enzymes of pathways are not always optimal. These results can also explain inconsistencies in the timing of the activation of the arginine biosynthetic pathway reported earlier 19. In the analysis of the timing of the activation of the enzymes of the arginine operon, it was observed that ArgG, which catalyzes the seventh step in arginine biosynthesis, is activated prior to earlier enzymes of that pathway (cf. Fig. 3 in 19 and Supplementary Figure S4). Analyzing abundance data from E. coli grown in glucose minimal medium 6, we found that ArgG is indeed the most abundant enzyme of this pathway, which can explain why it is activated much earlier than the preceding steps of the pathway (Supplementary Figure S4). In their work, Zaslaver et al. 19 argued that this discrepancy might be due to the topology of the metabolic pathway since ArgG catalyzes the formation of argino-succinate from citrulline, a product of the preceding arginine biosynthetic pathway, and aspartate (Supplementary Figure S4). However, using our optimisation routine, we found that the optimal strategy for the activation of a pathway with unit kinetic parameters comprising two converging branches is a sequential activation of enzymes along the individual branches and a subsequent activation of the enzyme condensing the products of the pathway branches (see Supplementary Figure S3). Thus, the abundance of ArgG, rather than the topology of the pathway, can explain the early activation of this enzyme. Moreover, CarAB an enzyme which produces carbamoylphosphate from bicarbonate, is a very abundant enzyme and is activated much earlier than ArgE (cf. Fig. 3 in 19 and Supplementary Figure S4) even though the products of both enzymes are required at the same time for the next step of arginine biosynthesis catalyzed by ArgF and ArgI. Supplementary References 46. Bremer, H. & Dennis, P.P. (eds.) Modulation of chemical composition and other parameters of the cell by growth rate, Vol. 2. (American Society for Microbiology, Washington, DC, Georgetown; 1996). 47. Vieira-Silva, S. & Rocha, E.P. The systemic imprint of growth and its uses in ecological (meta)genomics. PLoS Genet 6, e (2010). 48. Cho, B.K., Knight, E.M., Barrett, C.L. & Palsson, B.O. Genome-wide analysis of Fis binding in Escherichia coli indicates a causative role for A-/AT-tracts. Genome Res 18, (2008). 11
12 49. Najafabadi, H.S., Lehmann, J. & Omidi, M. Error minimization explains the codon usage of highly expressed genes in Escherichia coli. Gene 387, (2007). 50. Conrad, T.M. et al. RNA polymerase mutants found through adaptive evolution reprogram Escherichia coli for optimal growth in minimal media. Proc Natl Acad Sci U S A 107, (2010). 51. Feist, A.M. et al. A genome-scale metabolic reconstruction for Escherichia coli K-12 MG1655 that accounts for 1260 ORFs and thermodynamic information. Mol Syst Biol 3, 121 (2007). 52. Felsenstein, J. Phylogenies and the comparative method. American Naturalist, 1-15 (1985). 53. Freckleton, R., Harvey, P. & Pagel, M. Phylogenetic analysis and comparative data: a test and review of evidence. The American Naturalist 160, (2002). 54. Pagel, M.D. A method for the analysis of comparative data. J. Theor. Biol. 156, (1992). 55. Munoz, R. et al. Release LTPs104 of the All-Species Living Tree. Systematic and applied microbiology 34, (2011). 12
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