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Publication date: 2021
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Taborda Saldida Alves, J. (2021). On a quest for metabolic fluxes: sampling and inference tools using thermodynamics, metabolome and labelling data. University of Groningen.
https://doi.org/10.33612/diss.157440136
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SUMMARY OF THE THESIS
In this thesis, we focussed on developing tools for inference of steady-state metabolic fluxes. In chapter 2, we established a step-wise framework that included: 1) fitting a thermodynamic and stoichiometric model to exchange fluxes and metabolite concen-trations and calculation of bounds of the flux solution space according to the model constraints; 2) sampling of the flux solution space to produce flux distributions; 3) ranking of the flux solutions according to how well they fit to 13C-labelling data
us-ing an isotopomer model. This step-wise approach allowed the estimation of fluxes without use of assumptions on reaction direction or reversibility. In chapter 2, we also developed a new sampling approach that was able to tackle the non-convexity and high-dimensionality of the solution space defined by the thermodynamic and stoichiometric model. To this end, we split the non-convex space in flux and concentration polytopes to sample the flux solution space conditional on the existence of a non-empty concentra-tion polytope for each flux point. Addiconcentra-tionally, to increase the efficiency of the sampler, we split the flux polytope into sectors to avoid thermodynamically infeasible loops, we used an ellipsoid approximation to scale the flux polytope and we implemented a parallel tempering scheme. By applying our framework to two strains of S. cerevisiae, we quantified the uncertainty in flux estimation through sampling and we found new metabolic patterns that differed between the strains. Furthermore, we found that, with this set of constraints and data, labelling data did not significantly further constrain the solution space of fluxes. Using the new sampling method developed in chapter 2, in chapter 3 we explored three different sampling options based on the split of the space in convex polytopes: Rejection Sampler, Conditional Sampler, and Conditional Sampler with Parallel Tempering. Specifically, we evaluated the efficiency of the three
273 In chapter 2 we show how to infer metabolic fluxes using stoichiometry, thermodynam-ics and isotopomer balances allied to diverse types of experimental data. We envision that the generation of more thorough data sets would help decrease flux variability. Particularly, if we can have accurate measurements of exchange rates of gases (oxygen and carbon dioxide) and if we can also extend the number of metabolites for which we have concentration measurements, the flux bounds after the variability analysis step would be narrower. Additionally, the use of labelling data on more parallel labelling experiments, or even different stable isotopes, to fit the isotopomer model would help to refine the fit score generated for each sampled net flux point (Antoniewicz, 2015; Henry et al., 2006; Jacobson et al., 2019; Leighty & Antoniewicz, 2013; Park et al., 2019). In the future, an analysis of the effect of increased data sets in the flux estima-tions should be performed. Of utmost importance for accurate flux estimation, beyond increasing the data sets used, is the more experimentally challenging measurement of metabolite concentrations and labelling patterns compartment-specific (Lu et al., 2017; Töpfer et al., 2015). As opposed to the current common approach of averag-ing concentrations or assumaverag-ing most of the labellaverag-ing occurs in one compartment, if we can measure compartment-specific quantities it would immensely increase the prediction power of our flux inference method.
New opportunities in sampling non-convex spaces
While in chapter 2 we show the first application of the new sampling method de-veloped in this thesis, in chapter 3 we further explore the method and its efficiency when applied to different networks and under different levels of constraint complex-ity. One of the most essential points that allowed the sampling of non-convex high-dimensional flux solution spaces was the split of the non-convex space into convex
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flux and concentration polytopes. We envision that this technique will open doors for new mathematical formulations of non-convex sampling problems, particularly as an alternative to the commonly employed rejection sampling (Price et al., 2006; Saa & Nielsen, 2016), where points that violate the constraints are eliminated after sam-pling, for which the efficiency decreases with network size and the addition of more constraints (Kiatsupaibul et al., 2011), as also shown in chapter 3. This mathematical formulation could potentially be used in the computation of states of integrated meta-bolic and regulatory networks (Yu & Blair, 2019) or to sample the space of detailed thermodynamically consistent kinetic models (Saa & Nielsen, 2015) including more complex kinetic constraints.
Quantification of flux uncertainty
The flux inference pipeline developed in chapter 2 brings a new approach to quantify uncertainty in flux estimates. Specifically, because we sample the thermodynamically constrained flux solution space and only then provide thermodynamically feasible flux points to be fitted and scored with labelling data, we take into account the un-certainty of the fluxes created by general lack of information, after using exchange fluxes, metabolite concentrations and thermodynamic principles to decrease the size of the solution space. Future studies should consider the non-linearities of the solution space, whether from thermodynamics, isotopomer balances or both, on the applied methods to quantify flux uncertainty. Explicit sampling as in this work or Bayesian analysis (Theorell et al., 2017) can be used to this end. Furthermore, evaluation of different methods to quantify uncertainty in fluxes should be evaluated in parallel in order to assess if they produce similar results.
Validation of fundamental hypotheses
A general prospect for the new flux inference method developed in chapter 2 is to validate current hypotheses on fundamental metabolism. It is highly relevant to avoid potentially erroneous assumptions on flux directions when comparing different metabolic modes, as some fundamental mechanisms may be inferred from differences in reaction direction. Even though here we estimate fluxes for a yeast strain that fer-ments and another that mainly respires, if we repeat the study with more strains of intermediate levels of energy producing routes, we can attempt to validate theories on the mechanism that causes the metabolic shift between oxidative and fermentative operation (Frick & Wittmann, 2005). Another genre of hypothesis that can potentially
275 if applied to more complex metabolism, e.g. mammalian cells that grow on complex medium, we envision that our pipeline will help in identifying metabolic changes that characterise environmental and genetic modifications. Although, computation time and memory in the fitting of the isotopomer model to labelling data greatly increases with number of fluxes in the network. Thus, the development of new computational methods is beneficial to increase computation efficiency of optimisations with the isotopomer model in the near future.
CONCLUSION
The framework developed in this work for inference of metabolic fluxes puts together diverse types of experimental data – exchange fluxes, metabolite concentrations, labelling data -, physical constraints – stoichiometry, thermodynamics, isotopomer balances -, and methodology – optimisation, sampling - in a step-wise approach that allows such complex problem to be solved in reasonable time. We avoid any a priori assumptions on reaction direction and reversibility, contrarily to current metabolic flux analysis methods. We envision that our method will be highly relevant to esti-mate accurate fluxes and their uncertainty for fundamental research and metabolic engineering.
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