Abstract

   Turnover numbers characterize a key property of enzymes, and their
   usage in constraint-based metabolic modeling is expected to increase
   the prediction accuracy of diverse cellular phenotypes. In vivo
   turnover numbers can be obtained by integrating reaction rate and
   enzyme abundance measurements from individual experiments. Yet, their
   contribution to improving predictions of condition-specific cellular
   phenotypes remains elusive. Here, we show that available in vitro and
   in vivo turnover numbers lead to poor prediction of condition-specific
   growth rates with protein-constrained models of Escherichia coli and
   Saccharomyces cerevisiae, particularly when protein abundances are
   considered. We demonstrate that correction of turnover numbers by
   simultaneous consideration of proteomics and physiological data leads
   to improved predictions of condition-specific growth rates. Moreover,
   the obtained estimates are more precise than corresponding in vitro
   turnover numbers. Therefore, our approach provides the means to correct
   turnover numbers and paves the way towards cataloguing kcatomes of
   other organisms.

   Subject terms: Biochemical reaction networks, Computational models,
   Metabolic engineering, Bioinformatics
     __________________________________________________________________

   The construction of protein-constrained genome-scale metabolic models
   depends on the integration of organism-specific enzyme turnover
   numbers. Here, the authors show that correction of turnover numbers by
   simultaneous consideration of proteomics and physiological data leads
   to improved predictions of condition-specific growth rates.

Introduction

   Genome-scale metabolic models (GEMs) together with advances in
   constrained-based modeling have led to an improved understanding of how
   cellular resources are used to fulfill different cellular
   tasks^[30]1–[31]3. Recent advances are largely propelled by the
   development of protein-constrained GEMs (pcGEMs) in which the catalytic
   capacities of individual enzymes are linked to the allocation of enzyme
   abundances^[32]4. Such models have led to more accurate predictions of
   maximum specific growth rates on different carbon sources^[33]5–[34]7,
   flux distributions^[35]7, and other complex phenotypes^[36]8 in
   Escherichia coli and Saccharomyces cerevisiae. However, the development
   of pcGEMs critically depends on the integration of organism-specific
   enzyme turnover numbers,
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   , comprising the kcatome of an organism^[37]9.

   Measuring the kcatome of an organism based on in vitro characterization
   is limited due to the impossibility to purify specific enzymes, lack of
   availability of substrates, and knowledge of required cofactors, such
   that their relevance for studies of in vivo phenotypes remains
   questionable^[38]10,[39]11. Proxies for in vivo turnover numbers also
   termed maximal apparent catalytic rates, can be estimated by combining
   constraint-based approaches for flux prediction with measurements of
   protein abundance under different growth conditions or genetic
   modifications^[40]12–[41]14. The results from this approach, which
   entails ranking condition-specific estimates that use individual data
   sets, have shown that the proxies for in vivo turnover numbers
   generally concur with in vitro
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values in E. coli^[42]12. However, applications with data from S.
   cerevisiae^[43]15 and A. thaliana^[44]16 indicated that these proxies
   for in vivo turnover numbers do not reflect in vitro measurements.
   Another approach to estimate the kcatome relies exclusively on the
   machine and deep learning methods that use a variety of features of
   enzymes (e.g. network-based, structure-based, and
   biochemical)^[45]17–[46]19, resulting in predictive models that can
   explain up to 70% of the variance in turnover numbers obtained in
   vitro.

   The estimates of turnover numbers are integrated into metabolic models
   by different constraint-based approaches that have been grouped into
   coarse-grained (e.g. MOMENT^[47]5, sMOMENT^[48]20, eMOMENT^[49]21, and
   GECKO^[50]7, which all result in the same feasible space in protein
   limited growth scenarios) and fine-grained (e.g. resource balance
   analysis^[51]1 and ME-models^[52]2,[53]3). Of these, GECKO^[54]7 has
   been adopted in several recent studies due to the elegantly structured
   formulation of the protein constraints. In addition, GECKO allows for
   the integration of protein contents and correction factors that account
   for the mass fraction of enzymes (
   [MATH: <mi>f</mi> :MATH]
   ) included in the model as well as the average in vivo saturation (
   [MATH: <mi>σ</mi> :MATH]
   ) of all enzymes, facilitating the development of condition-specific
   models. While data-driven estimation of in vivo turnover numbers
   improves the coverage of
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values in pcGEMs, the available estimates usually lead to
   over-constrained models when using the allocation of total protein
   mass, not considered in flux balance analysis (FBA)^[55]22,[56]23.

   Here, we propose PRESTO (for protein-abundance-based correction of
   turnover numbers), a scalable constraint-based approach to correct
   turnover numbers by matching predictions from pcGEMs with measurements
   of cellular phenotypes— simultaneously—over multiple conditions. As a
   constraint-based approach, PRESTO facilitates the investigation of the
   variability of the proposed corrections. We show that predictions of
   growth by pcGEMs of S. cerevisiae with turnover numbers corrected by
   PRESTO are more accurate than those based on the models that include
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values corrected based on a contending heuristic that relies on enzyme
   control coefficients^[57]22. We also demonstrate that the same
   conclusions hold when enzyme abundances are integrated into the E. coli
   pcGEM using PRESTO. Therefore, PRESTO paves the way to broaden the
   applicability of pcGEMs for organisms with biotechnological
   applications and to arrive at genotype-specific estimates of the
   kcatome.

Results

Protein-abundance-based correction of turnover numbers

   For a given data set of protein abundances over a set of conditions,
   the enzymes with turnover numbers in a pcGEM can be partitioned into
   three groups. For instance, a data set of protein abundances that was
   recently used to estimate in vivo turnover numbers in S.
   cerevisiae^[58]15 includes 45%, 41%, and 14% measured overall, at least
   one (but not all), and none of the 27 used conditions, respectively.
   Therefore, there is then different data support for correcting the
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values of these classes of proteins. PRESTO relies on solving a linear
   program that minimizes a weighted linear combination of the average
   relative error for predicted specific growth rates and the correction
   of the initial turnover numbers integrated into the pcGEM (Fig. [59]1,
   see the “Methods” section). It further employs K-fold cross-validation
   (here, K = 3) with 10 repetitions while ensuring a steady state and
   integrating protein constraints for proteins measured overall
   conditions (Fig. [60]1, see the “Methods” section). The training set of
   conditions is used to generate a single set of corrected in vitro
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values, by using the respective in vivo protein abundances. The
   resulting corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values are in turn used to determine the relative error of the
   predicted specific growth rate for each condition in the test set using
   flux balance analysis with the pcGEM, while only constraining the total
   protein content and measured uptake rates. The relative error of the
   predicted specific growth rate along with the sum of introduced
   corrections is lastly used to select the value for the tuning parameter
   [MATH: <mi>λ</mi> :MATH]
   in the objective function of PRESTO, as done in machine learning
   approaches that rely on regularization.

Fig. 1. Schematic overview of the PRESTO approach for k[cat] correction.

   [61]Fig. 1
   [62]Open in a new tab

   The approach uses a GECKO-formatted pcGEM^[63]7 containing turnover
   numbers from BRENDA^[64]48. Using available data from n experimental
   conditions,
   [MATH: <mi>n</mi> :MATH]
   condition-specific models are generated using nutrient uptake rates and
   protein contents. PRESTO then uses data on abundances for the enzymes
   measured across the
   [MATH: <mi>n</mi> :MATH]
   investigated conditions and solves a linear program that minimizes a
   weighted sum of two objectives, the relative error to measured specific
   growth rates and the sum of positive
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   corrections, δ. The optimal weighting factor, λ, which modulates the
   trade-off between the two objectives, is then determined by
   cross-validation (Tr: training set; Ts: test set), choosing the
   parameter, which is associated with the lowest average relative error.
   Using the optimal value for λ, PRESTO combines all models for the
   experimental conditions to find a
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   correction for each enzyme with measured abundance. Last, the precision
   of δ values is assessed by variability analysis as well as by sampling
   and corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values are validated by comparing them to values obtained from other
   approaches.

PRESTO outperforms a contending heuristic in S. cerevisiae

   To determine the performance of PRESTO and compare it to that of
   contending heuristics, we used a data set comprising protein abundances
   and exchange fluxes from 27 diverse conditions, as supported by the
   principal component analysis (Supplementary Fig. [65]1). Application of
   PRESTO with a pcGEM of S. cerevisiae with initial in vitro turnover
   numbers obtained from BRENDA resulted in a mean relative error of 0.68
   from the cross-validation procedure, yielding a correction of on
   average 213 turnover numbers (Supplementary Fig. [66]2a). For the S.
   cerevisiae pcGEM, we found a value of
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>7</mn></mrow></msup>
   :MATH]
   for the parameter λ in the PRESTO objective provides the optimal
   trade-off between both the relative error and the sum of introduced
   corrections (see the “Methods” section). Moreover, we observed a high
   overlap between the sets of proteins with corrected turnover numbers in
   the cross-validation (average Jaccard distance of 0.07 (Supplementary
   Fig. [67]2b, c)), suggesting that the integrated data from different
   conditions point to a specific subset of enzymes that need to be
   corrected to improve performance of growth prediction.

   Unlike PRESTO, GECKO implements a heuristic for the correction of
   turnover numbers that are based on the objective control coefficient
   calculated for each protein in a given condition (Supplementary
   Fig. [68]3)^[69]22. The control coefficient of a protein is determined
   by increasing the turnover number by 1000-fold and scoring the effect
   on the predicted specific growth rate. The proteins are then ranked in
   decreasing order of their control coefficients, and the turnover number
   of the first enzyme in the list is changed to the maximum value found
   in BRENDA for this enzyme across all organisms. This procedure is
   repeated with the remaining enzymes until the pcGEM predicts a growth
   rate that is at most 10% smaller than the measured specific growth rate
   for that condition or no additional
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   value that strongly constrains the solution can be found (Supplementary
   Fig. [70]3). This leads to condition-specific sets of corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   with large intersections or full containment over a considered order of
   conditions (Supplementary Fig. [71]4a).

   In contrast to this procedure, PRESTO corrects at once the turnover
   numbers of multiple enzymes that are measured in all investigated
   conditions by simultaneously leveraging the data from the different
   conditions, considerably reducing runtime and the number of solved
   problems. As a result, rather than deriving condition-specific
   corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values, which are difficult to use in making predictions for unseen
   scenarios or for building large-scale kinetic metabolic
   models^[72]24,[73]25, PRESTO results in a single set of corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values.

   We compared the performance of PRESTO with the heuristic implemented in
   GECKO in three modeling scenarios that consider: (i) only
   condition-specific total protein content, (ii) both total protein
   content and uptake constraints, and (iii) additional constraints from
   abundances of enzymes measured in all conditions (Fig. [74]2). For
   corrections of turnover number from PRESTO, we observed that the
   relative error spans the range from 0.15 to 0.88 in the least
   constrained scenario (i) (Fig. [75]2a) and from 0.69 to 0.98 in the
   most constrained scenario (iii) (Fig. [76]2c). In contrast, the
   relative error with the corrections of turnover numbers from the GECKO
   heuristic is in the range from 0.96 to 1.00 in scenario (iii)
   (Fig. [77]2c). In addition, in scenario (iii), the median relative
   error in the case of the GECKO heuristic for each condition is larger
   than the relative error of the PRESTO predicted specific growth rate
   (Fig. [78]2c). We observed that predictions from FBA, considering
   enzyme abundances, without a constraint on the total protein content,
   led to an average relative error of 0.70 with
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values corrected according to PRESTO and 0.99 with
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values corrected according to GECKO (Supplementary Table [79]1).

Fig. 2. Comparison of predicted growth of S. cerevisiae from pcGEMs with
k[cat] corrections from GECKO and PRESTO.

   [80]Fig. 2
   [81]Open in a new tab

   Condition-specific pcGEMs with corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values generated by the GECKO heuristic were used to predict the
   specific growth rate for each condition (n = 27, a, b). The boxplots
   indicate the distribution of the relative error resulting from each set
   of condition-specific corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values obtained from the GECKO heuristic. Relative prediction error
   from each set is indicated by a circle. The red diamonds show the
   relative error of the predicted specific growth rate from the PRESTO
   model
   [MATH:
   <mrow><mo>(</mo><mrow><mi>λ</mi><mo>=</mo><msup><mrow><mn>10</mn></mrow
   ><mrow><mo>−</mo><mn>7</mn></mrow></msup></mrow><mo>)</mo></mrow>
   :MATH]
   by using the single set of corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values in the respective pcGEM. a Only the measured total protein pool
   was used to constrain the solution and condition-specific uptake rates
   were bounded by 1000
   [MATH: <mfrac><mrow><mi mathvariant="normal">mmol</mi></mrow><mrow><mi
   mathvariant="normal">h</mi><mspace width="0.16em"></mspace><mi
   mathvariant="normal">gDW</mi></mrow></mfrac> :MATH]
   ; b measured uptake rates were also considered; c abundances of enzymes
   measured in all conditions were used as additional constraints. The
   compared pcGEMs in each condition (n = 19) used the same respective
   biomass reaction, GAM,
   [MATH: <mi>σ</mi> :MATH]
   , and
   [MATH: <msub><mrow><mi>P</mi></mrow><mrow><mi
   mathvariant="normal">tot</mi></mrow></msub> :MATH]
   values (see the “Methods” section). L: Lahtvee et al.^[82]34, D: Di
   Bartolomeo et al.^[83]37, Y20: Yu et al.^[84]35, Y21: Yu et al.^[85]36.
   Middle line and boxes in the box charts in panels a–c indicate the
   median and 25th and 75th percentiles, respectively. Outlier values
   (circles outside the whisker range) are more than 1.5× the
   interquartile range away from the top or bottom of the box, and
   whiskers connect the lower or upper quartiles with the non-outlier
   minimum or maximum. Source data are provided as a Source Data file.

   We also performed a sensitivity analysis by investigating a smaller
   value, of
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup>
   <mo>,</mo> :MATH]
   for the weighting factor λ used in the PRESTO objective. We found that
   when the weighting factor is
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup>
   :MATH]
   (at which the total corrections of the initial
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values plateaus), the relative errors from PRESTO cross-validation can
   be further decreased to 0.69 considering the constraint on the total
   protein content, with no effects on the other findings (Supplementary
   Fig. [86]2a). We also note that the relative error lies in the range
   from 0.35 to 0.80 over the considered weighting factors in the range
   from
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>14</mn></mrow></msup>
   :MATH]
   to
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup>
   :MATH]
   . Together, these results demonstrated that
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values corrected according to PRESTO provide better model performance
   than the values obtained by the contending heuristic in the case of S.
   cerevisiae in the scenarios where all available data are integrated
   into the model constraints.

PRESTO provides precise corrections of turnover numbers

   In the following, we investigated the precision of the corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values from the application of PRESTO to data and a pcGEM model of S.
   cerevisiae. To this end, we determined the range that the correction of
   the
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   value of each enzyme can take while fixing the relative error in
   specific growth rate and total corrections from the optimum of PRESTO
   (see the “Methods” section). Moreover, we complemented this analysis by
   sampling corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values that achieve the optimum of PRESTO with two values of the
   weighting factor λ of
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>7</mn></mrow></msup>
   :MATH]
   and
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup>
   :MATH]
   .

   In the case of the corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values for S. cerevisiae with a weighting factor of 10^−7, we found
   that the
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values with the largest corrections are more precisely determined
   (Supplementary Fig. [87]5). In addition, the sampled corrections per
   enzyme show an average Euclidean distance to the respective mean of
   4.88 s^−1, indicating that the corrected values are more precise than
   the values in BRENDA, exhibiting an average Euclidean distance of
   27.54 s^−1 to the mean per EC number (Supplementary Fig. [88]6).
   Importantly, while
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values with smaller correction showed larger variability, the 25 and 75
   percentiles of the sampled corrections for 42 enzymes are concentrated
   around those resulting from PRESTO. Repeating the analysis with a
   weighting factor of 10^−10 showed that the larger total corrections of
   the initial
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values resulted in also larger variability for the corrections over all
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   (Supplementary Fig. [89]7). Here, too, for 62 enzymes the 25 and 75
   percentiles of the sampled corrections are concentrated around those
   resulting from PRESTO. Therefore, we concluded that the corrections
   from PRESTO are precise and can be used in downstream analyses.

Pathways enrichment for corrected turnover numbers

   In pcGEMs generated by the GECKO toolbox^[90]7, turnover numbers are
   assigned to each of the enzymes in the GEM using a fuzzy matching
   algorithm. It takes into account the organism, substrate, and EC number
   of a BRENDA entry. When we investigated the magnitude of the turnover
   number correction dependent on the quality of the match between BRENDA
   entry and the corresponding enzyme, we found that
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values measured in S. cerevisiae were associated with smaller
   corrections than those from other organisms (Supplementary
   Fig. [91]8a).

   To check which metabolic processes are more likely to require
   correction of in vitro
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values, we next conducted an enrichment analysis based on the KEGG
   pathway terms linked to corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values (see the “Methods” section). The most prominent pathway in this
   analysis was the synthesis of secondary metabolites, particularly the
   synthesis of cofactors and terpenoids (Fig. [92]3a). However, several
   terms linked to central carbon metabolisms, such as the tricarboxylic
   acid cycle and oxidative phosphorylation, were also significantly
   enriched. Interestingly, amino acid synthesis was the only term linked
   to nitrogen metabolism that came up in this analysis, although many
   pathways of nitrogen metabolism were among the tested terms. This
   analysis suggested that particularly in vitro turnover numbers in
   carbon metabolism need to be corrected, due to the underestimation of
   in vitro assays.

Fig. 3. Comparison of enzymes with corrected k[cat] values by both GECKO and
PRESTO.

   [93]Fig. 3
   [94]Open in a new tab

   a KEGG Pathway terms significantly enriched in the set of enzymes
   corrected by PRESTO
   [MATH:
   <mrow><mo>(</mo><mrow><mi>λ</mi><mo>=</mo><msup><mrow><mn>10</mn></mrow
   ><mrow><mo>−</mo><mn>7</mn></mrow></msup></mrow><mo>)</mo></mrow>
   :MATH]
   in the S. cerevisiae pcGEM. The x-axis gives the number of corrected
   enzymes linked to the given term. The one-sided p-values were
   calculated using the hypergeometric density distribution and corrected
   for multiple hypothesis testing using the Benjamini–Hochberg
   procedure^[95]45. b Venn diagram showing the overlap of enzymes whose
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values were manually corrected^[96]7 (“Manual”), automatically
   corrected by the GECKO heuristic in any of the conditions (“GECKO”), or
   corrected by PRESTO (“PRESTO”). c Log-transformed
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values corrected using both the GECKO heuristic and PRESTO are not
   associated (Spearman correlation coefficient of 0.166, p-value = 0.45).
   Source data are provided as a Source Data file.

Comparison of turnover number corrections from GECKO

   Next, we aimed to identify the extent to which the corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values differ between PRESTO and the GECKO approach. To this end, we
   determined the intersection of enzymes with
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values corrected manually^[97]7, by PRESTO, and by the GECKO heuristic.
   For this comparison, we considered the union of all condition-specific
   corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values from the GECKO approach. With the weighing factor
   [MATH:
   <mi>λ</mi><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>7
   </mn></mrow></msup> :MATH]
   , PRESTO adapted the
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values of 48% of enzymes corrected by the GECKO heuristic (Fig. [98]3b,
   Supplementary Data [99]1). We did not find a significant Spearman
   correlation (
   [MATH: <msub><mrow><mi>ρ</mi></mrow><mrow><mi
   mathvariant="normal">S</mi></mrow></msub><mo>=</mo><mn>0.17</mn> :MATH]
   ,
   [MATH: <mi>P</mi><mo>=</mo><mn>0.45</mn> :MATH]
   ) between the log-transformed
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values in this intersection (Fig. [100]3c), owing to the different
   principles employed in the two procedures. To determine the pathways
   that comprise enzymes whose turnover number are corrected by GECKO and
   PRESTO, we next repeated the pathway enrichment analysis for the
   enzymes in the overlap. Among the significant terms, like in PRESTO, we
   again found 2-Oxocarboxylic acid, amino acid, and secondary metabolism
   to be enriched (Fig. [101]3a, S9). However, the more specific pathway
   terms were associated with pathways that are part of carbohydrate
   metabolism and aromatic amino acid metabolism corrected by both
   approaches (Supplementary Fig. [102]9, Supplementary Data [103]2). In
   addition, the intersection between enzymes with manually corrected
   values and those corrected by the GECKO heuristic is higher than with
   PRESTO. This is expected since the manual curation is partly aimed at
   correcting the most constraining turnover numbers^[104]7.

   We also compared the
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values adjusted by GECKO against estimates of in vivo
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values obtained by parsimonious FBA (pFBA) using the same proteomics
   data^[105]15 (Supplementary Fig. [106]10a, b). We confirmed the low
   correspondence (
   [MATH: <msub><mrow><mi>ρ</mi></mrow><mrow><mi
   mathvariant="normal">S</mi></mrow></msub><mo>=</mo><mn>0.23</mn> :MATH]
   ) between the
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values obtained from BRENDA, included in the GECKO model without manual
   modifications, and the in vivo
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   estimates. As expected, the correspondence of the estimated in vivo
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values to the turnover numbers corrected based on PRESTO was higher (
   [MATH: <msub><mrow><mi>ρ</mi></mrow><mrow><mi
   mathvariant="normal">S</mi></mrow></msub><mo>=</mo><mn>0.34</mn> :MATH]
   ). To investigate how these estimates perform as model parameters, we
   also generated a pcGEM in which BRENDA values were substituted by in
   vivo
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values from pFBA^[107]15, whenever available. In scenarios without
   enzyme abundance values, this model performed worse than that including
   the
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values corrected by PRESTO as well as the model combining the maximum
   of all condition-specific GECKO corrections (Supplementary
   Fig. [108]11a, b). In the enzyme abundance-constrained scenario, the
   model with in vivo turnover numbers estimated by pFBA performed
   slightly better than GECKO but still only achieved a minimum relative
   error of 0.93, which is larger than 0.69 resulting from PRESTO
   (Supplementary Fig. [109]11c). These results demonstrated the value of
   PRESTO in combining the genome-scale coverage of BRENDA with in vivo
   proteomics chemostat measurements to obtain less biased estimates of
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values.

PRESTO with protein-constrained model of E. coli metabolism

   To demonstrate the applicability of PRESTO across species, we deployed
   it with a pcGEM of E. coli (eciML1515)^[110]22,[111]26. To this end, we
   used a large dataset comprising 31 different growth
   conditions^[112]12,[113]27–[114]29. Due to the lack of data on nutrient
   exchange rates, the same GAM value (i.e., 75.55
   [MATH: <mfrac><mrow><mi mathvariant="normal">mmol</mi></mrow><mrow><mi
   mathvariant="normal">gDWh</mi></mrow></mfrac> :MATH]
   ) was used across all conditions. Similarly, we used the same value for
   total protein content since condition-specific measurements were not
   available (see the “Methods” section).

   By applying three-fold cross-validation, we found the optimal value for
   the λ parameter to be
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>5</mn></mrow></msup>
   :MATH]
   (Supplementary Fig. [115]12a). This value was associated with an
   average relative error of 1.95 (average over all λ: 3.32) and 73
   corrected turnover numbers, while on average 156
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values were corrected across all explored values for λ. On average, the
   Jaccard distance between cross-validation folds was 0.13 (Supplementary
   Fig. [116]12b), while the average Jaccard distance between unique sets
   of enzymes with corrected turnover numbers for each λ parameter was
   three-fold larger (0.4, Supplementary Fig. [117]12c). Thus, the
   corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values among cross-validation folds for each λ are more similar
   (maximum Jaccard distance of 0.29). Moreover, the union of the set of
   enzymes with corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values can remain similar over a range of chosen λ parameters up to
   four orders of magnitude (Supplementary Fig. [118]12c), demonstrating
   the robustness of the method.

   The performance of PRESTO was assessed and compared to GECKO using
   scenarios (i) and (iii) since no condition-specific uptake rates were
   available. With default uptake rates, the relative error for predicted
   growth ranged between 0.01 and 8.56 in the less constrained scenario
   (i) (Fig. [119]4a). Further, we obtained relative errors between 0.01
   and 0.88 for the more constrained scenario (iii), when using the
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values corrected by PRESTO (Fig. [120]4b). In contrast, when using the
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values from the GECKO approach, the relative error was in the range
   between 0.01 and the 4.89 for scenario (i) and between 0.89 and 0.99
   for scenario (iii). In this scenario, too, we observed that the
   relative error using
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values corrected by GECKO was consistently larger than the relative
   error resulting from the single set of corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values obtained by PRESTO (Fig. [121]4b).

Fig. 4. Comparison of predicted growth of E. coli from pcGEMs with k[cat]
corrections from GECKO and PRESTO.

   [122]Fig. 4
   [123]Open in a new tab

   Condition-specific pcGEMs with corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values generated by the GECKO heuristic were used to predict the
   specific growth rate for each condition (a: n = 31, b: n = 27). The
   boxplots indicate the distribution of the relative error resulting from
   each set of condition-specific corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values obtained from the GECKO heuristic. Relative prediction error
   from each set is indicated by a circle. The red diamonds show the
   relative errors of predicted specific growth rates from the PRESTO
   model
   [MATH:
   <mrow><mo>(</mo><mrow><mi>λ</mi><mo>=</mo><msup><mrow><mn>10</mn></mrow
   ><mrow><mo>−</mo><mn>5</mn></mrow></msup></mrow><mo>)</mo></mrow>
   :MATH]
   by using the single set of corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values in the respective pcGEM. a Only the measured total protein pool
   was used to constrain the solution and condition-specific uptake rates
   were bounded by 1000
   [MATH: <mfrac><mrow><mi mathvariant="normal">mmol</mi></mrow><mrow><mi
   mathvariant="normal">h</mi><mspace width="0.16em"></mspace><mi
   mathvariant="normal">gDW</mi></mrow></mfrac> :MATH]
   ; b abundances of enzymes measured in all conditions were used as
   additional constraints. Missing data points originate from the
   infeasibility of the respective models. The compared pcGEMs in each
   condition used the same respective biomass coefficients, GAM
   [MATH: <mi>σ</mi> :MATH]
   , and
   [MATH: <msub><mrow><mi>P</mi></mrow><mrow><mi
   mathvariant="normal">tot</mi></mrow></msub> :MATH]
   values (see the “Methods” section). P: Peebo et al.^[124]28, V:
   Valgepea et al.^[125]27, S: Schmidt et al.^[126]29. Middle line and
   boxes in the box charts in panels a and b indicate the median and 25th
   and 75th percentiles, respectively. Outlier values (circles outside the
   whisker range) are more than 1.5× the interquartile range away from the
   top or bottom of the box, and whiskers connect the lower or upper
   quartiles with the non-outlier minimum or maximum. Source data are
   provided as a Source Data file.

   Since we observed high relative errors in the less-constrained scenario
   (Fig. [127]4a), we added a second step to PRESTO, which introduces
   negative corrections that lead to the same relative errors with
   consideration of proteomics data. This is an optional step that a user
   can choose to perform in addition to the positive corrections (i.e.
   relaxation of turnover numbers), introduced in the first step
   (Supplementary Method [128]1). As a result, we found 170 negative
   corrections that reduced the relative error in scenario (i)
   (Supplementary Fig. [129]13).

   We do not perform a simultaneous search for positive and negative
   corrections because negative corrections can only reduce the relative
   error when the current
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values lead to an overprediction of growth, which is not the case when
   considering proteomics data. Therefore, no negative corrections are
   found if the absolute value of introduced positive and negative
   corrections are to be considered in a single step.

   Importantly, the aim of PRESTO is to correct turnover numbers, which
   represent upper limits on the catalytic efficiency of enzymes.
   Therefore, we can assume that in vitro turnover numbers that lead to
   underprediction of specific growth rates when paired with protein
   abundance data are too low. However, an overprediction of specific
   growth rates in the same scenario can be caused by thermodynamic,
   temperature effects, or in-vivo-specific effects. Thus, a reduction of
   in vitro turnover numbers results in average apparent catalytic rates
   for the considered conditions, rather than corrected turnover numbers.

   Considering the models with positive
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   corrections, the sum of corrections reached a plateau at
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup>
   :MATH]
   for the weighting factor
   [MATH: <mi>λ</mi> :MATH]
   in the PRESTO objective. We found that the relative cross-validation
   error at this value was 5.26, which is 2.7-fold larger than the
   relative error obtained using the optimal λ. Hence, allowing for more
   and larger corrections in PRESTO leads to a decrease in the overall
   relative error within the PRESTO program at the cost of highly biased
   parameters. The predictions with the highly biased parameters are worse
   in the test conditions and result in a larger specific growth rate when
   no enzyme abundance constraints are enforced. This observation is in
   line with the small number of corrections introduced by the GECKO
   approach, where only the pool constraint is considered. We conclude
   that the prediction performance of the eciML1515 model was improved by
   using turnover numbers corrected by PRESTO only when enzyme abundances
   are integrated.

   To assess the precision of the introduced
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   corrections, we performed variability analysis and sampling (see the
   “Methods” section) of the introduced corrections to the initial
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values for two values of the weighting factor
   [MATH: <mi>λ</mi> :MATH]
   , namely
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>5</mn></mrow></msup>
   :MATH]
   and
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup>
   :MATH]
   . We observed that the 25th and 75th percentiles enclose a narrow
   interval around the values resulting from PRESTO (Supplementary
   Fig. [130]14) and are thus not evenly distributed across the respective
   interval determined by the variability analysis. We further noted that
   here, the predictions of smaller δ are generally more precise than the
   large corrections (
   [MATH:
   <mi>δ</mi><mo>≥</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>50</mn></mro
   w></msub> :MATH]
   ), which span ~2 orders of magnitude (small δ (<p[50]): 1.83,
   Supplementary Fig. [131]14). However, we also observed that the
   precision decreased when more corrections were allowed in PRESTO. This
   further justified our choice for the optimal parameter λ, which results
   in a lower number of 73 corrections compared to 204 at
   [MATH:
   <mi>λ</mi><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>1
   1</mn></mrow></msup> :MATH]
   ; moreover, this value guarantees more precise estimates (Supplementary
   Fig. [132]17). In conclusion, the application of PRESTO is not limited
   to a single species but presents a versatile tool for the correction of
   turnover numbers across species.

   In contrast to the observations made in S. cerevisiae we found that a
   model parameterized with in vivo turnover numbers estimated by
   pFBA^[133]12 outperformed both PRESTO and GECKO in the modeling
   scenario where no enzyme abundance constraints are taken into account
   (Supplementary Fig. [134]11d). This is due to the fact, that pFBA, in
   contrast to PRESTO and GECKO, can generate estimates lower than the in
   vitro
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values, in turn leading to more accurate predictions. However, in the
   scenario with enzyme abundance constraints, PRESTO predicts specific
   growth rates closer to the experimental observation in 87% of the
   conditions (Supplementary Fig. [135]11e). Thus, in this scenario the
   integration of information from different modeling conditions achieved
   in PRESTO serves to obtain
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   value that performs better than the pFBA approach applied by^[136]12.

   Finally, we compare the resulting flux distributions and predicted
   protein abundances by models that are parameterized with
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values that were corrected using either GECKO or PRESTO. Overall, the
   feasible ranges (
   [MATH:
   <msup><mrow><mi>v</mi></mrow><mrow><mi>max</mi></mrow></msup><mo>−</mo>
   <msup><mrow><mi>v</mi></mrow><mrow><mi>min</mi></mrow></msup> :MATH]
   ) for both approaches resulted in Pearson correlation coefficients of
   0.985 across all conditions (Supplementary Method [137]2). The
   difference between both flux distributions is manifested in a smaller
   interquartile range in feasible ranges with PRESTO (Supplementary
   Fig. [138]15). More specifically, there are fewer reactions with highly
   constrained flux after introducing
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   corrections with PRESTO compared to GECKO. Moreover, we used the models
   that were corrected using GECKO and PRESTO to predict protein
   abundances (Supplementary Method [139]3), which were then compared to
   the measured proteomics data using Spearman correlation. Since PRESTO
   only considers abundances of proteins that were measured across all
   considered conditions, we computed the correlations for (1) the set of
   proteins that are measured across all conditions and (2) all protein
   abundances per condition. In the first scenario, the correlation was
   higher after correction with PRESTO in 70% of conditions, compared to
   the median correlation with GECKO and outperformed all GECKO pcGEMs in
   30% of conditions (Supplementary Fig. [140]16a). When all measured
   proteins were considered, PRESTO only performed better in 54% of
   conditions (Supplementary Fig. [141]16b). Similar to the predicted
   specific growth rate we also observed better performance for PRESTO
   models that were subjected to an additional
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   down correction step (Supplementary Method [142]1). However, we note
   that the reduced
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   cannot strictly be considered condition-independent
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values because there may exist physiological states where these enzymes
   may achieve the efficiency given by the original
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   value (see the “Discussion” section). Since PRESTO considers protein
   abundances for the correction, which is not the case for GECKO, we
   expected to find the increased prediction performance with PRESTO
   compared to GECKO; however, we still observe low overall predictability
   of protein abundances using the resulting models. Recently, a more
   sophisticated protein abundance prediction approach using pcGEMs was
   introduced that can be used to predict more reliable values and might
   further be improved by considering corrections introduced by
   PRESTO^[143]30.

   Interestingly, in contrast to S. cerevisiae, we did not observe larger
   corrections by PRESTO for organism-unspecific in vitro
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values (Supplementary Fig. [144]8b). For the condition-specific sets of
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   corrections introduced by GECKO we observe that all smaller sets were
   proper subsets of larger sets (Supplementary Fig. [145]4b) The low
   number of corrections introduced by GECKO leads to an overlap of only
   three (75%) enzymes whose
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values were also corrected by PRESTO (Supplementary Data [146]3,
   Supplementary Fig. [147]18a). These three enzymes catalyze reactions in
   three distinct metabolic pathways: Phosphoribosylformylglycinamidine
   synthase acts in the synthesis of purines, while serine
   acetyltransferase and NADP dependent Ketol-acid reductoisomerase are
   involved in the synthesis of sulfur amino acids and hydrophic amino
   acids, respectively (Supplementary Data [148]4). The pathway enrichment
   analysis for all PRESTO corrections at
   [MATH:
   <mi>λ</mi><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>5
   </mn></mrow></msup> :MATH]
   , indeed also identified amino acid and secondary metabolite synthesis
   as significantly enriched terms among the enzymes with corrected
   turnover numbers (Supplementary Fig. [149]18b). These results argue for
   a systematic underestimation of in vivo turnover numbers in the
   pathways in in vitro experiments, irrespective of the investigated
   organism. However, the lower-order KEGG pathway terms enriched in E.
   coli do not overlap with the ones found in S. cerevisiae. Here, fatty
   acid metabolism and the synthesis of hydrophobic amino acids are among
   the pathways requiring the correction of turnover numbers.

Robustness of turnover number corrections

   All of the approaches for estimation of in vivo turnover numbers rely
   on predicted (or estimated) fluxes and protein abundances from multiple
   conditions^[150]12,[151]14,[152]15, but have not investigated the
   robustness of the estimates to the number of conditions used.
   Therefore, next, we investigate the difference in the sets of enzymes
   with corrected turnover numbers and the concordance of their
   corrections when ten randomly sampled subcollections of M experimental
   conditions (M = 3, 5, 10, 15) were used instead of all experiments. The
   differences and concordance were quantified with respect to the
   estimates obtained by considering data from all available experiments
   using the Jaccard index and the Pearson correlation coefficient,
   respectively. In the case of S. cerevisiae, we found that the smallest
   Jaccard difference over 200 scenarios was 0.36, while for E. coli this
   was 0.41 (Supplementary Figs. [153]19 and [154]20). In addition, the
   Pearson correlation coefficient between the (log-transformed) corrected
   turnover numbers with consideration of all versus a subcollection of M
   experiments in S. cerevisiae ranged from 0.99 to 1.00 (for M = 15) to
   0.11 and 1.00 (for M = 3) (Supplementary Fig. [155]19). Repeating the
   analysis in the case of E. coli, we found that that the Pearson
   correlation coefficient ranged from 0.15 to 1.00 (for M = 15) to 0.14
   and 1.00 (for M = 3) (Supplementary Fig. [156]20). This is in line with
   the expectation that the corrections stabilize with an increasing
   number of experiments. Altogether, these findings pointed out the
   robustness of turnover number corrections derived from PRESTO with the
   number of available experiments.

Discussion

   Characterization of enzyme parameters that can inform models of
   reaction rates is key to expanding and further propelling the usage of
   metabolic models in diverse biotechnological applications. While the
   generation of pcGEMs has facilitated the integration of more
   biophysically relevant constraints, it necessitates access to estimates
   of turnover numbers as key enzyme parameters. We assessed the bias in
   the available in vitro and in vivo estimates of turnover numbers as the
   discrepancy between measured and predicted growth in the ultimate
   validation scenario when they are combined with constraints from
   protein abundances. We use the modeling scenario that considers
   measured protein abundances as the ultimate validation scenario not
   only for the prediction of metabolic fluxes but also for the prediction
   of specific growth rates as it contains considerably more biochemically
   relevant constraints. Indeed for this scenario, we showed that
   condition-specific growth rates cannot be reliably predicted with
   pcGEMs of S. cerevisiae and E. coli when available in vitro
   (Figs. [157]2c and [158]4b) and in vivo estimates (Supplementary
   Fig. [159]11c, e) of turnover numbers are used. GECKO resolves this
   issue by flexibilities measured protein abundances without considering
   physiological information during the procedure. To overcome this
   limitation, we developed PRESTO, which corrects turnover numbers and
   facilitates the integration of enzyme abundance constraints.

   In contrast to PRESTO, GECKO uses measured total protein content from a
   single condition to achieve specific growth rates in the process of
   correcting the turnover numbers. As a result, the corrected turnover
   numbers vary between different experiments. Like all existing
   approaches for the estimation of in vivo turnover numbers based on
   GEMs, we integrated protein abundance data directly to correct turnover
   numbers. Following this strategy in PRESTO is further justified by the
   observation that the turnover numbers included in pcGEMs are often
   neither from the same enzyme (i.e., EC number), substrate, nor
   organism. While in vivo turnover number estimates can be adjusted by
   considering recently proposed Bayesian statistical learning^[160]18,
   this approach has not considered protein abundance information from
   proteomics measurements.

   We employed PRESTO with the largest data set of these measurements
   available to date for S. cerevisiae^[161]15 and E. coli^[162]12.
   Through a series of comparative analyses, we demonstrated that the
   corrections of turnover numbers from PRESTO ultimately increase the
   prediction accuracy of condition-specific growth for the two organisms
   when enzyme abundance data are integrated into the corresponding
   pcGEMs.

   Since PRESTO generates a condition-independent
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   set it is bound to correct parameters that lead to the underprediction
   of biological fluxes. The same reasoning is applied when obtaining in
   vivo
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   estimates from pFBA by taking the maximum of apparent catalytic rates,
   and overall conditions^[163]12,[164]15. Nevertheless, we included an
   optional step, that additionally allows for the reduction of in vitro
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values, which can be considered average apparent catalytic rate
   estimates and result in better performance in the E. coli experiments
   (Supplementary Figs. [165]13 and [166]16). Moreover, we showed that in
   vivo turnover number proxies, obtained by the ranking of
   condition-specific estimates that use proteomics and fluxomics data,
   are more highly (but modestly) correlated to estimates from PRESTO than
   to in vitro turnover numbers. Owing to the constraint-based formulation
   of PRESTO, we also determined the precision of the correction of
   turnover numbers. Previous studies have shown that even for the
   well-studied model organism Saccharomyces cerevisiae, only 52% of
   enzyme turnover numbers in the pcGEM can be obtained from
   organism-specific in vitro measurements^[167]22. Using organism
   unspecific
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values for parameterization and correction of pcGEMs, as done in the
   GECKO pipeline, assumes that enzyme kinetic properties are comparable
   within one EC number class^[168]31,[169]32. However, we did not
   identify clear differences between EC classes, down to the second
   digit, when considering the distribution of
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   similarities within EC classes (Supplementary Fig. [170]21). Indeed, it
   has been reported that EC class plays only a minor role in the
   prediction of turnover numbers^[171]19 and show stronger similarity
   with concordant GO categories^[172]33. Interestingly, our findings show
   that the turnover number corrections obtained from PRESTO are more
   precise than EC class-based corrections. (Supplementary Fig. [173]6).
   Together, these findings demonstrated PRESTO can be readily used to
   decrease the bias of turnover numbers. This paves the way for employing
   the outcome of PRESTO and future extensions toward effectively
   predicting the kcatome from available protein sequences.

Methods

Experimental data

   For S. cerevisiae, we made use of a dataset gathered by^[174]15 from
   four different studies^[175]34–[176]37, which included protein
   abundance data (
   [MATH: <msup><mrow><mi mathvariant="normal">mmol</mi><mspace
   width="0.16em"></mspace><mi
   mathvariant="normal">gD</mi><mi>W</mi></mrow><mrow><mo>−</mo><mn>1</mn>
   </mrow></msup> :MATH]
   ) as well as measured growth or dilution rates (
   [MATH: <msup><mrow><mi
   mathvariant="normal">h</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></ms
   up> :MATH]
   ) and nutrient exchange fluxes (
   [MATH: <msup><mrow><mi mathvariant="normal">mmol</mi><mspace
   width="0.16em"></mspace><mi
   mathvariant="normal">gD</mi><mi>W</mi></mrow><mrow><mo>−</mo><mn>1</mn>
   </mrow></msup><msup><mrow><mi
   mathvariant="normal">h</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></ms
   up> :MATH]
   ). Exchange fluxes missing in certain conditions were set to
   [MATH: <mn>1000</mn><mspace width="0.25em"></mspace><msup><mrow><mi
   mathvariant="normal">mmol</mi><mspace width="0.16em"></mspace><mi
   mathvariant="normal">gD</mi><mi>W</mi></mrow><mrow><mo>−</mo><mn>1</mn>
   </mrow></msup><msup><mrow><mi
   mathvariant="normal">h</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></ms
   up> :MATH]
   if the nutrient was present in the used culture media. We further
   extended this data set by total protein content measurements (
   [MATH: <msup><mrow><mi mathvariant="normal">g</mi><mspace
   width="0.16em"></mspace><mi
   mathvariant="normal">gD</mi><mi>W</mi></mrow><mrow><mo>−</mo><mn>1</mn>
   </mrow></msup> :MATH]
   ) from the original studies. For subsequent analyses, we used the
   maximum abundance of each protein over all replicates per experimental
   condition. Similarly, we used the average value for specific growth
   rates and nutrient exchange rates. Since no measurement of total
   protein content was available for the two conditions evaluated in the
   Di Bartolomeo study^[177]37, we used the maximum protein content
   measured across the remaining conditions for these conditions (i.e.,
   [MATH: <mn>0.67</mn><mspace width="0.25em"></mspace><mi
   mathvariant="normal">g/gDW</mi> :MATH]
   ). Moreover, we excluded three temperature stress conditions (i.e.,
   Lahtvee2017_Temp33, Lahtvee2017_Temp36, Lahtvee2017_Temp38) from the
   analysis since the temperature can have a large effect on the catalytic
   activity of an enzyme. Gene names in the proteomics dataset were
   translated to UniProt identifiers using the batch retrieval service of
   the UniProt REST API^[178]38.

   For E. coli, we used a dataset comprising 31 experimental conditions,
   which was gathered by Davidi and colleagues and augmented by Xu et
   al.^[179]12,[180]14 from three publications^[181]27–[182]29. Here, too,
   we used the maximum protein abundance over all replicates (in
   [MATH: <msup><mrow><mi mathvariant="normal">mmol</mi><mspace
   width="0.16em"></mspace><mi
   mathvariant="normal">gD</mi><mi>W</mi></mrow><mrow><mo>−</mo><mn>1</mn>
   </mrow></msup> :MATH]
   ). Due to the absence of total protein content measurements in two of
   the original studies, we relied on the maximum protein content measured
   in the Valgepea study (i.e.,
   [MATH: <mn>0.61</mn><mspace width="0.25em"></mspace><mi
   mathvariant="normal">g/gDW</mi> :MATH]
   ) to be used for all conditions. Since precise data on nutrient uptake
   rates were only given for a few conditions, we assigned a default upper
   bound of
   [MATH: <mn>1000</mn><mspace width="0.25em"></mspace><msup><mrow><mi
   mathvariant="normal">mmol</mi><mspace width="0.16em"></mspace><mi
   mathvariant="normal">gD</mi><mi>W</mi></mrow><mrow><mo>−</mo><mn>1</mn>
   </mrow></msup><mspace width="0.25em"></mspace><msup><mrow><mi
   mathvariant="normal">h</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></ms
   up> :MATH]
   to all nutrients contained in the minimal medium (Supplementary
   Table [183]2) with additional carbon sources as specified. Gene
   identifiers were translated to UniProt similar as for S. cerevisiae.

Model preparation

   The proposed approach aims at parsimonious correction of turnover
   values in genome-scale enzyme-constraint metabolic models using
   measured protein abundances. Therefore, it is important to consider the
   differential association between enzymes and reactions, i.e., isozymes,
   enzyme complexes, and promiscuous enzymes. We decided to use the GECKO
   formalism^[184]7, which deals with these problems elegantly by directly
   encoding the required information in the stoichiometric matrix. The
   genome-scale metabolic models for S. cerevisiae (YeastGEM v.8.5.0) and
   E. coli (iML1515) were obtained from the yeast-GEM and ecModels GitHub
   repository, respectively^[185]22,[186]39
   [[187]https://github.com/SysBioChalmers; accessed on 22.08.2021]. For
   subsequent steps, functions of the COBRA v3.0 toolbox^[188]40 and
   GECKO2.0 toolbox^[189]22 were employed, of which several functions were
   adapted for our purposes.

   To arrive at raw protein-constrained models for both organisms, the
   GECKO2.0 model enhancement pipeline was adapted to allow the
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   correction procedure to be omitted. Moreover, any manual corrections of
   turnover numbers were excluded from model generation. In the process of
   adapting the raw pcGEM to the respective experimental conditions for
   both organisms, the GAM value per condition was fitted using the
   scaleBioMass function of GECKO2.0, based solely on the
   condition-specific nutrient exchange rates, and returning the minimum (
   [MATH: <mn>9</mn><mfrac><mrow><mi
   mathvariant="normal">mmol</mi></mrow><mrow><mi
   mathvariant="normal">gDWh</mi></mrow></mfrac> :MATH]
   ) or maximum (161
   [MATH: <mfrac><mrow><mi mathvariant="normal">mmol</mi></mrow><mrow><mi
   mathvariant="normal">gDWh</mi></mrow></mfrac> :MATH]
   ) interval boundary if reached (only S. cerevisiae). Furthermore, we
   omitted enzyme abundances, which were not measured across all
   experiments as the approach proposed here is only applicable for
   enzymes in the set with measured abundances (
   [MATH: <mi>M</mi> :MATH]
   ).

PRESTO approach

   In the design of PRESTO, we modified the enzyme mass-balance
   constraints of the augmented stoichiometric matrix, created by GECKO,
   from
   [MATH:
   <mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msubsup><mrow><mi>k</mi>
   </mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></
   msubsup></mrow></mfrac><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></m
   row></msub><mo>+</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mro
   w></msub><mo>=</mo><mn>0</mn> :MATH]
   1

   to inequality constraints that use the measured protein abundance
   directly. The variable
   [MATH: <mi>e</mi> :MATH]
   denotes the predicted protein abundance in the pcGEM, while
   [MATH: <mi>E</mi> :MATH]
   represents the vector of measured enzyme abundances. Further, we assume
   a single turnover number per enzyme
   [MATH: <mi>i</mi> :MATH]
   over all catalyzed reactions (
   [MATH: <msubsup><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi
   mathvariant="normal">cat</mi></mrow><mrow><mi>i</mi></mrow></msub></mro
   w><mrow><mi
   mathvariant="normal">min</mi></mrow></msubsup><mo>=</mo><munder><mrow><
   mi>arg</mi><mspace
   width="0.25em"></mspace><mi>min</mi></mrow><mrow><mi>j</mi></mrow></mun
   der><msubsup><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></
   msubsup> :MATH]
   ):
   [MATH:
   <mo>∀</mo><mi>r</mi><mo>∈</mo><mi>R</mi><msub><mrow><mo>∑</mo></mrow><m
   row><mi>i</mi><mo>∈</mo><mi mathvariant="normal">GPR</mi><mfenced
   close=")"
   open="("><mrow><mi>r</mi></mrow></mfenced></mrow></msub><msub><mrow><mi
   >v</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>≤</mo><msubsup><mrow><m
   i>k</mi></mrow><mrow><msub><mrow><mi
   mathvariant="normal">ca</mi><mi>t</mi></mrow><mrow><mi>i</mi></mrow></m
   sub></mrow><mrow><mi>min</mi></mrow></msubsup><mo>⋅</mo><msub><mrow><mi
   >e</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>.</mo> :MATH]
   2

   GPR stands for gene–protein-reaction rule that associates reactions (
   [MATH: <mi>R</mi> :MATH]
   ) with underlying genes and proteins. The variable
   [MATH: <mi>E</mi> :MATH]
   denotes the measured protein abundance in
   [MATH: <msup><mrow><mi mathvariant="normal">mmol</mi><mspace
   width="0.16em"></mspace><mi
   mathvariant="normal">gD</mi><mi>W</mi></mrow><mrow><mo>−</mo><mn>1</mn>
   </mrow></msup> :MATH]
   . We justify making the assumption for Eq. ([190]2) based on our
   observation that most enzymes in the S. cerevisiae model are associated
   with no more than four reactions (Supplementary Fig. [191]22a, c).
   Further, the vast majority of enzymes are assigned a single unique
   turnover number even though they catalyze multiple reactions
   (Supplementary Fig. [192]22b, d).

   We then introduced a correction factor δ, which is added to each
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   if the protein abundances for the underlying enzyme were available:
   [MATH:
   <mo>∀</mo><mi>r</mi><mo>∈</mo><mi>R</mi><msub><mrow><mo>∑</mo></mrow><m
   row><mi>i</mi><mo>∈</mo><mi mathvariant="normal">GPR</mi><mfenced
   close=")"
   open="("><mrow><mi>r</mi></mrow></mfenced></mrow></msub><msub><mrow><mi
   >v</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>≤</mo><mfenced
   close=")"
   open="("><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi
   mathvariant="normal">ca</mi><mi>t</mi></mrow><mrow><mi>i</mi></mrow></m
   sub></mrow><mrow><mi>min</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi
   >δ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mfenced
   close="]"
   open="["><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></ms
   ub></mrow></mfenced><mo>.</mo> :MATH]
   3

   To find a biologically relevant minimal set of adaptations with respect
   to the sum of δ, we minimized the weighted sum of the average absolute
   relative errors, ω, between measured (
   [MATH: <msup><mrow><mi>μ</mi></mrow><mrow><mi>exp</mi></mrow></msup>
   :MATH]
   ) and predicted specific growth rates (
   [MATH: <msub><mrow><mi>v</mi></mrow><mrow><mi
   mathvariant="normal">bio</mi></mrow></msub> :MATH]
   ) overall experimental conditions
   [MATH: <mi>C</mi> :MATH]
   , and the average δ:
   [MATH:
   <munder><mrow><mi>min</mi></mrow><mrow><mi>v</mi><mo>,</mo><mi>δ</mi><m
   o>,</mo><mi>ω</mi></mrow></munder><mfrac><mrow><mn>1</mn></mrow><mrow><
   mfenced close="∣"
   open="∣"><mrow><mi>C</mi></mrow></mfenced></mrow></mfrac><munder><mrow>
   <mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi>C</mi></mrow></munder><m
   sub><mrow><mi>ω</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>+</mo><mfr
   ac><mrow><mi>λ</mi></mrow><mrow><mfenced close="∣"
   open="∣"><mrow><mi>M</mi></mrow></mfenced></mrow></mfrac><munder><mrow>
   <mo>∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>M</mi></mrow></munder><m
   sub><mrow><mi>δ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>.</mo>
   :MATH]
   4

   Finally, the linear programming formulation of the
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   correction in PRESTO is the following:
   [MATH:
   <munder><mrow><mi>min</mi></mrow><mrow><mi>v</mi><mo>,</mo><mi>δ</mi><m
   o>,</mo><mi>ω</mi></mrow></munder><mfrac><mrow><mn>1</mn></mrow><mrow><
   mfenced close="∣"
   open="∣"><mrow><mi>C</mi></mrow></mfenced></mrow></mfrac><munder><mrow>
   <mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi>C</mi></mrow></munder><m
   sub><mrow><mi>ω</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>+</mo><mfr
   ac><mrow><mi>λ</mi></mrow><mrow><mfenced close="∣"
   open="∣"><mrow><mi>M</mi></mrow></mfenced></mrow></mfrac><munder><mrow>
   <mo>∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>M</mi></mrow></munder><m
   sub><mrow><mi>δ</mi></mrow><mrow><mi>i</mi></mrow></msub> :MATH]

   subject to
   [MATH: <mi>N</mi><msup><mrow><mi>v</mi></mrow><mrow><mspace
   width="0.25em"></mspace><mi>j</mi></mrow></msup><mo>=</mo><mn>0</mn><mo
   >,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>j</mi><mo>∈</mo><mi>C</mi> :MATH]
   5
   [MATH: <munder><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mspace
   width="0.25em"></mspace><mo>∈</mo><mi
   mathvariant="normal">GPR</mi><mfenced close=")"
   open="("><mrow><mi>r</mi></mrow></mfenced></mrow></munder><msubsup><mro
   w><mi>v</mi></mrow><mrow><mi>r</mi></mrow><mrow><mspace
   width="0.25em"></mspace><mi>j</mi></mrow></msubsup><mo>≤</mo><mfenced
   close=")" open="("><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>min</
   mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>i<
   /mi></mrow></msub></mrow></mfenced><mfenced close="]"
   open="["><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow><
   mrow><mi>j</mi></mrow></msubsup></mrow></mfenced><mo>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>r</mi><mo>∈</mo><mi>R</mi><mo>,</
   mo><mspace
   width="0.25em"></mspace><mi>i</mi><mo>∈</mo><mi>M</mi><mo>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>j</mi><mo>∈</mo><mi>C</mi> :MATH]
   [MATH:
   <msubsup><mrow><mi>v</mi></mrow><mrow><mi>min</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>≤</mo><msup><mro
   w><mi>v</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msup><mo>≤</mo><msubsup><mro
   w><mi>v</mi></mrow><mrow><mi>max</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>;</mo><mo>∀</mo>
   <mi>j</mi><mo>∈</mo><mi>C</mi> :MATH]
   6
   [MATH:
   <msub><mrow><mi>δ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><m
   fenced close=")"
   open="("><mrow><mi>ε</mi><mo>−</mo><mn>1</mn></mrow></mfenced><mo>⋅</mo
   ><msubsup><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>min</
   mi></mrow></msubsup><mo>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>i</mi><mo>∈</mo><mi>M</mi> :MATH]
   7
   [MATH: <msubsup><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>min</
   mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>i<
   /mi></mrow></msub><mo>≤</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>max<
   /mi></mrow></msup><mo>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>i</mi><mo>∈</mo><mi>M</mi> :MATH]
   8
   [MATH: <msubsup><mrow><mi>v</mi></mrow><mrow><mi
   mathvariant="normal">bio</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>⋅</mo><msub><mro
   w><mi>ω</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≥</mo><msubsup><mr
   ow><mi>μ</mi></mrow><mrow><mi>exp</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>−</mo><msubsup><
   mrow><mi>v</mi></mrow><mrow><mi
   mathvariant="normal">bio</mi></mrow><mrow><mi>j</mi></mrow></msubsup><m
   o>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>j</mi><mo>∈</mo><mi>C</mi> :MATH]
   9
   [MATH: <msub><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mi
   mathvariant="normal">bio</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>⋅</mo><mi>ω</mi>
   </mrow><mrow><mi>j</mi></mrow></msub><mo>≥</mo><msubsup><mrow><mi>v</mi
   ></mrow><mrow><mi mathvariant="normal">bio</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>−</mo><msubsup><
   mrow><mi>μ</mi></mrow><mrow><mi>exp</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>j</mi><mo>∈</mo><mi>C</mi> :MATH]
   10
   [MATH: <mtable><mtr><mtd columnalign="center"><mi
   mathvariant="normal">ω</mi><mo>≤</mo><mi
   mathvariant="normal">θ</mi><mo>,</mo></mtd><mtd
   columnalign="center"><mi
   mathvariant="normal">δ</mi><mo>≥</mo><mn>0</mn><mo>.</mo></mtd></mtr></
   mtable> :MATH]

   The value for ω was bound from above by a value θ, which was set to
   0.6. Constraints that enforce metabolic steady state are captured in
   Eqs. ([193]5) and ([194]6) represent the lower and upper bounds in the
   flux through each reaction in each condition, respectively. The
   constraints in Eqs. ([195]7) and ([196]8) impose an upper bound on δ,
   which is the minimum of the allowed fold change in
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values, ε, and a cut-off value
   [MATH: <msup><mrow><mi>K</mi></mrow><mrow><mi>max</mi></mrow></msup>
   :MATH]
   , which denotes the maximum allowed
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   value. The value for ε was set to
   [MATH: <msup><mrow><mn>10</mn></mrow><mrow><mn>5</mn></mrow></msup>
   :MATH]
   since lower values did not yield solutions and
   [MATH: <msup><mrow><mi>K</mi></mrow><mrow><mi>max</mi></mrow></msup>
   :MATH]
   was set to 57,500,000 s^−1 (5.3.1.1, Pyrococcus furiosus^[197]41).
   Equations ([198]9) and ([199]10) ensure that ω is equal to
   [MATH: <mfrac><mrow><mfenced close="∣"
   open="∣"><mrow><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>j</mi></mrow><
   mrow><mi>exp</mi></mrow></msubsup><mo>−</mo><msub><mrow><mi>v</mi></mro
   w><mrow><mi
   mathvariant="normal">bio</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><
   /mfenced></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi
   mathvariant="normal">bio</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><
   /mfrac> :MATH]
   .

   The parameter λ controls the trade-off between both minimization
   objectives (see Eq. ([200]4)). As λ is unknown and may also be
   condition- and model-specific, it was fitted using a 3-fold
   cross-validation scheme, which was repeated for 10 iterations. To this
   end, we scanned a log-scale interval between
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>14</mn></mrow></msup>
   :MATH]
   and
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup>
   :MATH]
   . In each iteration, we performed
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   corrections on two folds of condition-specific models and validated the
   obtained corrections on the remaining fold of condition-specific
   models. The validation was done by predicting growth only with a
   constraint on total protein content, without constraints from measured
   protein abundances. This was done to counteract overprediction in the
   scenario without constraints from proteomics data. The relative errors
   (ω) and the sum of δ (i.e., Δ) were then used to calculate the scores
   [MATH: <msub><mrow><mi>s</mi></mrow><mrow><mi>λ</mi></mrow></msub>
   :MATH]
   , which helped us choose the optimal value for λ:
   [MATH:
   <msub><mrow><mi>s</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>=</mo><m
   frac><mrow><mn>1</mn></mrow><mrow><mn>10</mn></mrow></mfrac><munderover
   accent="false"
   accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><m
   n>1</mn></mrow><mrow><mn>10</mn></mrow></munderover><mfrac><mrow><msub>
   <mrow><mi>ω</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>τ</mi></mrow></msu
   b><mo>−</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>λ</mi><mo>,</mo><
   mi>τ</mi></mrow><mrow><mi>min</mi></mrow></msubsup></mrow><mrow><msubsu
   p><mrow><mi>ω</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>τ</mi></mrow><mr
   ow><mi>max</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi>ω</mi></mr
   ow><mrow><mi>λ</mi><mo>,</mo><mi>τ</mi></mrow><mrow><mi>min</mi></mrow>
   </msubsup></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msub><mrow><mi>log</mi
   ></mrow><mrow><mn>10</mn></mrow></msub><mfrac><mrow><msub><mrow><mi
   mathvariant="normal">Δ</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>τ</mi><
   /mrow></msub></mrow><mrow><msubsup><mrow><mi
   mathvariant="normal">Δ</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>τ</mi><
   /mrow><mrow><mi>min</mi></mrow></msubsup></mrow></mfrac></mrow><mrow><m
   sub><mrow><mi>log</mi></mrow><mrow><mn>10</mn></mrow></msub><mfrac><mro
   w><msubsup><mrow><mi
   mathvariant="normal">Δ</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>τ</mi><
   /mrow><mrow><mi>max</mi></mrow></msubsup></mrow><mrow><msubsup><mrow><m
   i
   mathvariant="normal">Δ</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>τ</mi><
   /mrow><mrow><mi>min</mi></mrow></msubsup></mrow></mfrac></mrow></mfrac>
   <mo>.</mo> :MATH]
   11

   The score can be described as the average product of min-max-scaled ω
   and Δ across the 10 cross-validation iterations per explored λ. The
   optimal value was then determined by finding the first sign change in
   the second numerical gradient over
   [MATH: <msub><mrow><mi>s</mi></mrow><mrow><mi>λ</mi></mrow></msub>
   :MATH]
   , starting from the maximum value for λ. In addition to the optimal λ,
   we also compared our results to a second λ, where the sum of δ reached
   a plateau (
   [MATH:
   <mi>λ</mi><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>1
   0</mn></mrow></msup> :MATH]
   for S. cerevisiae and
   [MATH:
   <mi>λ</mi><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>1
   1</mn></mrow></msup> :MATH]
   for E. coli, Supplementary Fig. [201]2a and [202]12a). The presented
   approach and analysis scripts were implemented using MATLAB^[203]42.

Variability analysis for δ

   While PRESTO considers multiple experimental conditions to find a set
   of universal corrections for
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values, it does not provide an exhaustive view over all possible
   solutions to this problem. To assess the precision of the corrections,
   we first performed a variability analysis for δ to find the minimum and
   maximum possible values. To guarantee that a solution of equal quality
   is found with respect to the previously determined sum of δ and the
   relative errors to experimentally measured specific growth rates (i.e.,
   [MATH: <msup><mrow><mi>ω</mi></mrow><mrow><mi
   mathvariant="normal">opt</mi></mrow></msup> :MATH]
   ), corresponding constraints were added to arrive at the following
   linear programming problem:
   [MATH:
   <munder><mrow><mi>min</mi></mrow><mrow><mi>v</mi><mo>,</mo><mi>δ</mi><m
   o>,</mo><mspace
   width="0.25em"></mspace><mi>ω</mi></mrow></munder><mo>/</mo><munder><mr
   ow><mi>max</mi></mrow><mrow><mi>v</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi
   >ω</mi></mrow></munder><msub><mrow><mi>δ</mi></mrow><mrow><mi>i</mi></m
   row></msub> :MATH]

   s.t.
   [MATH: <mi>N</mi><msup><mrow><mi>v</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msup><mo>=</mo><mn>0</mn><mo
   >,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>j</mi><mo>∈</mo><mi>C</mi> :MATH]
   [MATH: <munder><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi
   mathvariant="normal">GPR</mi><mfenced close=")"
   open="("><mrow><mi>r</mi></mrow></mfenced></mrow></munder><msubsup><mro
   w><mi>v</mi></mrow><mrow><mi>r</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>≤</mo><mfenced
   close=")" open="("><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>min</
   mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>i<
   /mi></mrow></msub></mrow></mfenced><mfenced close="]"
   open="["><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow><
   mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup></mrow></mfenced><mo
   >,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>r</mi><mo>∈</mo><mi>R</mi><mo>,</
   mo><mspace
   width="0.25em"></mspace><mi>i</mi><mo>∈</mo><mi>M</mi><mo>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>j</mi><mo>∈</mo><mi>C</mi> :MATH]
   [MATH:
   <msubsup><mrow><mi>v</mi></mrow><mrow><mi>min</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>≤</mo><msup><mro
   w><mi>v</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msup><mo>≤</mo><msubsup><mro
   w><mi>v</mi></mrow><mrow><mi>max</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>j</mi><mo>∈</mo><mi>C</mi> :MATH]
   [MATH:
   <msub><mrow><mi>δ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><m
   fenced close=")"
   open="("><mrow><mi>ε</mi><mo>−</mo><mn>1</mn></mrow></mfenced><mo>⋅</mo
   ><msubsup><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>min</
   mi></mrow></msubsup><mo>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>i</mi><mo>∈</mo><mi>M</mi> :MATH]
   [MATH: <msubsup><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>min</
   mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>i<
   /mi></mrow></msub><mo>≤</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>max<
   /mi></mrow></msup><mo>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>i</mi><mo>∈</mo><mi>M</mi> :MATH]
   [MATH: <msubsup><mrow><mi>v</mi></mrow><mrow><mi
   mathvariant="normal">bio</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>⋅</mo><msub><mro
   w><mi>ω</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≥</mo><msubsup><mr
   ow><mi>μ</mi></mrow><mrow><mi>exp</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>−</mo><msubsup><
   mrow><mi>v</mi></mrow><mrow><mi
   mathvariant="normal">bio</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>j</mi><mo>∈</mo><mi>C</mi> :MATH]
   [MATH: <msub><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mi
   mathvariant="normal">bio</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>⋅</mo><mi>ω</mi>
   </mrow><mrow><mi>j</mi></mrow></msub><mo>≥</mo><msubsup><mrow><mi>v</mi
   ></mrow><mrow><mi mathvariant="normal">bio</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>−</mo><msubsup><
   mrow><mi>μ</mi></mrow><mrow><mi>exp</mi></mrow><mrow><mspace
   width="0.16em"></mspace><mi>j</mi></mrow></msubsup><mo>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>j</mi><mo>∈</mo><mi>C</mi> :MATH]
   [MATH:
   <mn>0.99</mn><mo>⋅</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>j</mi>
   </mrow><mrow><mi
   mathvariant="normal">opt</mi></mrow></msubsup><mo>≤</mo><msub><mrow><mi
   >ω</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≤</mo><mn>1.01</mn><mo>
   ⋅</mo><msup><mrow><mi mathvariant="normal">ω</mi></mrow><mrow><mi
   mathvariant="normal">opt</mi></mrow></msup><mo>,</mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>j</mi><mo>∈</mo><mi>C</mi> :MATH]
   12
   [MATH: <msup><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi
   mathvariant="normal">opt</mi></mrow></msup><mo>−</mo><msup><mrow><mn>10
   </mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup><mo>≤</mo><munder><
   mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>M</mi></mrow></mund
   er><msub><mrow><mi>δ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo
   ><msup><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi
   mathvariant="normal">opt</mi></mrow></msup><mo>+</mo><msup><mrow><mn>10
   </mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup> :MATH]
   13
   [MATH: <mtable><mtr><mtd columnalign="center"><mi
   mathvariant="normal">ω</mi><mo>≤</mo><mi
   mathvariant="normal">θ</mi><mo>,</mo></mtd><mtd
   columnalign="center"><mi
   mathvariant="normal">δ</mi><mo>≥</mo><mn>0</mn><mo>.</mo></mtd></mtr></
   mtable> :MATH]

   The minimal relative error determined for each condition
   [MATH: <mi>j</mi> :MATH]
   was fixed within a narrow tolerance (±1%, Eq. ([204]12)) and the
   minimum sum of corrections
   [MATH: <mi mathvariant="normal">Δ</mi> :MATH]
   was fixed with a tolerance of
   [MATH:
   <mo>±</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mro
   w></msup><msup><mrow><mi
   mathvariant="normal">h</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></ms
   up> :MATH]
   (Eq. ([205]13)).

   As the distribution within the obtained min/max intervals can be
   skewed, we sampled 10,000 points within the obtained intervals. For
   uniform random sampling, we created random vectors of corrections
   [MATH: <msup><mrow><mi>δ</mi></mrow><mrow><mo>*</mo></mrow></msup>
   :MATH]
   within the determined intervals and projected them onto the solution
   space by minimizing the distance of δ to the respective random vector.
   Therefore, we updated the objective of the program above:
   [MATH:
   <munder><mrow><mi>min</mi></mrow><mrow><mi>v</mi><mo>,</mo><mi>δ</mi><m
   o>,</mo><mi>ω</mi></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow>
   <mi>i</mi><mo>∈</mo><mi>M</mi></mrow></munder><mfenced close="∣"
   open="∣"><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi>i</mi></mrow></ms
   ub><mo>−</mo><msubsup><mrow><mi>δ</mi></mrow><mrow><mi>i</mi></mrow><mr
   ow><mo>*</mo></mrow></msubsup></mrow></mfenced><mo>.</mo> :MATH]
   14

   To ensure reproducibility and compatibility with the COBRA
   toolbox^[206]40, we solved all optimization problems using the
   optimizeCbModel of the COBRA toolbox. Within this environment, we used
   the Gurobi solver v9.1.1^[207]43 but we note that any other supported
   solver can also be used. As we observed numerical instability of the
   problems in some cases, we decreased the feasibility tolerance (i.e.,
   feasTol parameter) for the COBRA solver to
   [MATH:
   <msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>9</mn></mrow></msup>
   :MATH]
   for all predictions. The results were visualized using MATLAB^[208]42.

Validation of corrected models

   We used the adapted GECKO pipeline (fitting a condition-specific GAM;
   excluding manual
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   adaptions) to obtain models with
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values adapted according to the objective control coefficient
   heuristic. We note that, when no manual modifications were introduced
   to the S. cerevisiae models, the
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   adaption of the GECKO pipeline would stop because no objective control
   coefficient above the threshold of 0.001 could be found, and corrected
   models would still be below the predicted growth error tolerance of
   10%. To compare the predictive performance of PRESTO and GECKO
   corrected models, the models were adapted with the same
   condition-specific GAM, biomass reaction, and total protein content,
   [MATH: <msub><mrow><mi>P</mi></mrow><mrow><mi
   mathvariant="normal">tot</mi><mo>.</mo></mrow></msub> :MATH]
   Additionally, PRESTO models were constrained using the same
   condition-specific saturation rate,
   [MATH: <mi>σ</mi> :MATH]
   and enzyme mass fraction, f, as obtained from the GECKO pipeline. In
   contrast to the GECKO formulation, we did not subtract the mass of
   measured enzymes from the total protein pool constraint but instead
   introduced the measured protein concentration as the upper bound on the
   enzyme usage reaction,
   [MATH: <msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub>
   :MATH]
   , in the respective scenario. This formulation still guarantees that
   the mass of all used enzymes is lower or equal to the approximated
   cellular protein pool according to
   [MATH:
   <munder><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>r</mi><mi
   mathvariant="normal">GPR</mi><mrow><mo>(</mo><mrow><mi>r</mi></mrow><mo
   >)</mo></mrow></mrow></munder><msub><mrow><mi>e</mi></mrow><mrow><mi>i<
   /mi><mo>,</mo><mspace
   width="0.25em"></mspace><mi>j</mi></mrow></msub><mo>⋅</mo><msub><mrow><
   mi
   mathvariant="normal">MW</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤<
   /mo><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi><mi>o</mi><mi>t</mi><m
   o>,</mo><mspace
   width="0.25em"></mspace><mi>j</mi></mrow></msub><mo>⋅</mo><mi>f</mi><mo
   >⋅</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,
   </mo><mspace
   width="0.25em"></mspace><mo>∀</mo><mi>j</mi><mo>∈</mo><mi>C</mi> :MATH]
   15

   where
   [MATH: <mi mathvariant="normal">MW</mi> :MATH]
   is the respective molecular weight of the protein. By considering
   measured and unmeasured enzymes in Eq. ([209]15) we do not have to
   change f and use the same factor as for the scenario where no protein
   abundance measures are used^[210]7. Maximum growth was predicted in
   three different constraint scenarios: (i) using only the protein pool
   constraint and default uptake rates (1000 mmol/gDW/h), (ii) using the
   pool constraint and experimentally measured uptake rates, (iii) using
   the previous constraints plus the absolute enzyme abundance.

   The two studies which generated in vivo
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values from pFBA^[211]12,[212]15 calculated a single value per reaction
   irrespective of the presence of isoenzymes. Thus, to parameterize the
   raw pcGEM (containing only uncorrected BRENDA values) we substituted
   the in vitro
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values of all isoenzyme reactions with the respective estimate provided
   in the study. Reactions catalyzed by complexes were not corrected.
   Since PRESTO and the pFBA studies provide a single
   condition-independent model, we generated a condition-independent GECKO
   model by following the maximum overall conditions approach: For the
   comparisons, shown in Supplementary Figs. [213]11 and [214]15, the
   condition-wise GECKO models were aggregated into a single union model
   in which for each reaction the maximum
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   value was used.

Pathway enrichment analysis

   The KEGG pathway terms^[215]44, associated with each enzyme that was
   measured in all conditions, were acquired using the KEGG REST API. The
   one-sided p-value, p, for significant enrichment of a pathway term
   among the enzymes with corrected
   [MATH: <msub><mrow><mi>k</mi></mrow><mrow><mi
   mathvariant="normal">cat</mi></mrow></msub> :MATH]
   values was calculated using the hypergeometric density distribution:
   [MATH: <mi>p</mi><mfenced close=")"
   open="("><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><mn>1</mn><mo>−</mo
   ><munderover accent="false"
   accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><m
   n>1</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></munderover>
   <mfrac><mrow><mfenced close=")" open="("><mrow><mtable><mtr><mtd
   columnalign="center"><mi>K</mi></mtd></mtr><mtr><mtd
   columnalign="center"><mi>i</mi></mtd></mtr></mtable></mrow></mfenced><m
   fenced close=")" open="("><mrow><mtable><mtr><mtd
   columnalign="center"><mi>M</mi><mo>−</mo><mi>K</mi></mtd></mtr><mtr><mt
   d
   columnalign="center"><mi>N</mi><mo>−</mo><mi>i</mi></mtd></mtr></mtable
   ></mrow></mfenced></mrow><mrow><mfenced close=")"
   open="("><mrow><mtable><mtr><mtd
   columnalign="center"><mi>M</mi></mtd></mtr><mtr><mtd
   columnalign="center"><mi>N</mi></mtd></mtr></mtable></mrow></mfenced></
   mrow></mfrac><mo>.</mo> :MATH]
   16

   Only KEGG pathway terms associated with at least two corrected enzymes
   were taken into consideration. The p-values associated with all tested
   pathway terms were corrected for a false discovery rate of 0.05 using
   the Benjamini–Hochberg correction^[216]45.

Reporting summary

   Further information on research design is available in the [217]Nature
   Portfolio Reporting Summary linked to this article.

Supplementary information

   [218]Supplementary Information^ (2.8MB, pdf)
   [219]Peer Review File^ (482.3KB, pdf)
   [220]41467_2023_37151_MOESM3_ESM.pdf^ (114KB, pdf)

   Description of Additional Supplementary Files
   [221]Supplemental Data 1^ (14.6KB, xlsx)
   [222]Supplemental Data 2^ (14.5KB, xlsx)
   [223]Supplemental Data 3^ (13.5KB, xlsx)
   [224]Supplemental Data 4^ (12.9KB, xlsx)
   [225]Reporting Summary^ (1.4MB, pdf)

Acknowledgements