Dynamics of co-substrate pools can constrain and regulate metabolic fluxes

  1. Robert West
  2. Hadrien Delattre
  3. Elad Noor
  4. Elisenda Feliu
  5. Orkun S Soyer

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    This manuscript presents an important mathematical analysis on metabolic "co-substrates" and how their cycling can affect metabolic fluxes. Through mathematical analysis of simple network motifs, it shows the impact on constraining metabolic fluxes and the applied mathematical modeling/simulation approaches and the statistical analysis to compare predictions with data from previous studies offer convincing support for the potential biological relevance of co-substrate cycling. The work will be of interest to researchers who study microbial metabolism and metabolic engineering. However, part of this analysis remains unclear and would benefit from clarification.

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Abstract

Cycling of co-substrates, whereby a metabolite is converted among alternate forms via different reactions, is ubiquitous in metabolism. Several cycled co-substrates are well known as energy and electron carriers (e.g. ATP and NAD(P)H), but there are also other metabolites that act as cycled co-substrates in different parts of central metabolism. Here, we develop a mathematical framework to analyse the effect of co-substrate cycling on metabolic flux. In the cases of a single reaction and linear pathways, we find that co-substrate cycling imposes an additional flux limit on a reaction, distinct to the limit imposed by the kinetics of the primary enzyme catalysing that reaction. Using analytical methods, we show that this additional limit is a function of the total pool size and turnover rate of the cycled co-substrate. Expanding from this insight and using simulations, we show that regulation of these two parameters can allow regulation of flux dynamics in branched and coupled pathways. To support these theoretical insights, we analysed existing flux measurements and enzyme levels from the central carbon metabolism and identified several reactions that could be limited by the dynamics of co-substrate cycling. We discuss how the limitations imposed by co-substrate cycling provide experimentally testable hypotheses on specific metabolic phenotypes. We conclude that measuring and controlling co-substrate dynamics is crucial for understanding and engineering metabolic fluxes in cells.

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  1. Version published to 10.7554/elife.84379 on eLife
  2. eLife

    Author Response

    Reviewer 1 (Public Review):

    The authors in this manuscript investigate the effect of co-substrate cycling on the metabolic flow. The main finding is that this cycling can limit the flux through a pathway. The authors examine implications of this effect in different simple configurations to highlight the potential impact on metabolic pathways. Overall, the manuscript follows logical steps and is accessible. Once the main point-reduction in flux of a pathway with limited pool of a cycled co-substrate-is established, some of the following steps become expected (e.g. the fraction of the flux in a branched pathway). Nevertheless, it is understandable that the authors have picked a few simple examples of the metabolic network motifs to highlight the implications. The results presented in the manuscript overall support the conclusions. One weakness is that some of the details of the assumptions (e.g. the choices of rates) are not explicitly spelt out in the manuscript. This work is impactful because it brings into light how cycling of some of the intermediates in a pathway can influence metabolic fluxes and dynamics. This is a factor in addition to (and separate from) reaction rates which are often considered as the main driver of metabolic fluxes.

    We thank the reviewer for this accurate summary. Regarding the effect of parameters on the presented results, we note that the first part of the results are based on analytical solutions provided in the Appendix (formerly the SI). These results are given as inequalities comprising parameters, allowing direct evaluation of parameter effects. We have now made this point explicit in the presentation of the results.

    In the second part of the results, we utilise numerical simulations and in this case, the observed results can possibly depend on parameters. We have explored effects of key parameters, that is kin and total substrate concentration through presented 'phase diagram' style figures - see Figure 2 and 4. For additional parameters, we have now included additional simulations exploring their effects - e.g. see Appendix - Figure 11 and Appendix – Figure 13.

    Reviewer 2 (Public Review):

    The cycling of "co-substrates" in metabolic reactions is possibly a very important but often overlooked determinant of metabolic fluxes. To better understand how the turnover dynamics of co-substrates affect metabolic fluxes the authors dissect a few metabolic reaction motifs. While these motifs are necessarily much simpler than real metabolic networks with dozens or hundreds of reactions, they still include important characteristics of the full network but allow for a deeper mathematical analysis. I found this mathematical approach of the manuscript convincing and an important contribution to the field as it provides more intuitive insights how co-substrate cycling could affect metabolic fluxes. In the manuscript, the authors stress particularly how the pool sizes of co-substrates and the enzymes involved in the cycling of those can constrain metabolic fluxes but the presented results also go substantially beyond this statement as the authors further illustrate how turnover characteristics of substrates in branches/coupled reactions can affect the ratio of produced substrates.

    The authors further present an analysis of previously published experimental data (around Figure 3). This is a very nice idea as it can in principle add more direct proof that the cycling of co-substrates is indeed an important constraint shaping fluxes in real metabolic networks and (instead of being merely a theoretical phenomena which occurs only in unphysiological parameter regimes). However, the way currently presented, it remained unclear to which extent the data analysis is adding convincing support that co-cycling substantially constrains metabolic fluxes. Particularly, it remains unclear for which organisms and conditions the used experimental dataset holds, how it has been generated, and with what uncertainty different measured values come. For example, the comparison requires an estimation of v_max. How can these values determined in-vivo? Are (expected) uncertainties sufficiently low to allow for the statement that fluxes are higher than what enzyme kinetics predict? Furthermore, I am wondering to which extent the correlations between co-substrate pool levels and flux is supporting the idea that co-substrate cyling is important. The positive relation between ATP/AMP/ADP levels for example, is a nice observation. However, it remains a correlation which might occur due to many other factors beyond the limitations of cosubstrate cycling and which might change with provided conditions.

    We thank the reviewer for this accurate summary. Although, we would like to clarify that we do not observe nor analyse any relation between ATP/AMP/ADP levels. Rather, in the analysis presented in Fig. 3B-D, we are looking at the relation between fluxes in co-substrate utilising reactions and the pool size of that co-substrate (e.g. total ATP, AMP, and ADP level for reactions utilising any one of these three co-substrates).

    In their summary, the reviewer raises several valid points about the data analysis and its possible limitations. We address them here point by point:

    How are Vmax values gathered/estimated? We have now added more information regarding how the Vmax values were gathered and from which organisms and conditions. Specifically, we used previously published values of Vmax from (Davidi et al. 2016) where it was estimated by multiplying the in vitro determined kcat by the concentration of the enzyme from proteomic measurement under different conditions - all for model organism Escherichia coli. See also below, reply to recommendation 2.

    Are (expected) uncertainties sufficiently low? It is difficult to have an estimate for the uncertainty since much of the error in the previous analysis probably comes from the fact that the kinetic parameters determined in vitro are used to estimate fluxes under in vivo conditions - the main source of error is expected to be this discrepancy, which is hard to estimate. However, since the plot is in log-scale, we highlight only gaps that are more than 1 order of magnitude (dashed diagonal lines) and hopefully the uncertainty is lower than that. Furthermore, high uncertainty would probably contribute equally to over- and under-estimating the maximal flux, while we can clearly see that the flux rarely exceeds the Vmax. We have now included a statement in the revised text capturing this point.

    Correlations offer weak evidence. Unfortunately, as we do not have measurements on co-substrate pool sizes and cycling kinetics under all conditions, our analyses of experimental data from cycling-involving reactions are admittedly limited. However, they do show that (1) measured fluxes are lower than those predicted by kinetics of the primary enzyme (i.e. enzyme involved in co-substrate and substrate conversion) alone, and (2) there is - for some cycling-involving reactions - a correlation between flux and co-substrate pool size. Both observations could indicate co-substrate pool sizes and/or co-substrate cycling dynamics being limiting. As the reviewer points out, we cannot state this as a certainty.

    Other possible limitations include thermodynamic effects, i.e. limitation by the concentration of both substrate or product, or substrate saturation. We already explored the latter possibility and found that there is still a lower flux when taking into account the primary substrate saturation (see Fig. S6). The former effect is very difficult to analyse without more data, as calculating reaction thermodynamics requires knowledge of concentrations for all substrates and products, as well as enzyme Michaelis-Menten constants in both forward and backward directions. This information is currently not available except for few of the reactions among the ones we analysed. Nevertheless, to give as much insight as possible on the thermodynamic effect, we added a new figure (Appendix – Figure 8) where we plot the physiological Gibbs free energy (is calculated assuming that all reactants are at 1 mM and pH=7) against the normalized flux. The plot shows that although in few cases, such as malate dehydrogenase (MDH), the normalised flux seems to be greatly reduced by the thermodynamic barrier, the general picture is that there is little correlation between physiological Gibbs free energy and normalised flux. We have now included the resulting figure and associated discussion in the revised manuscript.

    In relation to all these points on data-based support of the theory, we would also like to point out the comments from reviewer 3 and the fact that our theoretical work provides motivation for further future experimental studies of co-substrate cycling dynamics. Our main analysis about co-substrate dynamics becoming limiting is based on analytical solutions. These solutions provide an inequality of system parameters relating pathway influx, co-substrate pool size, and co-substrate related enzymatic parameters. When this inequality is satisfied, there will be flux limitation due to cosubstrate cycling. Future experimental studies can now be devised to explore this inequality under different conditions by measuring the key parameters more explicitly. This key point and aspects of the above replies are incorporated at the relevant points in the main text. In addition, we have included a new paragraph in the Discussion section (see reply to second recommendation of reviewer 3) and the following paragraph at the end of the Results section:

    In summary, these results show that for reactions involving co-substrate cycling (1) measured fluxes are lower than those predicted by kinetics of the primary enzyme (i.e. enzyme involved in substrate conversion) alone, and (2) there is - for some reactions - a correlation between flux and co-substrate pool size. Both observations could indicate co-substrate pool sizes and/or co-substrate cycling dynamics being a main limiting factor for flux. We can not state this as a certainty, however, as there are possibly other factors acting as the extra limitation, including thermodynamic effects. These points call for further experimental analysis of co-substrate cycling within the study of metabolic system dynamics.

    Reviewer 3 (Public Review):

    In the study, the authors present a mathematical framework and data analysis approach that revisits an "old" idea in cell physiology: The role of co-substrate cycling as potential key determinant of reaction flux limits in enzyme-catalyzed reaction systems. The aim of the study is to identify metabolic network properties that indicate potential global flux regulatory capacities of co-substrate cycling.

    The authors approached this aim in two steps. First, a mathematical framework, which is based on ODEs was developed and which reflects small abstract metabolic pathways including kinetic parameters of the involved reactions. While the modeled pathways are abstract, the considered pathway motifs are motivated by structures of known existing pathways such as glycolysis (as example of a linear pathway) and certain amino acid biosynthesis pathways (as example of branched pathways). The developed ODE-based models were used for steady state analysis and symbolic and numerical simulations of flux dynamics. As a main result of the first step, the authors highlight that co-substrate cycling can act as mechanism which limits specific metabolic fluxes across the metabolic network and that co-substrate cycling can facilitate flux regulation at branching points of the network. Second, the authors re-analyzed data on flux rates (experimental measurements and flux-balance-analysis predictions) from previous publications in order to assess whether the predicted role of co-substrate cycling could explain the observed flux distributions. In this data analysis, the author provide evidence that the fluxes of specific reactions in central metabolism could be constrained by co-substrate cycling, because their observed fluxes are often lower than expected by the kinetics of the corresponding enzymes.

    A particular strength of the study is that the authors highlight that co-substrates are not limited to ATP and NAD(P)H, but could include a range of other metabolites and which could also be organism-specific. Building on this broad definition of cosubstrates, the authors developed an abstract mathematical framework that can be used to study the general potential 'design principle' of co-substrate cycling in cellular metabolism and to adapt the framework to study different co-substrates in specific organisms in future works.

    Experimental data (i.e. measured fluxes using mass-spectrometry data and labeled substrates) that is available to date is limited and therefore also limits the broad evaluation of the developed mathematical framework across various different organisms and environmental conditions. However, with advances in metabolomics and derived metabolic flux measurements, the mathematical framework will serve as a valuable resource to understand the potential role of co-substrate cycling in more biological systems. The framework might also guide new experiments that generate data for a systematic evaluation of when and to what extent co-substrate cycling governs flux distributions, e.g. depending on growth rates or response to environmental stress.

    We thank the reviewer for this accurate summary. We agree with the reviewer's final comments on limitations of current testing of our theory, due to limitations in existing data, and that this analysis will now motivate further experimental study of co-substrate dynamics. We have already included revisions of the manuscripts to further highlight and discuss limitations of the data-based analysis.

  3. eLife

    eLife assessment

    This manuscript presents an important mathematical analysis on metabolic "co-substrates" and how their cycling can affect metabolic fluxes. Through mathematical analysis of simple network motifs, it shows the impact on constraining metabolic fluxes and the applied mathematical modeling/simulation approaches and the statistical analysis to compare predictions with data from previous studies offer convincing support for the potential biological relevance of co-substrate cycling. The work will be of interest to researchers who study microbial metabolism and metabolic engineering. However, part of this analysis remains unclear and would benefit from clarification.

  4. eLife

    Reviewer #1 (Public Review):

    The authors in this manuscript investigate the effect of co-substrate cycling on the metabolic flow. The main finding is that this cycling can limit the flux through a pathway. The authors examine implications of this effect in different simple configurations to highlight the potential impact on metabolic pathways. Overall, the manuscript follows logical steps and is accessible. Once the main point-reduction in flux of a pathway with limited pool of a cycled co-substrate-is established, some of the following steps become expected (e.g. the fraction of the flux in a branched pathway). Nevertheless, it is understandable that the authors have picked a few simple examples of the metabolic network motifs to highlight the implications. The results presented in the manuscript overall support the conclusions. One weakness is that some of the details of the assumptions (e.g. the choices of rates) are not explicitly spelt out in the manuscript. This work is impactful because it brings into light how cycling of some of the intermediates in a pathway can influence metabolic fluxes and dynamics. This is a factor in addition to (and separate from) reaction rates which are often considered as the main driver of metabolic fluxes.

  5. eLife

    Reviewer #2 (Public Review):

    The cycling of "co-substrates" in metabolic reactions is possibly a very important but often overlooked determinant of metabolic fluxes. To better understand how the turnover dynamics of co-substrates affect metabolic fluxes the authors dissect a few metabolic reaction motifs. While these motifs are necessarily much simpler than real metabolic networks with dozens or hundreds of reactions, they still include important characteristics of the full network but allow for a deeper mathematical analysis. I found this mathematical approach of the manuscript convincing and an important contribution to the field as it provides more intuitive insights how co-substrate cycling could affect metabolic fluxes. In the manuscript, the authors stress particularly how the pool sizes of co-substrates and the enzymes involved in the cycling of those can constrain metabolic fluxes but the presented results also go substantially beyond this statement as the authors further illustrate how turnover characteristics of substrates in branches/coupled reactions can affect the ratio of produced substrates.

    The authors further present an analysis of previously published experimental data (around Figure 3). This is a very nice idea as it can in principle add more direct proof that the cycling of co-substrates is indeed an important constraint shaping fluxes in real metabolic networks and (instead of being merely a theoretical phenomena which occurs only in unphysiological parameter regimes). However, the way currently presented, it remained unclear to which extent the data analysis is adding convincing support that co-cycling substantially constrains metabolic fluxes. Particularly, it remains unclear for which organisms and conditions the used experimental dataset holds, how it has been generated, and with what uncertainty different measured values come. For example, the comparison requires an estimation of v_max. How can these values determined in-vivo? Are (expected) uncertainties sufficiently low to allow for the statement that fluxes are higher than what enzyme kinetics predict? Furthermore, I am wondering to which extent the correlations between co-substrate pool levels and flux is supporting the idea that co-substrate cyling is important. The positive relation between ATP/AMP/ADP levels for example, is a nice observation. However, it remains a correlation which might occur due to many other factors beyond the limitations of co-substrate cycling and which might change with provided conditions.

  6. eLife

    Reviewer #3 (Public Review):

    In the study, the authors present a mathematical framework and data analysis approach that revisits an "old" idea in cell physiology: The role of co-substrate cycling as potential key determinant of reaction flux limits in enzyme-catalyzed reaction systems. The aim of the study is to identify metabolic network properties that indicate potential global flux regulatory capacities of co-substrate cycling.

    The authors approached this aim in two steps. First, a mathematical framework, which is based on ODEs was developed and which reflects small abstract metabolic pathways including kinetic parameters of the involved reactions. While the modeled pathways are abstract, the considered pathway motifs are motivated by structures of known existing pathways such as glycolysis (as example of a linear pathway) and certain amino acid biosynthesis pathways (as example of branched pathways). The developed ODE-based models were used for steady state analysis and symbolic and numerical simulations of flux dynamics. As a main result of the first step, the authors highlight that co-substrate cycling can act as mechanism which limits specific metabolic fluxes across the metabolic network and that co-substrate cycling can facilitate flux regulation at branching points of the network. Second, the authors re-analyzed data on flux rates (experimental measurements and flux-balance-analysis predictions) from previous publications in order to assess whether the predicted role of co-substrate cycling could explain the observed flux distributions. In this data analysis, the author provide evidence that the fluxes of specific reactions in central metabolism could be constrained by co-substrate cycling, because their observed fluxes are often lower than expected by the kinetics of the corresponding enzymes.

    A particular strength of the study is that the authors highlight that co-substrates are not limited to ATP and NAD(P)H, but could include a range of other metabolites and which could also be organism-specific. Building on this broad definition of co-substrates, the authors developed an abstract mathematical framework that can be used to study the general potential 'design principle' of co-substrate cycling in cellular metabolism and to adapt the framework to study different co-substrates in specific organisms in future works.

    Experimental data (i.e. measured fluxes using mass-spectrometry data and labeled substrates) that is available to date is limited and therefore also limits the broad evaluation of the developed mathematical framework across various different organisms and environmental conditions. However, with advances in metabolomics and derived metabolic flux measurements, the mathematical framework will serve as a valuable resource to understand the potential role of co-substrate cycling in more biological systems. The framework might also guide new experiments that generate data for a systematic evaluation of when and to what extent co-substrate cycling governs flux distributions, e.g. depending on growth rates or response to environmental stress.

  7. Version published to 10.1101/2022.09.05.506656v1 on bioRxiv