Measurement uncertainty of isotopologue fractions in fluxomics determined via mass spectrometry

Abstract

Metabolic flux analysis implies mass isotopomer distribution analysis and determination of mass isotopologue fractions (IFs) of proteinogenic amino acids of cell cultures. In this work, for the first time, this type of analysis is comprehensively investigated in terms of measurement uncertainty by calculating and comparing budgets for different mass spectrometric techniques. The calculations addressed amino acids of Pichia pastoris grown on 10 % uniformly ¹³C labeled glucose. Typically, such experiments revealed an enrichment of ¹³C by at least one order of magnitude in all proteinogenic amino acids. Liquid chromatography–time-of-flight mass spectrometry (LC-TOFMS), liquid chromatography–tandem mass spectrometry (LC-MS/MS) and gas chromatography–mass spectrometry (GC-MS) analyses were performed. The samples were diluted to fit the linear dynamic range of the mass spectrometers used (10 μM amino acid concentration). The total combined uncertainties of IFs as well as the major uncertainty contributions affecting the IFs were determined for phenylalanine, which was selected as exemplary model compound. A bottom-up uncertainty propagation was performed according to Quantifying Uncertainty in Analytical Measurement and using the Monte Carlo method by considering all factors leading to an IF, i.e., the process of measurement and the addition of ¹³C-glucose. Excellent relative expanded uncertainties (k = 1) of 0.32, 0.75, and 0.96 % were obtained for an IF value of 0.7 by LC-MS/MS, GC-MS, and LC-TOFMS, respectively. The major source of uncertainty, with a relative contribution of 20–80 % of the total uncertainty, was attributed to the signal intensity (absolute counts) uncertainty calculated according to Poisson counting statistics, regardless which of the mass spectrometry platforms was used. Uncertainty due to measurement repeatability was of importance in LC-MS/MS, showing a relative contribution up to 47 % of the total uncertainty, whereas for GC-MS and LC-TOFMS the average contribution was lower (30 and 15 %, respectively). Moreover, the IF actually present also depends on the isotopic purity of the carbon sources. Therefore, in the uncertainty calculation a carbon source purity factor was introduced and a minor contribution to the total uncertainty was observed. The results obtained by uncertainty calculation performed according to the Monte Carlo method were in agreement with the uncertainty value of the Kragten approach and showed a Gaussian distribution.

Introduction

Metabolic engineering addresses the improvement or alteration of the performances and properties of cells via modification of their genetic material for industrial and/or scientific purposes. Nowadays, transcriptomic and proteomic techniques are widely used to analyze cellular behavior and for elucidation of targets for genetic engineering. However, these technologies can only indirectly reflect the metabolic state of an organism. A rather new and global technique is offered by another omics approach aiming at the analysis of metabolic fluxes in vivo named fluxomics [1]. In metabolic flux analysis (MFA) experiments, for instance, substrate uptake and CO₂ secretion rates are measured and implemented together with information on stable isotopes tracers (e.g., ¹³C) in the stoichiometric network, creating a constrained-based model of metabolic fluxes in the cell. The distribution of the stable isotope in the metabolites of an organism is used to calculate the cellular fluxes by either using isotopomer balancing and iterative fitting [2] or locally calculating flux ratios [3]. An essential part of MFA as well as of all other fluxomics experiments is to accurately determine the isotope-enriched species distributions (i.e., mass isotopomer distributions, MIDs, or isotopologue fractions) resulting from labeling experiments with target cellular metabolites.

Mass spectrometry in MFA

The isotope-enriched analytes generated in MFA experiments are called isotopomers—isobaric substances with the same number of labeling isotopes (e.g., ¹³C) in different positions—and isotopologues—substances with different numbers of isotopes (Fig. 1). The role of the analytical chemist in fluxomics workflows is to provide a vector assigning the relative abundance to each isotope-containing species within the population of the selected analyte. This vector—also called mass isotopomer distribution (MID)—is then employed for isotopomer balancing [4].

Fig. 1

Theoretical isotopologue and isotopomer species obtained after ¹³C labeling of a compound containing three carbon atoms

Mass spectrometry (MS) and nuclear magnetic resonance are the techniques of choice for measurement of MIDs in flux experiments [4]. Besides its significantly higher sensitivity, MS is advantageous thanks to its routine compatibility with gas chromatography (GC) or liquid chromatography (LC) as well as capillary electrophoresis, allowing selective preseparation of the metabolites of interest. However, regarding isotopomer analysis it is mandatory that MS-based platforms provide a fragmentation process allowing positional assessment of the ¹³C-labeling. Accordingly, besides nuclear magnetic resonance, comprehensive MID analysis can only be achieved by using either a GC-MS system with electron impact fragmentation or MS instrumentation equipped with collision-induced dissociation/fragmentation capability [5].

The analytical quality of the measured MIDs can be evaluated by two parameters: accuracy (or trueness) and precision. The accuracy of the MID is the bias between the measured and the true value, which can be assessed if the theoretical ¹³C fraction in the total amount of carbon of the metabolites is known. Precision can be expressed as (1) measurement repeatability, where “the measurements are being made on portions of the same material by a single analyst, using the same procedure, under the same operating conditions over a short time period,” (2) intermediate measurement precision, and (3) measurement reproducibility [6]. In this work measurement repeatability is considered in the uncertainty budgeting. The precision of MS data plays a fundamental role in the isotopomer-balancing procedure, as the standard deviation (SD) of the MID directly influences the quality of the calculated model [4]. For this reason it is mandatory to implement measures for quality control to accurately quantify and assess the uncertainty of MID data.

Measurement uncertainty

In the recent years two approaches for the determination of measurement uncertainty have stood out in the large field of quality assurance and quality control: the top-down approach, which is focused on the determination of measurement uncertainty from measurement reproducibility, and the bottom-up approach—reported in the EURACHEM/CITAC guide for measurement uncertainty Quantifying Uncertainty in Analytical Measurement [7], which focuses on the determination of measurement uncertainty by assessment and combination of different uncertainty sources. In the present work a measurement uncertainty budget according to the bottom-up approach was employed to assess the different uncertainty contributions to the total combined uncertainty of measurement of isotopologue fractions done by three different mass spectrometers. In metrology measurement uncertainty is associated with each quantity value, also called measurand. In Terminology in Analytical Measurement – Introduction to VIM 3 [6] it is defined as follows: “A non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used.”

It is noteworthy that the Guide to the Expression of Uncertainty in Measurement (GUM) bottom-up approach does not take in account any bias affecting the measurand because the measurand has to fulfill the requirement of traceability. The definition of traceability according to Terminology in Analytical Measurement – Introduction to VIM 3 is as follows: “The property of a measurement result, whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty.”

According to this definition, all measurements results, which are traceable, can be corrected for a possible bias. The original and detailed bottom-up approach protocol is listed in the original guide [7] and is discussed in several excellent reviews [8–10]. Basically the bottom-up method is a protocol which implies the identification and quantitation of all uncertainty sources present in a procedure from sampling and sample preparation to data evaluation. The variability of those sources has to be first transformed into a standard uncertainty (uncertainty expressed as SD) and subsequently entered into an appropriate mathematical model in order to calculate the total combined standard uncertainty (u _c) which characterizes the dispersion of the measurand during the experiment. The combined standard uncertainty is then multiplied by a coverage factor k (generally a value between 2 and 3 according to the desired level of confidence) in order to calculate the expanded uncertainty U. The expanded uncertainty is the interval in which the true measurand value is present with probability corresponding to the level of confidence chosen. The Kragten spreadsheet [11] and informatics platform as the GUM workbench [12] are fit-for-purpose tools to calculate uncertainty and the relative contribution of the individual uncertainties of the different input quantities.

Monte Carlo method

In 2008 the Joint Committee for Guides in Metrology published a supplement to Quantifying Uncertainty in Analytical Measurement [13] which details the application of the Monte Carlo method (MCM) in measurement uncertainty calculation. The Kragten approach suffers from several limitations regarding the approximation of the mathematical model used to combine the uncertainties recovered from nonlinear input quantities. The MCM applied to spreadsheets allows random changing of the input parameter values within their standard uncertainties. Through the model equation, the randomly generated input quantities are used to calculate a wide measurand distribution (10⁵–10⁶ simulations). The mean value and SD represent the measurand value, y, and the corresponding standard uncertainty, u(y), respectively. The final distribution of the measurement results is the result of the interaction of all probability distribution functions (PDFs) of the input quantities involved [14] and its graphical representation is a powerful tool for understanding the corresponding uncertainty contributions. Furthermore, the direct study of MCM simulation results allows one to design a coverage factor that can afterwards be employed to expand the uncertainty obtained via the Kragten approach in order to accurately encompass the desired fraction of measurand observations.

Aim of study

The present study aims at the creation of an easy and straightforward workflow for quality assessment of MS-based data in fluxomics experiments. When a mixture of labeled/unlabeled substrates is fed to an organism, the resulting labeled pattern of molecules is a black box, which is investigated by MS. Although the relative abundance of isotope-containing species within a population is not significantly affected by any process during sample preparation, it is prone to mass interference and precision problems characteristic of the MS platform. The uncertainty determination via bottom-up approach studies of the physiological and random fluctuations of a measurement is the appropriate method for this purpose. We have tested the workflow developed using phenylalanine as a model compound, since it can be, theoretically, isotopically enriched at ten positions, producing a large number of isotopologues, some of which are challenging in terms of detection owing to their low abundance in the biological experiment investigated. The analysis of ¹³C-labeled phenylalanine (¹³C-Phe) extract from Pichia pastoris protein lysates with an unknown labeling pattern was performed by employing three different MS platforms, i.e., LC–time-of-flight (TOF) MS (TOFMS), LC-MS/MS and GC-MS. Uncertainty was calculated via the Kragten approach and the MCM, and the resulting uncertainty budgets are discussed for all MS instruments.

Experimental

Chemicals

LC-MS grade acetonitrile and water were purchased from Sigma-Aldrich (Vienna, Austria), and hydrochloric acid (30 % ultrapure) and formic acid 98–100 % Suprapur® where purchased from Merck (Darmstadt, Germany). The reagents used to produce P. pastoris media were all purchased from Merck, except FeSO₄·7H₂O, which was purchased from Carl Roth (Karlsruhe, Germany). The unlabeled glucose and uniformly ¹³C labeled glucose ([U-¹³C]glucose) (99 % purity grade) were from Carl Roth and Cortecnet (Voisins-Le-Bretonneux, France), respectively. The derivatizing materials anhydrous dimethylformamide and N-tert-butyldimethylsilyl-N-methyltrifluoroacetamide with 1 % tert-butyldimethylchlorosilane were purchased from Sigma-Aldrich (Vienna, Austria). ¹³C₁-Phe (99 % purity grade) was purchased from Cambridge Isotope Laboratories (Andover, MA, USA).

Instrumental

Cell precipitation was performed with a Sorvall RC 6+ centrifuge from Thermo Scientific (Waltham, MA, USA). The samples were brought to dryness with an SD131DDA SpeedVac concentrator equipped with a Savant RVT400, both from Thermo Scientific (Waltham, MA, USA).

For LC-MS analysis, a G1312A 1200 series binary pump from Agilent Technologies together with an Agilent G1367B high-performance autosampler and an Agilent G1316A column compartment was employed for high-performance LC. The MS analyzer was a 6410 triple-quadrupole system from Agilent Technologies (Palo Alto, CA, USA). Alternatively, a G1312A 1200 series binary pump from Agilent Technologies together with an Agilent G1316A column compartment equipped with a CTC HTC PAL50 from CTC Analytics (Zwingen, Scheiz) was combined with a 6210 TOF system from Agilent Technologies. The SeQuant 4.6 mm × 150 mm ZIC®-HILIC separation column with 3.5-μm particle size and the 4.6 mm × 20 mm ZIC®-HILIC guard column with 3.5-μm particle size were from Merck (Darmstadt, Germany).

For GC-MS analysis, an Agilent Technologies 6890 N gas chromatograph with a 5975 mass spectrometer detector was used. The derivatives were separated on a Phenomenex (Aschaffenburg, Germany) Zebron ZB-5MS GC-column (30 m × 0.25-mm inner diameter, 0.25-μm film thickness, 5 % diphenyl 95 % dimethyl polysiloxane stationary phase) equipped with a Phenomenex GC guard column (3 mm × 0.25 mm inner diameter).

Chemostat cultivation

One hundred milliliters of preculture medium (10 g yeast extract, 20 g peptone, and 10 g glycerol per liter) was inoculated from the working cell bank (750 μL cryostock of P. pastoris CBS7435) and grown at 28 °C and 150 rpm overnight. This culture was used for inoculation of the bioreactor at an optical density at 600 nm of 1.0. After a batch phase of approximately 24 h, the feed and harvest for the continuous chemostat cultivation were started. To achieve a controlled steady-state in chemostat cultivations, the cells were grown under glucose-limited conditions with a dilution rate of 0.1 h^-1. For six residence times, cells were grown on minimal medium until they reached a metabolic steady state. After the steady state had been reached, 10 % of the naturally labeled glucose in the feed medium was replaced by [U-¹³C]glucose. Samples were taken after one residence time. During fermentation, temperature, pH, and dissolved oxygen concentration were maintained at 25 °C, 5.0 (with 8 M KOH), and 20 % (by controlling the stirred speed and inlet gas flow), respectively. The batch medium contained 39.9 g glycerol, 1.8 g citric acid, 12.6 g (NH₄)₂HPO₄, 0.022 g CaCl₂·2H₂O, 0.9 g KCl, 0.5 g MgSO₄·7H₂O, 2 mL biotin (0.2 g L^-1), and 4.6 mL trace salts stock solution per liter. The pH was set to 5.0 with 32 % (w/w) HCl. The chemostat medium contained 55 g glucose, 2.3 g citric acid, 21.75 g (NH₄)₂HPO₄, 0.04 g CaCl₂·2H₂O, 2.5 g KCl, 1.0 g MgSO₄·7H₂O, 2 g biotin (0.2 g L^-1), and 2.43 g trace salts stock solution per liter. The pH was set to 5.0 with 32 % (w/w) HCl. The trace salts stock solution contained 6.0 g (CuSO₄)₅H₂O, 0.08 g NaI, 3.0 g (MnSO₄)H₂O, 0.2 g Na₂MoO₄·2H₂O, 0.02 g H₃BO₃, 0.5 g CoCl₂, 20.0 g ZnCl₂, 5.0 g FeSO₄·7H₂O, and 5.0 mL H₂SO₄ (95–98 % w/w) per liter.

Sampling and quenching of metabolism

Samples for analysis of intracellular metabolites were taken rapidly by using a peristaltic pump. Approximately 50 mL fermentation broth was quenched in 200 mL of 60 % (v/v) methanol at −27 °C. Then, 2 mL of the quenched cell suspension (corresponding to approximately 10 mg biomass) was filtered on cellulose acetate filters (0.45 μm, Sartorius Biolab Products) by applying a vacuum. Cell dry weight (CDW) was determined by drying duplicates of 2 mL chemostat culture to constant weight at 105 °C in preweighed glass tubes.

Acidic protein hydrolysis

The harvested P. pastoris cells were hydrolyzed overnight (16 h) using 1 mL of 6 M HCl per 10 mg CDW. The debris was precipitated by centrifugation and the supernatant was transferred to a new tube. There the HCl solution was brought to dryness via SpeedVac centrifugation and the metabolites were resuspended in 0.1 M HCl, resulting in an average concentration of proteinogenic amino acids of 5 μM. The theoretical concentration was calculated assuming that 40 % of the P. pastoris CDW is composed of proteins [15].

Standard preparation

The synthetic standard mixtures were prepared by accurate weighing and dissolving amino acid standards in a 0.1 M HCl solution. The following solutions of monolabeled ¹³C-Phe and naturally labeled phenylalanine (^natPhe) were prepared: pure ¹³C-Phe, 10:1 ¹³C-Phe/^natPhe, 1:2 ¹³C-Phe/^natPhe, 1:10 ¹³C-Phe/^natPhe, and pure ^natPhe. This set of solutions was prepared at three concentrations: 0.100, 1.00, and 10.0 μM.

Direct infusion MS

To assess the purity factor specified by the manufacturers of the ^natPhe and ¹³C-Phe, two direct-infusion experiments were performed by connecting a syringe pump to the TOFMS system and MS/MS analyzer. Three concentrations, 0.100, 1.00, and 10.0 μM, and a blank solution were measured at three different scan times (250, 500, and 1,000 ms) to evaluate the effect of concentration and scan time on isotope ratio accuracy and precision.

Amino acid analysis via LC-TOFMS and LC-MS/MS

The separation of proteinogenic amino acids was done a 4.6 mm × 150 mm ZIC-HILIC separation column (3.5-μm particle size; Merck, Darmstadt, Germany) with a 4.6 mm × 20 mm ZIC-HILIC guard column (3.5-μm particle size). The mobile phase was formed by mixing eluent A (98 % v/v H₂O, 1 % v/v acetonitrile, 1 % v/v HCOOH) and eluent B (98 % v/v acetonitrile, 1 % v/v H₂O, 1 % v/v HCOOH) and applying the following gradient: 90 % eluent B was maintained for 1 min, then eluent B was reduced to 10 % within 4 min and was this held for 1 min. Subsequent reconstitution of the starting conditions within 0.1 min and reequilibration with 90 % eluent B for 10 min resulted in a total analysis time of 15 min. The flow rate was set to 1 mL/min. The column temperature was 40 °C.

For electrospray ionization (ESI) TOFMS the parameters were as follows: dual ESI mode with the standard spacer equipped, positive polarity, drying gas 10 L/min heated at 350 °C, nebulizer pressure 55 psig, and capillary voltage held at 4,000 V. The fragmentor, skimmer, and first octupole voltages were 150, 60, and 250 V, respectively. The scan rate was set to 4 spectra per second (equivalent to a dwell time of 250 ms) and the TOF system was run in 4-Ghz acquisition mode.

The ESI-MS/MS parameters comprised single ESI mode with the standard spacer equipped, positive polarity, drying gas 10 L/min heated at 350 °C, nebulizer 50 psig, and capillary voltage held at 4,000 V. The analysis was performed in multiple reaction monitoring mode (MRM), and the selected transitions for phenylalanine analysis were as follows: ^natPhe 166.1→120; ¹³C-Phe 167.1→120, 167.1→121; ¹³C₂-Phe 168.1→121, 168.1→122; ¹³C₃-Phe 169.1→120, 169.1→121, 169.1→122, 169.1→123; ¹³C₄-Phe 170.1→120, 170.1→121, 170.1→122, 170.1→123, 170.1→124. The fragmentor voltage and collision energy set for MRM transitions were 80 V and 9 V, respectively, and the dwell time was set to 15 ms.

Amino acid analysis via GC-MS

Immediately after complete drying of the proteinogenic amino acids, they were dissolved in 40 μL anhydrous dimethylformamide per milligram CDW. Subsequently, an equal volume of derivatizing agent (N-tert-butyldimethylsilyl-N-methyltrifluoroacetamide) was added. The samples were derivatized for 1 h at 85 °C. The samples were then transferred to GC vials and analyzed within 24 h. Aliquots of 1 μL were injected in split mode (split ratio 1:10) at a temperature of 230 °C, and the carrier gas flow rate was set to 1.5 mL/min. The separation was performed with the following oven gradient: hold for 1 min at 160 °C, ramp to 310 °C at 20 °C/min, hold for 0.5 min at 310 °C. The quadrupole was operated in full-scan mode from 180 to 550 m/z with a scan rate of seven scans per second. The source temperature was set to 280 °C and the solvent delay time was set to 2.7 min. Tuning of the instrument was performed as described by Zamboni et al. [15].

Measurement of phenylalanine isotopologue ratios

The individual isotopologue fractions were obtained by dividing the measured area of the respective isotopologues by the sum of all measured isotopologue areas of the target compound within the same chromatogram (Eq. 1). For each sample, six repetitive injections were performed, the isotopologues fractions derived from each injection were then averaged, and the repeatability precision was calculated. To compare GC-MS data with the LC-MS data, the software MS correction tool [16] was employed to correct the measured isotope abundances for the contribution coming from the derivatizing agent.

Uncertainty calculation

The uncertainty budget was built following the official protocol given in Quantifying Uncertainty in Analytical Measurement [7].

Definition of measurand

The measurands are the relative abundances of the isotopologue species IF⁺⁰, IF⁺¹, IF⁺², IF⁺³ and IF⁺⁴ of phenylalanine (isotopologue fractions).

The model equation used for uncertainty calculation was established as follows:

$$ \mathrm{IF}_{\mathrm{meas}}^{+x }=\left( {\frac{{{A^{+x }}}}{{\sum\nolimits_{x=0}^n {{A^{+x }}} }}} \right)K_{{\mathrm{rep},}}^{+x } $$

(1)

where $\mathrm{IF}_{\mathrm{meas}}^{+ \it x }$ is the measured isotopologue fraction, n is the metabolite number of carbon, and x is the number of ¹³C atoms present in the backbone of the metabolite. A ⁺⁰ to A ⁺⁴ represent the peak areas of the isotopologues which have been considered. It is noteworthy that these areas correspond to the total number of ions which have been measured by the respective instruments. $K_{\mathrm{rep}}^{+x }$ is the measurement repeatability precision of each single isotopologue fraction (IF⁺⁰→$K_{\mathrm{rep}}^{+0 }$, IF⁺¹→$K_{\mathrm{rep}}^{+1 }$, etc.).

As the uncertainty of the number of ions counted for each measured isotopologue u(A ⁺ⁿ) depends not only on counting statistics but also on varying instrumental conditions, there is correlation between all A ⁺ⁿ and the respective IF⁺ⁿ. Accordingly, a correlation study regarding A ⁺ⁿ and $K_{\mathrm{rep}}^{+n }$ was performed, and the correlation coefficients were implemented in the uncertainty calculation together with the covariance of the correlated uncertainty sources (see Eq. 6).

In addition to measurement precision of isotopologue fractions, the isotopic purity of the carbon sources and the weight of added substrate are two relevant sources of error. Accordingly, the isotopologue fractions of the isotopologues containing no ¹³C atoms or one ¹³C atom are biased by the purity factor of the feed substrates. To correct for this contribution, correction factors for IF⁺⁰ (Eq. 2) and IF⁺¹ (Eq. 3) are introduced:

$$ \mathrm{c}{{\mathrm{f}}_{{12\mathrm{C}}}}=\frac{{\mathrm{P}{{\mathrm{f}}^{12 }}\mathrm{C}\times W_{\mathrm{sub}}^{12 }}}{{\mathrm{P}{{\mathrm{f}}^{12 }}\mathrm{C}\times W_{\mathrm{sub}}^{12 }+\left( {1-\mathrm{P}{{\mathrm{f}}^{13 }}\mathrm{C}} \right)\times W_{\mathrm{sub}}^{13 }}}, $$

(2)

$$ \mathrm{c}{{\mathrm{f}}_{{13\mathrm{C}}}}=\frac{{\mathrm{P}{{\mathrm{f}}^{13 }}\mathrm{C}\times W_{\mathrm{sub}}^{13 }}}{{\mathrm{P}{{\mathrm{f}}^{13 }}\mathrm{C}\times W_{\mathrm{sub}}^{13 }+\left( {1-\mathrm{P}{{\mathrm{f}}^{12 }}\mathrm{C}} \right)\times W_{\mathrm{sub}}^{12 }}}. $$

(3)

The definition of the correction factors cf_12C and cf_13C in Eqs. 2 and 3 is based on the purity factors of the labeled and the unlabeled substrate (Pf¹³C and Pf¹²C, respectively), which are defined as the ¹³C content in ^natC-glucose and the ^natC content in [U-¹³C]glucose, respectively. The correction factors are implemented in the model because of the impurity of the substrate, i.e., ^natC-glucose with one ¹³C atom or [U-¹³C]glucose with one ¹²C atom. These isotopes enter the flux and produce isotopologues that differ by ±1 Da from the expected molecular mass. Evidently, this effect is only significant for a mass shift of ±1, and can be disregarded for mass shifts of more than 1 Da (e.g., ±2/±3). Therefore, only the shifts +1 and −1 of IF⁺⁰ and IF⁺¹ were considered in the model. Moreover, the natural abundance of other isotopes such as deuterium, ¹⁵N, and ¹⁸O randomly appears without modifying the isotopologue fractions of ¹³C. To the best of our knowledge, this is the first time that the atomic purity factor of substrates has been considered in the context of measurement uncertainty in fluxomics.

Additionally, the uncertainty of the weight of labeled and unlabeled substrate ($W_{\mathrm{sub}}^{12 }$ and $W_{\mathrm{sub}}^{13 }$) fed to the microorganism is considered.

The corrected IF⁺⁰ and IF⁺¹ can then be calculated by using the final model equations (Eqs. 4, 5):

$$ \mathrm{IF}_{\mathrm{corr}}^{+0 }=\mathrm{IF}_{\mathrm{meas}}^{+0}\times \mathrm{c}{{\mathrm{f}}_{{12\mathrm{C}}}}+\mathrm{IF}_{\mathrm{meas}}^{+1}\times \left( {1-\mathrm{c}{{\mathrm{f}}_{{13\mathrm{C}}}}} \right), $$

(4)

$$ \mathrm{IF}_{\mathrm{corr}}^{+1 }=\mathrm{IF}_{\mathrm{meas}}^{+1}\times \mathrm{c}{{\mathrm{f}}_{{13\mathrm{C}}}}+\mathrm{IF}_{\mathrm{meas}}^{+0}\times \left( {1-\mathrm{c}{{\mathrm{f}}_{{12\mathrm{C}}}}} \right). $$

(5)

The correction factors are designed to correct the isotopologue fractions for the carbon contribution deriving from the substrates. All other possible contributions (¹⁸O, ¹⁵N, and ²H) are not taken into account either because their relative abundances affect all isotopologue fractions equally or because their natural relative abundance is negligible. The bias introduced by the isotopic purity of the substrate is usually corrected by the bioinformatic platforms. However, the uncertainty linked to this bias is then not evaluated. To be able to assess the contribution of the purity of the substrate to the total combined uncertainty, the correction factors are implemented in the model equation and their uncertainty is quantified.

Identification of uncertainty sources

Table 1 lists all sources of uncertainty contributing to the total combined uncertainty of isotopologue fractions as considered in the model equation.

Table 1 Input quantities considered in the model equation of the isotopologue fractions (IFs) as a measurand for mass spectrometry based techniques

Quantification of uncertainty sources related to input quantities

The uncertainty sources linked to the intensity of the measured isotopologues are calculated according to Poisson counting statistics (type A uncertainty) criteria, which express the SD as the square root of the total counts, to which is added the covariance of signal intensity and measurement repeatability. The repeatability precision was calculated from the results of six repetitive injections on all three MS systems (type A uncertainty); this contribution groups all the uncertainty sources that affect the measurand. The purity factors and their uncertainties were specified by the manufacturers (type B uncertainty). The uncertainty linked to the weight of labeled and unlabeled substrates fed to the organism is specified in the manuals of the balances used (type B uncertainty). For all the type B uncertainty involved in the calculation, a rectangular distribution was assumed.

Calculation of total combined uncertainty via the Kragten spreadsheet

The total combined uncertainty was calculated by using the standard equation (Eq. 6) provided in Quantifying Uncertainty in Analytical Measurement:

$$ {u_c}\left( {{y_{{\left( {{x_1},{x_2}..,,{x_n}} \right)}}}} \right)=y\sqrt{{\sum\nolimits_{x=1,n } {{{{\left( {\frac{u(x) }{{{y_x}}}} \right)}}^2},} }} $$

(6)

where u _c is the total combined uncertainty of the measurand (y) depending on the input quantities (x ₁ , x ₂ ,..,x _n ), and y is the measurand result multiplied by the square root of the sum of the squares of the relative contributions to the input quantities u(x). The expanded uncertainty was then calculated using a coverage factor of k = 1.

The nonindependent uncertainty sources were implemented in the Kragten spreadsheet by adding to the single uncertainty contribution the covariance between the correlated sources.

The covariance was expressed according to GUM as follows:

$$ u\left( {{x_1},{x_2}} \right)=u\left( {{x_1}} \right)u\left( {{x_{12 }}} \right){r_{{{x_1},{x_2}}}}. $$

(7)

In Eq. 7 the covariance, u(x ₁, x ₂), between uncertainty sources x ₁ and x ₂ is calculated by multiplying the uncertainties of the two sources [u(x ₁) and u(x ₂), respectively] by the correlation coefficient $\left( {{r_{{{x_1},{x_2}}}}} \right)$, which expresses the relationship between the two variables. In this case the sensitivity coefficient was assumed to be equal to 1 because the effect of the uncertainty source on the final measurand result (isotopologue fraction) was evaluated directly.

The uncertainties obtained were then propagated according to the Kragten approach in an Excel spreadsheet [8, 13] in order to evaluate their relative contribution.

Calculation of total combined uncertainty via Monte Carlo simulation

In parallel to the Kragten approach, the uncertainty calculation was also performed by using the MCM in an Excel spreadsheet.

The PDFs of the associated input quantities were the same as those of the Kragten uncertainty propagation. According to the model equation, 2 × 10⁴ random simulations were performed and the measurand data distribution was subsequently displayed in a frequency chart.

From the measurand distribution, a mean value Y and the respective combined standard uncertainty u(y) expressed as SD were calculated.

Comparison of the Kragten spreadsheet calculation and the MCM

The measurand interval (Y ± U _Kragten or Y ± Ku _(y)) obtained from the Kragten calculation was compared with the measurand distribution (Y ± SD_PDF) calculated with the MCM. Whenever the measurement result simulations performed via the MCM show a Gaussian distribution, the mathematical hypothesis and models assumed in the bottom-up approach are assessed and thus the uncertainty calculation is validated. Otherwise, different models for combining standard uncertainties have to be selected as well as the conversion from uncertainty to equivalent standard uncertainties.

Results and discussion

Direct infusion analysis

Precision of ESI-TOFMS and ESI-MS/MS signals according to the Poisson distribution

Pure standard solutions were continuously infused into the ESI-MS systems (MS/MS and TOFMS) in order to evaluate the influence of scan time and analyte concentration on accuracy and precision of the measured isotopologue fractions. Solutions containing phenylalanine with natural isotopic composition (^natPhe) at different concentrations (0.100, 1.00, and 10.0 μM) were delivered at a constant flow rate of 10 μL/min. The TOF scan rate was increased from one spectrum per second to two spectra per second and further to four spectra per second by reducing the scan time from 1,000 ms to 500 and 250 ms, respectively. Four spectra per second was the upper limit regarding the ESI-TOFMS system, as higher scan rates resulted in unacceptably high baseline noise. These scan times were set as corresponding dwell times in the ESI-MS/MS system. Figure 2 depicts the theoretical SD calculated assuming a Poisson distribution of the total ion counts obtained with the two systems. Both mass spectrometers show a theoretical relative SD (RSD) below 5 %, with the RSD of the signals calculated for the ESI-MS/MS system being up to five times higher than for the ESI-TOFMS system. This clearly indicates that counting statistics are a limitation regarding precision of ESI-MS/MS signals, where a high fraction of analyte is lost in the collision cell and only a low amount of the final analyte, i.e., ionic fragments, is transmitted to the detector. Accordingly, increasing the analyte concentration leads to higher precision (lower RSD), whereas a decrease of the scan time from 1,000 to 250 ms does not significantly reduce precision, since this is clearly not the lower dwell time limit of the system employed.

Fig. 2

Relative standard deviation (RSD) of the Poisson distribution calculated from a electrospray ionization (ESI) tandem mass spectrometry (MS/MS) and b ESI time-of flight mass spectrometry (TOFMS) signals measured at different scan rates and analyte concentrations

The low Poisson RSD of the ESI-TOFMS system is related to the higher transmission and total ion counts compared with the ESI-MS/MS system. As can be seen, both reduction of scan time and decreasing analyte concentration have a strong impact on the SD.

Precision of isotopologue ratios

To evaluate the accuracy and precision of isotopologue ratios obtained by the two mass spectrometers with regard to the signal intensity varied via infusion of different analyte concentrations or different scan times, the relative abundance of ¹²C in a ^natPhe standard was determined (six replicates). As can be seen in Fig. 3, the relative abundance of ¹²C-Phe measured by direct infusion ESI-TOFMS and ESI-MS/MS shows a maximum bias of 1 % regarding the theoretical value, whereas precisions below 1 % RSD were achieved. Only at the lowest concentration examined was the precision of LC-TOFMS in the range of 3 %.

Fig. 3

Relative abundance of ¹²C-phenylalanine (¹²C-Phe) in natural phenylalanine (^nat Phe) determined by a direct infusion ESI-TOFMS and b ESI-MS/MS at different scan times and analyte concentrations. The red line represents the natural relative abundance of ¹²C-Phe in ^natPhe for the ESI-TOFMS and ESI-MS/MS systems (fragments m/z 166 to 120 and m/z 167 to 121). The theoretical values obtained with the two platforms are different since the ESI-MS/MS analysis causes the loss of the carbonyl group of the phenylalanine; therefore, the relative abundance of ¹³C was calculated for eight carbon atoms instead of nine

Again LC-TOFMS shows a strong influence of counting statistics with pronounced improvement of precision with increasing analyte concentration and increasing scan time. The LC-MS/MS data reveal high precision even at low concentrations independent of the concentration and scan rate, showing only a moderate effect of signal intensity and ion counts. This clearly contradicts the observation above, where the theoretical precision of the signal obtained with 0.100 μM ^natPhe solution was in the range of only 5 % and can be explained by the high scan rate of the MS/MS system leading to a quasi-simultaneous measurement of the adjacent m + 0 and m + 1 isotopologues with a precise isotopologue ratio.

The accuracy of the relative abundance of ¹²C-Phe in ^natPhe changes by approximately 1 % for the LC-TOFMS system when analyte concentration is increased from 0.100 to 1.00 μM. These results indicate that the isotopologue ratios calculated at low concentrations are systematically biased by the background, which interferes with the less abundant isotopologue peak. The decrease of the relative abundance measured in the 10 μM solution can be explained by a saturation effect of the TOF detector. Regarding the different scan times adopted, it is noteworthy that with LC-TOFMS a reduction of the scan time produces a less stable signal that is translated into a high variability (RSD), whereas LC-MS/MS does not suffer from this effect. Hence, it can be concluded that the accuracy and precision of isotopologue fractions obtained by LC-TOFMS are limited by Poisson statistics and the narrow dynamic range of the detector.

As the direct infusion experiment confirmed that concentration and scan rate have a significant impact on the accuracy and precision of isotope ratio measurement, the optimum scan rates and concentration ranges were used according to the experiments described earlier (see “Experimental”). The GC-MS settings were chosen according to Zamboni et al. [15].

Distribution of phenylalanine isotopologues measured by LC-TOFMS, LC-MS/MS, and GC-MS

The isotopologue fractions of the amino acids obtained from P. pastoris cell hydrolysates from a ¹³C MFA experiment were measured by applying the optimum concentrations and scan rates obtained from the direct infusion experiment to the LC–MS combination. Because of the limited signal-to-noise ratio of the LC-TOFMS system, only five isotopologue fractions were considered in our study. The results shown in Table 2 are the average relative abundances of the target compound phenylalanine calculated as the mean of six repetitive injections (isotopologue fraction), the RSD of each fraction, and the expanded uncertainties derived via Kragten spreadsheet calculation (U _Kragten) and the MCM (U _MCM), which additionally consider the uncertainty contribution of the isotopic purity of the carbon sources and the uncertainty of the weight of the added substrate.

Table 2 Phenylalanine IFs for Pichia pastoris cell hydrolysates (cells grown on 10 % [U-¹³C]glucose) derived by gas chromatography–mass spectrometry (GC-MS), liquid chromatography–time-of-flight mass spectrometry (LC-TOFMS), and liquid chromatography–tandem mass spectrometry (LC-MS/MS) measurements (n = 6). Three types of data dispersion are reported. The relative standard deviation (RSD; as a percentage) of the IFs was calculated for six repetitive injections. The relative uncertainty was calculated according to the model equation given in “Experimental” via the Kragten spreadsheet (U _Kragten) and the Monte Carlo method (U _MCM). The former uncertainty is expanded with a coverage factor of k = 1

Accuracy and precision of isotopologue fractions

The IFs given in Table 2 show good agreement regarding the relative distribution of isotopologues obtained via GC-MS and LC-MS/MS. LC-TOFMS resulted in an underestimation of the unlabeled species because of the high signal intensity of IF⁺⁰ (total counts 1.4 × 10⁶), which causes a detector saturation effect owing to the limited linear dynamic range of the TOF detector (as mentioned earlier). The consequence of this is the constant overestimation of the relative abundances of all the other isotopologues (IF⁺¹ to IF⁺⁴) with respect to the other two MS systems. In accordance with the results obtained by the direct infusion experiments, the LC-MS/MS system shows a much higher precision and lower RSD compared with the LC-TOFMS system. This is also true for the values obtained by GC-MS, which offers the highest signal-to-noise ratio and lowest limit of detection in the case of IF⁺⁰.

It is noteworthy that the RSD of the isotopologue fractions obtained with the LC-MS/MS system do not significantly change with decreasing isotopologue fraction values. This is mainly due to the high number of ions counted and the high signal-to-noise ratio of the MRM mode.

Uncertainty of measurement of isotopologue fractions

The measurement uncertainty corresponding to the measured isotopologue fractions is reported in Table 2 as the relative expanded uncertainty calculated by both the Kragten approach (U _Kragten) and the MCM (U _MCM). As can be seen, the uncertainties calculated according to the two methods are consistent. Graphical evaluation of the MCM measurement result simulation revealed a clear Gaussian distribution, proving that the MCM is a good tool for calculating and/or assessing the overall uncertainty in an easy and straightforward way. The relative uncertainties of the first two isotopologue fractions (Phe and ¹³C₁-Phe) are comparable for all three systems, but become more heterogeneous with the decrease of the relative abundance of the ions investigated. Whereas U for the LC-MS/MS data is constantly below 3 %, U for the LC-TOFMS and GC-MS data constantly increases and reaches a maximum of about 9 % on the last detected isotopologue species.

Uncertainty contribution of different input quantities

In the following section a more detailed discussion of the single uncertainty sources and their contribution to the overall uncertainty of the isotopologue fractions is given to further assess and compare the three systems employed in terms of their analytical chemical performance.

Uncertainty contribution of signal intensity according to counting statistics

The peak area (or signal intensity expressed in total counts) basically carries two sources of uncertainty: the ones linked to the MS instrument, which are addressed here via counting statistics uncertainties, and the group linked to chromatographic separation (peak shape, signal-to-noise ratio, chromatographic resolution, repeatability of the integration process, etc.) [17]. The study of each of the above-mentioned chromatography-linked variables is not addressed here owing to high complexity of the equation needed for this purpose; however, several excellent reviews are available in this respect [17]. Moreover, these sources are anyway indirectly evaluated for their contribution in the repeatability precision that takes into account all the random variation occurring during acquisition and integration of the signal intensity of the isotopologues. The uncertainty linked to signal intensity plays a fundamental role, as it is one of the few parameters which can be directly assessed. The signal intensities and signal-to-noise ratios obtained for the different phenylalanine isotopologues are given in Table 3. The corresponding relative uncertainty contributions are summarized in Fig. 4.

Table 3 Averaged peak areas (total counts) and signal-to-noise ratios (S/N) of isotopologue species of phenylalanine from P. pastoris grown on 10 % [U-¹³C]glucose obtained with GC-MS, LC-TOFMS, and LC-MS/MS instruments (n = 6). With the LC-MS/MS instrument, the total IF was obtaining by summing the signal intensity of each multiple reaction monitoring transition

Fig. 4

Relative contribution of signal intensity uncertainty in the measured isotopologue fractions. The uncertainty of the isotopologue signal, i.e., absolute number of ions (counts) measured by the three systems, is highlighted in each isotopologue fraction. Each isotopologue signal intensity has its major contribution in the proper isotopologue fraction (e.g., ¹³C₀-Phe → IF⁺⁰, ¹³C₁-Phe → IF⁺¹, etc.). GC gas chromatography, LC liquid chromatography, MS mass spectrometry

In accordance with theory, for all three instruments the relative contribution of counting statistics increases with decreasing species abundance. Moreover, ion counting shows the highest contribution when it concerns its own isotopologue fraction, whereas the contribution of the signal intensities of the remaining isotopologues is minor or even negligible.

All the other peak areas (A ⁺¹ to A ⁺⁴) show a higher contribution in the isotopologue fractions where they appear as a numerator. The results are comparable for the three MS instruments. The relative contribution show a moderate impact (below 30 %) on the total combined uncertainty of IF⁺⁰, and the absolute count of signal intensity is systematically the major uncertainty source (between 60 and 90 %) of the uncertainties of the other isotopologue fractions (Fig. 5).

Fig. 5

Contribution of repeatability precision to the overall uncertainty of the isotopologue fractions

These results should not be misinterpreted in terms of possible unsuitability of the instruments to collect enough data points per peak since the less abundant signal intensity in the case of ¹³C₄-Phe is still on the order of 10⁴ total counts (Table 3).

The signal intensity uncertainty shows a high contribution because of the exceptionally low repeatability precision (K _rep) affecting LC-MS/MS data (RSD 1.1 %). On the other hand, these results indicate that a large number of MRM transitions leading to a short dwell time (or high scan rate) per ion species can lead to a more prominent contribution of counting statistics to the expanded uncertainty. Accordingly, during method development, MRM transitions and the dwell time have to be carefully optimized in order to generate a minimum of 15–20 data points per peak.

Repeatability precision of isotopologue fractions (K _rep)

The uncertainty contribution of repeatability precision is universally acknowledged as a prominent source of uncertainty in many studies; however, when the absolute counts per analyte are implemented in the measurement model, the magnitude of the contribution decreases. Generally, measurement repeatability is the SD of the mean value of the measurand from analysis performed in a short time period. Since the bottom-up approach sometimes results in a very detailed model equation, the precision term very often groups all the random uncertainty sources [6, 9, 10, 18] in order to simplify the budget [9, 17, 19]. Figure 5 clearly shows that repeatability precision impacts on overall uncertainty.

The major uncertainty sources affecting repeatability are linked to the chromatographic separation of the phenylalanine. Many publications have reviewed the different effects that chromatographic parameters (temperature, flow rate, efficiency, background intensity, etc.) have on the overall precision [10, 17, 18, 20]. Although, in general, these parameters affect all measured isotopologue species equally, it is fundamental to take into consideration how they are correlated with the concentration of the analytes.

Accordingly, the quantitative effect of the chromatographic parameters is negligible in the case of highly abundant species, but increases as soon as the isotopologue species tend toward their respective limit of detection. These contributions ultimately affect the peak integration procedure, resulting in poor reproducibility in the case of low-abundance species.

Correction factor: ¹³C and ^natC substrate weight and purity

The correction factor was built in order to study the uncertainty associated with the correction of the isotopologue fractions for the ¹³C and ^natC impurities of the substrates fed to the P. pastoris culture.

According to the selected labeling strategy, it is evident that the purity factors (Pf¹²C and Pf¹³C) and substrate weights ($W_{sub}^{12 }$ and $W_{sub}^{13 }$) affect only the isotopologue fractions containing no ¹³C atoms and one ¹³C atom. Within the expanded uncertainty it is possible to observe the role of the unlabeled glucose purity factor (Pf¹²C) as an uncertainty source (the relative contributions in GC-MS, LC-TOFMS, and LC-MS/MS are 5, 18, and 17 % respectively), whereas the Pf¹³C shows a negligible relative contribution (relative uncertainty below 1 %).

On the other hand, the [U-¹³C]glucose weight show a minimum contribution of 7.5 % for the LC-MS/MS platform, whereas it contribution is negligible for the other two instruments. The Pf¹²C uncertainty source affects mainly IF⁺¹ instead of IF⁺⁰. The same behavior can be observed in the synthetic isotopologue experiment results: Fig. 6 exemplarily shows the uncertainty contribution to the synthetic isotopologue ratio ^natPhe/¹³C-Phe equal to 2:1.

Fig. 6

Purity factor contribution to the overall uncertainties of synthetic isotopologue fractions. The relative uncertainty sources of the substrate purity factor affecting IF⁺⁰ and IF⁺¹ in a synthetic mixture with a 2:1 ratio of ^natPhe/¹³C₁-Phe are highlighted exemplarily. A ⁺⁰ and A ⁺¹ are the areas of the species containing no ¹³C atoms and one ¹³C atom, respectively, pF12 and pF13 are the isotope purity factors of unlabeled and labeled glucose, respectively, and K _rep is the repeatability precision. The contribution of the purity factors is inversely proportional to the measured ratio, which is approximately 0.64:0.36. This suggests that the larger uncertainty sources in the measurement model are the ones linked to minor input quantities

The impact of Pf¹³C uncertainty mainly concerns IF⁺⁰ and vice versa. The same relationship is observed in the amino acids ratios 10:1 and 1:10 (Fig. 7). The experiment confirms that the purity factor of the unlabeled substrate is the major uncertainty source of the labeled isotopologue fraction (i.e., Pf¹³C of IF⁺⁰) and vice versa (i.e., Pf¹²C of IF⁺¹).

Fig. 7

Purity factor contribution to the overall uncertainties of synthetic isotopologue fractions and relative uncertainty sources of the substrate purity factor affecting IF⁺⁰ and IF⁺¹ in the synthetic ratio ^natPhe/¹³C₁-Phe equal to 10:1 and 1:10. The uncertainties represented are the same as those reported in Fig. 6. Again it appears that the purity factor associated with the minor input quantity, in this case peak area (i.e., A ⁺⁰ or A ⁺¹), carries the higher uncertainty

Besides the mathematical interpretation of this new information—uncertainty in ratio measurement model [21]—it has been proved that the uncertainty affecting the bias corrections is not negligible. Therefore, whenever they are present further experiments are necessary in order to accurately assess the purity factor of the substrates employed.

Conclusions

Bottom-up approaches according to the Kragten method and the MCM were employed to assess accuracy and measurement uncertainty in an MS-based fluxomics experiment employing the amino acid phenylalanine as a model compound. The procedures are potentially applicable to MID, but in order to compare the performances of different MS systems, only five isotopologue fractions of phenylalanine were addressed. The uncertainties calculated according to the Kragten spreadsheet and the PDF study were consistent with each other, proving that the MCM is a versatile tool to calculate and/or assess the overall uncertainty in an easy and straightforward way.

In a comparison of two separation techniques (GC, LC) and three MS instruments (ESI-TOFMS, ESI-MS/MS, EI-MS) regarding accuracy, precision, and measurement uncertainty, LC-MS/MS provided the lowest uncertainty and is preferable regarding targeted analysis. However, analysis speed, i.e., the scan speed of the quadrupole, is limiting in terms of dwell time, points per chromatographic peak, and total ion counts. LC-TOFMS provides untargeted profiling and retrospective data analysis, although the limitation regarding precision and sensitivity has to be carefully evaluated in the case of low-abundance isotopologues. Lastly GC-MS analysis with an electron ionization source run in scan mode provides satisfactory results and MID or isotopologue fraction datasets that can be directly implemented in fluxomics software; however, the low number of ions collected is a serious limitation to take in account.