Using Heavy Mass Isotopomers for Protein Turnover in Heavy Water Metabolic Labeling

Abstract

Metabolic labeling followed by LC-MS based proteomics is a powerful tool to study proteome dynamics in high throughput both in vivo and in vitro. High mass resolution and accuracy allow differentiation in isotope profiles and quantification of partially labeled peptide species. Metabolic labeling duration introduces a time domain in which the gradual incorporation of labeled isotopes is recorded. Different stable isotopes are used for labeling. The labeling with heavy water has advantages since it is cost-effective and easy to use.

Protein degradation rate constant has been modeled using exponential decay models for relative abundances of mass isotopomers. The recently developed closed-form equations were applied to study the analytic behavior of the heavy mass isotopomers in the time domain of metabolic labeling. The predictions from the closed-form equations are compared with the practices that have been used for extracting degradation rate constants from the time course profiles of heavy mass isotopomers. It is shown that all mass isotopomers, except for the monoisotope, require data transformations to obtain exponential depletion, which serves as a basis for the rate constant model. Heavy mass isotopomers may be preferable choices for modeling high mass peptides or peptides with a high number of labeling sites. The results are also applicable to stable isotope labeling with other atom-based labeling agents.

Graphical Abstract

Introduction

Metabolic labeling with stable-isotopes followed by Liquid Chromatography coupled into Mass Spectrometry (LC-MS) is a powerful tool to study proteome dynamics in vivo on high throughput and large scale.^1–4 Living organisms are provided with a diet containing stable, non-radioactive isotopes. The isotopes incorporate into amino acids and subsequently into proteins. Tissue samples are collected at specific time points of label duration. They are analyzed using standard workflows of mass spectrometry-based proteomics: protein/peptide separation, LC-MS, and bioinformatics processing⁵. Atom-based heavy isotope labeling agents^6–9 (such as deuterium in heavy water, ¹⁵N, or ¹²C) can label most of the amino acids. The gradual incorporation of the “heavy” isotopes into peptides allows the determination of the protein replacement rate constants. Among the labeling agents, heavy water is cost-effective and easy to use. It is provided in the drinking water enriched in low concentrations (5% or less). The labeling with heavy water rapidly achieves equilibrium³. However, due to the low concentrations (high concentrations - 20%, or higher enrichments are toxic), the labeling is partial. Only a small number of all accessible hydrogens are replaced with the deuterium in heavy water. Therefore, the resulting complex mass isotope profile is the overlap of the isotope profiles of labeled and unlabeled peptides. Various numerical techniques^10–12 (such as multinomial probability distribution, fast-Fourier transforms) have been developed to generate the isotope profiles of partially labeled peptides. The complete isotope profiles are used to determine relative abundances (RAs) of mass isotopomers.

MIDA (mass isotopomer distribution analysis) was the first algorithmic approach to analyze mass spectral data for quantifying the changes in isotope distributions resulting from metabolic incorporation of stable-isotope labels (tracers) into peptides¹³. It uses combinatorial distributions of natural and enriched isotopes. The original application was developed for the incorporation of labeled amino acids, such as ³H₃ Leu. Later, it was extended into studies of labeling dynamics from metabolic labeling with heavy water¹⁴.

The time course of label incorporation was suggested to be modeled as an exponential curve. he current software has adopted this approach for analysis of protein turnover data from metabolic labeling and high-throughput LC-MS^7,15. Currently, there are three public software solutions^16–18 that have been reported for use in analyzing mass spectral data from heavy water metabolic labeling. Two of them (DeuteRater¹⁷ and d2ome¹⁸) are in the public domain via Github.

In a recent study¹⁹, exact equations of time course dynamics for the first three heavy mass isotopomers have been derived. They allowed new mechanistic insights into the RAs of these mass isotopomers. It follows that the RAs of heavy mass isotopomers do not exhibit an exponential depletion behavior. However, as it will be shown, with appropriate transformation for each mass isotopomer RA, the transformed quantities are of exponential decay form. Simulations demonstrate the accuracies of the rate constant estimations with and without the transformation.

Data Description.

The data set of the liver proteome of (low-density lipoprotein receptor knock-out) LDLR^−/− mice fed a normal diet¹⁸ was used in this study. There were six labeling times: 0, 3, 7, 11, 15, and 21 days. Peptide samples were prepared from eleven SDS-PAGE bands. Each sample was run in a duplicate. The body water enrichment with deuterium in these samples was 0.03. Mass spectral data were collected using Q Exactive™ Plus Hybrid Quadrupole-Orbitrap™ Mass Spectrometer (Thermo Scientific, CA). Mascot²⁰ database search engine was used for peptide identification. Mascot controls the global false discovery rate (which was set at 5%) by using matches to the reversed sequences. The data set is publicly available at ProteomeXChange site (PXD009493).

RA of the Monoisotope for Estimating Protein Degradation Rate Constant

MIDA was probably the first algorithm to model mass spectral data for protein turnover estimation from in vivo experiments¹³. It has described two features that can be used to extract the fractional synthesis rate (also termed degradation rate constant): as an enrichment of a particular mass isotopomer of the product (molar excess) or per atom enrichment (atom excess). The incorporation of deuterium from heavy water leads to the reduction of the RA of monoisotope. MIDA used the normalized (with respect to the plateau enrichment) changes in RAs of a mass isotopomer, ΔE_n, to model the label incorporation:

$$\mathrm{\Delta }{\mathrm{E}}_{\mathrm{n}}=\frac{{{\mathrm{I}}_{\mathrm{n}}\left(\mathrm{t}\right)-\mathrm{I}}_{\mathrm{n}}^{\mathrm{a}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{p}}}{{{\mathrm{I}}_{\mathrm{n}}\left(0\right)-\mathrm{I}}_{\mathrm{n}}^{\mathrm{a}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{p}}}={\mathrm{e}}^{-\mathrm{k}\mathrm{t}}$$ Eq. (1)

where I_n(t) is the RA of the n^th mass isotopomer (n = 0 for the monoisotope) at time point t, ${\mathrm{I}}_{\mathrm{n}}^{\mathrm{a}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{p}}$ is the RA of the n^th mass isotopomer at the plateau of enrichment. k is the degradation rate constant. A rearrangement of terms in Eq. (1) leads to (for the monoisotopic RA):

$${\mathrm{I}}_0\left(\mathrm{t}\right)={\mathrm{I}}_0^{\mathrm{a}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{p}}+\left({{\mathrm{I}}_0\left(0\right)-\mathrm{I}}_0^{\mathrm{a}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{p}}\right){\mathrm{e}}^{-\mathrm{k}\mathrm{t}}$$ Eq. (2)

where $I_0^{asymp}$ is the RA of the monoisotope at the enrichment plateau. The value of $I_0^{asymp}$ is obtained from the isotope model of the labeled species. The isotope distribution of a peptide in metabolic labeling with heavy water is modeled by separating the hydrogen atoms into two groups^8,21,22. In the first group are the hydrogens that are non-accessible to the deuterium. In the second group are the hydrogens that are accessible to the deuterium, N_EH. N_T denotes the total number of hydrogens in a peptide. The relevant probabilities of ²H for each group are p_H and (p_H + p_X(t)). Here, p_H is the relative abundance of ²H in nature, p_X(t) is the enrichment of a peptide in ²H incorporated from the heavy water in the diet at the labeling time t. Using these definitions, the RA of the monoisotopic peak, I₀(t), can be expressed as:

$${\mathrm{I}}_0\left(\mathrm{t}\right)={\mathrm{I}}_0\left(0\right){\left(1-\frac{{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)}{1-{\mathrm{p}}_{\mathrm{H}}}\right)}^{{\mathrm{N}}_{\mathrm{E}\mathrm{H}}}={\mathrm{I}}_0\left(0\right)\sum _{\mathrm{k}=0}^{{\mathrm{N}}_{\mathrm{E}\mathrm{H}}}\left(\begin{array}{c}{\mathrm{N}}_{\mathrm{E}\mathrm{H}}\\ \mathrm{k}\end{array}\right){{\left(-1\right)}^{\mathrm{k}}\left(\frac{{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)}{1-{\mathrm{p}}_{\mathrm{H}}}\right)}^{\mathrm{k}}$$ Eq. (3)

When the labeling of the corresponding protein reaches a plateau, (p_X(t)+p_H) is equal to the total body water enrichment (BWE), p_W. Thus, the asymptotic value of the RA of the monoisotope is (it is assumed that p_W ≥ p_H):

$${\mathrm{I}}_0^{\mathrm{a}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{p}\mathrm{t}}={{\mathrm{I}}_0\left(0\right)\left(1-\left({\mathrm{p}}_{\mathrm{W}}-{\mathrm{p}}_{\mathrm{H}}\right)/(1-{\mathrm{p}}_{\mathrm{H}})\right)}^{{\mathrm{N}}_{\mathrm{E}\mathrm{H}}}$$

Eq. (3) is an exact formula for expressing the time course of the monoisotope (under the model that divides the hydrogen atoms into two separate groups). It has a polynomial dependence on p_X(t) and exponential dependence on N_EH. As discussed below, using the recently derived closed-form equations for heavy mass isotopomers, it is seen that the RAs of all other mass isotopomers are not strictly exponential. However, certain (for each mass isotopomer) transformations are obtained that will behave as exponential decays.

RAs of Heavy Mass Isotopomers for Estimating Protein Turnover

As the deuterium-labeled amino acids are incorporated into proteins, mass isotopomer distributions of intact peptides exhibit a change in their profiles and increase RAs of heavy mass isotopomers. The study¹⁹ of the time course dynamics of three heavy mass isotopomers during labeling with heavy water has produced closed-form equations for their RAs. In particular, for I₁(t), I₂(t), and I₃(t), the following formulas were obtained:

$${\mathrm{I}}_1\left(\mathrm{t}\right)={(1-\frac{{\mathrm{p}}_{\mathrm{X}}(\mathrm{t})}{1-{\mathrm{p}}_{\mathrm{H}}})}^{{\mathrm{N}}_{\mathrm{E}\mathrm{H}}-1}\frac{{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)}{{\left(1-{\mathrm{p}}_{\mathrm{H}}\right)}^2}{\mathrm{N}}_{\mathrm{E}\mathrm{H}}{\mathrm{I}}_0\left(0\right)+{(1-\frac{{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)}{1-{\mathrm{p}}_{\mathrm{H}}})}^{{\mathrm{N}}_{\mathrm{E}\mathrm{H}}}{\mathrm{I}}_1\left(0\right)$$ Eq. (4)

$${\mathrm{I}}_2\left(\mathrm{t}\right)=\left\{{\mathrm{I}}_2\left(0\right)-{\mathrm{I}}_1\left(0\right){\mathrm{a}}_1\left(0\right)+{\mathrm{I}}_0\left(0\right)\left({\mathrm{a}}_2\left(0\right)-{\mathrm{a}}_2\left(\mathrm{t}\right)\right)\right\}{\left(1-\frac{{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)}{1-{\mathrm{p}}_{\mathrm{H}}}\right)}^{{\mathrm{N}}_{\mathrm{E}\mathrm{H}}}+{\mathrm{a}}_1\left(\mathrm{t}\right){\mathrm{I}}_1\left(\mathrm{t}\right)$$ Eq. (5)

$${\mathrm{I}}_3\left(\mathrm{t}\right)={\mathrm{a}}_3{\left(\mathrm{t}\right)\mathrm{I}}_0\left(\mathrm{t}\right)+{\mathrm{a}}_2\left(\mathrm{t}\right)\mathrm{*}\left\{{\mathrm{I}}_1\left(\mathrm{t}\right)-{\mathrm{a}}_1{\left(\mathrm{t}\right)\mathrm{I}}_0\left(\mathrm{t}\right)\right\}+{\mathrm{a}}_1\left(\mathrm{t}\right)\mathrm{*}\left\{{\mathrm{I}}_2\left(\mathrm{t}\right)-{\mathrm{a}}_1\left(\mathrm{t}\right)\left({\mathrm{I}}_1\left(\mathrm{t}\right)-{\mathrm{a}}_1{\left(\mathrm{t}\right)\mathrm{I}}_0\left(\mathrm{t}\right)\right)-{\mathrm{a}}_2{\left(\mathrm{t}\right)\mathrm{I}}_0\left(\mathrm{t}\right)\right\}+\left\{{\mathrm{I}}_3\left(0\right)-{\mathrm{a}}_1\left(0\right)\left[{\mathrm{I}}_2\left(0\right)-\frac{{(\mathrm{N}}_{\mathrm{E}\mathrm{H}}+1){\mathrm{p}}_{\mathrm{H}}}{2\left(1-{\mathrm{p}}_{\mathrm{H}}\right)}\left({\mathrm{I}}_1\left(0\right)-{\mathrm{I}}_0\left(0\right)\frac{{\mathrm{N}}_{\mathrm{T}}{\mathrm{p}}_{\mathrm{H}}}{1-{\mathrm{p}}_{\mathrm{H}}}\right)-{\mathrm{I}}_0\left(0\right)\left(3{\mathrm{N}}_{\mathrm{T}}{\mathrm{N}}_{\mathrm{E}\mathrm{H}}+3{\mathrm{N}}_{\mathrm{T}}-{\mathrm{N}}_{\mathrm{E}\mathrm{H}}^2-3{\mathrm{N}}_{\mathrm{E}\mathrm{H}}-2\right){\left(\frac{{\mathrm{p}}_{\mathrm{H}}}{1-{\mathrm{p}}_{\mathrm{H}}}\right)}^2\frac{1}{6}\right]\right\}{\left(1-\frac{{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)}{1-{\mathrm{p}}_{\mathrm{H}}}\right)}^{{\mathrm{N}}_{\mathrm{E}\mathrm{H}}}$$ Eq. (6)

where, a_n(t) is:

$${\mathrm{a}}_{\mathrm{n}}\left(\mathrm{t}\right)=\left(\begin{array}{c}{\mathrm{N}}_{\mathrm{E}\mathrm{H}}\\ \mathrm{n}\end{array}\right){\left(\frac{{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)+{\mathrm{p}}_{\mathrm{H}}}{1-{\mathrm{p}}_{\mathrm{H}}-{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)}\right)}^{\mathrm{n}},\mathrm{n}=1,2,3.$$

$\left(\begin{array}{c}{\mathrm{N}}_{\mathrm{E}\mathrm{H}}\\ \mathrm{n}\end{array}\right)$ is the binomial coefficient. Similar to the asymptote of I₀(t), the asymptotic values of the RAs of heavy mass isotopomers are obtained by equating (p_X(t) + p_H) to p_W. Compared to the original publication¹⁹, the equations have been recast to emphasize the exponential decay terms. The formula for each mass isotopomer RA contains only one term that depends on the natural RA of that mass isotopomer. Only this term forms the basis for the exponential depletion of the mass isotopomer.

Results and Discussions.

Figure 1 shows the relative abundances of the first five isotopomers of the mouse mitochondrial Carbamoyl-phosphate synthase (CPSM) peptide, DELGLNK, before the start of labeling and after 21 days of labeling¹⁸. The N_EH of this peptide sequence is 9. The BWE with deuterium after 21 days of labeling was only 3%. The relative changes of the mass isotopomer distribution at any given time point of labeling depend on the three factors: the rate of protein turnover, the number of exchangeable hydrogens in the peptides, and the BWE. The N_EH number and BWE determine the extent of possible changes (achievable at the plateau) to the isotope distribution. The protein turnover rate determines how soon, after the start of labeling, the changes will be observed.

Figure 1.

Labeling with low concentrations of heavy water results in overlapping (labeled and unlabeled) isotope profiles recorded in LC-MS. Relative Abundances of the mass isotopomers of CPSM peptide, DELGLNK, before the start and after 21 days of labeling. Shown are the relative abundances of the first 5 isotopomers obtained by integrating the volume of the +2 charged peptide’s signals in m/z and chromatographic time domains. After twenty-one days of labeling, the mass isotopomer profile of peptide is the result of the overlapping of labeled and unlabeled peptides.

As seen from Figure 1, the partial labeling results in a complex isotope profile, which is a mixture of profiles of labeled (heavy) and unlabeled (light) peptides. Several numerical techniques have been developed to simulate the isotope profiles. I have recently derived closed-form equations that described the time course of the RAs of three heavy mass isotopomers¹⁹.

Figure 2 shows the RAs of the four mass isotopomers of a mouse CPSM protein, TVLMNPNIASVQTNEVGLK (N_EH = 30, N_T = 151), as a function of peptide enrichment, p_X(t). It is seen from the figure that only I₀(t) and I₁(t) are decaying functions of p_X(t). In this example, I₂(t) and I₃(t) are not monotonically decreasing functions. The example shows that Eq. (1) does not apply to RAs of these mass isotopomers. However, as shown below, the equation is valid for the transformed values of the RAs. The transformed RAs are shown in the figure with the broken lines.

Figure 2.

RAs of heavy mass isotopomers are not exponential decay functions of deuterium enrichment. Shown are the RAs of the monoisotope and the first three heavy mass isotopomers for mouse CPSM peptide, TVLMNPNIASVQTNEVGLK. I₂(t) and I₃(t) clearly show non-exponential behavior. I₁(t) exhibits a mixed curve. Also shown are the transformed RAs (broken lines) of the first, ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$, and second, ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$, heavy mass isotopomers described further below in the text.

The time course equation of I₀(t), Eq. (3), is an exponential decay function. However, the RAs of the other mass isotopomers are non-exponential, Eqs. (4–6). Each of the RAs does contain an exponentially decaying component. The isolation of the exponential decay term from each of the RAs requires a specific transformation. In general, the exponential decay term for the n^th mass isotopomer will correspond to the following functional dependence of its RA:

$${\mathrm{I}}_{\mathrm{n}}\left(\mathrm{t}\right)={\mathrm{I}}_{\mathrm{n}}\left(0\right){\left(1-\frac{{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)}{1-{\mathrm{p}}_{\mathrm{H}}}\right)}^{{\mathrm{N}}_{\mathrm{E}\mathrm{H}}}$$ Eq. (7)

This functional form is present in each of the Eqs. (4–6). For example, I₁(t) is a sum of two terms, only one (the second term) of which is in the exponential decay form, Eq. (7). For small values of p_X(t), the first term in the expression for I₁(t), Eq. (4), is a linear growth function of p_X(t). The term accounts for the increase in the RA of the first heavy isotope due to the enrichment of the monoisotope with deuterium. The monoisotope is the only mass isotopomer contributing to the abundance of the first heavy mass isotopomer during the labeling with heavy water. The second term in Eq. (4) is the depletion of the first heavy isotope due to the incorporation of deuterium. Therefore, for exponential decay modeling of I₁(t), one needs a transformed quantity, ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$:

$${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)={\mathrm{I}}_1\left(\mathrm{t}\right)-{\left(1-\frac{{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)}{1-{\mathrm{p}}_{\mathrm{H}}}\right)}^{{\mathrm{N}}_{\mathrm{E}\mathrm{H}}-1}\frac{{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)}{{\left(1-{\mathrm{p}}_{\mathrm{H}}\right)}^2}{\mathrm{N}}_{\mathrm{E}\mathrm{H}}{\mathrm{I}}_0\left(0\right)={\left(1-\frac{{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)}{1-{\mathrm{p}}_{\mathrm{H}}}\right)}^{{\mathrm{N}}_{\mathrm{E}\mathrm{H}}}{\mathrm{I}}_1\left(0\right)$$ Eq. (8)

The right-hand side of Eq. (8) is an exponential decay form, similar to Eq. (3). At the start of the labeling (p_X(0) = 0.), its value is I₁(0). As p_X(t) increases, the right-hand side of the Eq. (8) decreases, and it is modeled as an exponential decay, similar to Eq. (2). The calculation of ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ requires the knowledge of p_X(t). p_X(t) is obtained from the raw abundances of monoisotope, A₀(t), and the first heavy mass isotopomer, A₁(t), at the label duration time t:

$${\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)=\frac{\left({\mathrm{A}}_1\left(\mathrm{t}\right)/{\mathrm{A}}_0\left(\mathrm{t}\right)-{\mathrm{I}}_1\left(0\right)/{\mathrm{I}}_0\left(0\right)\right){\left(1-{\mathrm{p}}_{\mathrm{H}}\right)}^2}{{\mathrm{N}}_{\mathrm{E}\mathrm{H}}+\left(1-{\mathrm{p}}_{\mathrm{H}}\right)\left({\mathrm{A}}_1\left(\mathrm{t}\right)/{\mathrm{A}}_0\left(\mathrm{t}\right)-{\mathrm{I}}_1\left(0\right)/{\mathrm{I}}_0\left(0\right)\right)}$$ Eq. (9)

The above discussion shows that the exponential decay is an appropriate model for ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ but not for I₁(t). The exponential decay forms can also be obtained for transformed quantities of I₂(t) and I₃(t). However, in their original forms, the RAs are not exponential decays.

To show the theoretical differences, an R code was written to simulate the RAs as a function of the enrichment in deuterium, p_X(t). For peptide TVLMNPNIASVQTNEVGLK the results are shown in Figure 2. It is seen that I₂(t) and I₃(t) are not monotonically decaying functions (a pre-requisite for exponential decay). At first, I₁(t) and ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ will be tested for use in Eq. (1) to extract the degradation rate constant. Below, it will be shown that a similar approach exists for I₂(t) and ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$. For the simulations, the experimentally estimated rate constant (k = 0.113 day⁻¹) was used to obtain I₀(t) using Eq. (2). Small Laplace distributed noise²³ (the scaling parameter of the distribution was exp(−10)) was added to I₀(t) at each labeling time point. Fast-Fourier transformations (FFTs) were used to generate RAs of the mass isotopomers using p_X(t) for the deuterium incorporation and natural isotope distributions for all other atom types. From the enriched (in deuterium) isotope profiles, I₁(t) was determined. The asymptotic value was obtained (from isotope profiles generated by FFT) by setting the enrichment equal to the BWE in the experiments, 0.03. Next, a non-linear least-squares fit²⁴ in Eq. (1) was applied to estimate the rate constants. Since the Laplace noise was small, the variation in the estimation of the rate constant was negligible. The estimated rate constant was approximately 0.063 day^-1. It is almost half of the true value of 0.113 day⁻¹ used in the simulation. The example clearly shows that the original I₁(t) is not accurate for extracting rate constants using Eq. (1).

Next, ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ is used for the estimation of the rate constant. For this procedure, p_X(t) at each time point was obtained using Eq. (9) and A₁/(t)/A₀(t) ratio. The ratio was obtained from the isotope profiles computed using FFTs, as described above. The asymptotic value was also calculated from ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ by setting (p_X(t)+p_H) equal to BWE, 0.03. The rate constant was estimated using non-linear least-squares fit in Eq. (1). This procedure returned the true value of the rate constant, 0.113 day^-1. Since the equations are exact, large-scale simulations to confirm the obtained result are redundant.

Figure 3 shows the predicted (lines) and actual (circles) data points for this simulation. The time course of the ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ (purple) follows that of the I₀(t) (black). I₁(t) (blue), on the other hand, differs from them. The figure shows that only ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ correctly reproduces the time course of label incorporation. The R codes used for the calculations and simulations are provided at the Github site: https://github.com/rgsadygov/Heavy-Mass-Isotopomers.

Figure 3.

Transformed RA of the first heavy mass isotopomer parallels the corresponding time course of the RA of the monoisotope. The transformed RA of the first heavy mass isotopomer, ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$, (purple line and circles) follows the dynamics of the RA of the monoisotope, I₀(t) (black line and circles). The time course of the original RA of the first heavy mass isotopomer, I₁(t) (blue line and circles) underestimates the true rate constant. The lines are the predictions, circles are data used for determining the rate constants. The mass isotopomer time courses are those of the peptide, TVLMNPNIASVQTNEVGLK.

As seen in Eq. (5), I₂(t) also contains an exponential decay term. The corresponding transformation is given in Eq. (10):

$${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)={\mathrm{I}}_2\left(\mathrm{t}\right)+\left\{\frac{{{\mathrm{a}}_1\left(0\right)\mathrm{I}}_1\left(0\right)}{{\mathrm{I}}_0\left(0\right)}-{\mathrm{a}}_2\left(0\right)+{\mathrm{a}}_2\left(\mathrm{t}\right)\right\}{\mathrm{I}}_0\left(\mathrm{t}\right)-{\mathrm{a}}_1\left(\mathrm{t}\right){\mathrm{I}}_1\left(\mathrm{t}\right)={\left(1-\frac{{\mathrm{p}}_{\mathrm{X}}\left(\mathrm{t}\right)}{1-{\mathrm{p}}_{\mathrm{H}}}\right)}^{{\mathrm{N}}_{\mathrm{E}\mathrm{H}}}{\mathrm{I}}_2\left(0\right)$$ Eq. (10)

Eq. (10) was used to calculate ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$ for another peptide sequence of CPSM: EPLFGISTGNIITGLAAGAK. I₁(t) and I₂(t) (for every value of p_X(t)) used in Eq. (10) were obtained using FFTs. The Laplace noise was added to ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$, and the noisy time course data of ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$ was used in non-linear least-squares fit to obtain the degradation rate constant, similar to Eq. (2). The asymptote of ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$ was obtained from Eq. (10) by setting p_X(t) equal to (p_W – p_H). The time courses of M₀-M₂ mass isotopomers and those of the transformed ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ and ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$ are shown in Figure 4. The original RA of M₂, I₂(t), is a growth function of time in the labeling duration used in the experiments. The transformed RA of M₂, ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$, is an exponential decay function. The degradation rate constant obtained using ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$ was equal to the true value, 0.110 day⁻¹, used in the simulations. The rate constant obtained from I₂(t) was twice larger, 0.220 day^-1.

Figure 4.

Transformed RA of M₂, ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$, yields the true rate constant for small noise (exp(−10)). Shown are the RAs of the monoisotope (black circles and line), M₁ (blue circles and line), M₂ (magenta circles and line), and the transformed RAs of the latter two; ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ purple circles line and ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$ cyan circles and line. The transformed RA of the second heavy mass isotopomer, ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$, follows the dynamics of the RA of the monoisotope. The time course of the original RA of M₂, I₂(t) is a growth function under the labeling conditions (deuterium enrichment in the diet water and labeling duration) used in the labeling experiments. The lines are the predictions, and circles are data used for determining the rate constants. The mass isotopomer time courses are those of the peptide, EPLFGISTGNIITGLAAGAK.

Simulations of more than 17000 peptide sequences (all distinct peptides identified and quantified in Band 4 of the data set) were analyzed for comparison of the rate constant estimations using the RAs described above. For small noise values, the differences between the actual rate constants and those obtained from using I₀(t), ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ and ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$ were small. These results are illustrated in Supplementary Materials Figures 1–3. As seen from the figures, except for large values of rate constants, the results from all three RAs agree with the actual rates. The rate constants obtained using I₁(t) and I₂(t) differed from the actual rates, even for the small noise (exp(−10)), for all values of the rate constants, Supplementary Materials Figures 4 and 5. The density plots of the relative errors from each of the RAs are shown in Supplementary Materials Figures 6 and 7. These results show that the time courses of I₁(t) and I₂(t) do not produce accurate results for rate constants.

As Eq. (7) shows, the only difference between the time courses of the transformed RAs and the RA of the monoisotope is in the I_n(0) term – the natural RA of the n^th mass isotopomer. Theoretically, for the transformed RAs, the RA with the largest I_n(0) will have the highest signal-to-noise ratio. It was observed in the simulations that used equal noise for each RA of a peptide. The parameters of this noise were estimated from the modeling of the experimental data set. Its parameters were −0.0035 and 0.0159 for the location and scale of the Laplace distribution, respectively. I₀(t) time courses produced a smaller absolute relative error than that of ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ for all peptides that had I₀(0) > I₁(0). Correspondingly, for peptides that had I₀(0) < I₁(0), the rates obtained from ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ had smaller absolute errors than those from I₀(t). These results are shown in Supplementary Material Figures 8 and 9. However, in mass spectral measurements, the monoisotope and the first heavy mass isotopomer tend to have different signal-to-noise ratios and spectral accuracy²⁵. Therefore, separate noises were sampled for I₀(t) and ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ from the Laplace distribution. In this case, the natural RAs of the mass isotopomers, I₀(0) and I₁(0), were not exclusive determining factors for choosing the better accuracy time courses. This can be seen from Figure 5, which depicts the absolute errors of the rate constant estimations using the peptides for which I₀(0) > I₁(0). For more than 5000 out of 13000 peptides, the errors obtained from ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ were less than those obtained from using I₀(t). This compares with just seven peptides in the case when the noise was equal, Supplementary Materials Figure 9. The results for the peptides with I₀(0) < I₁(0) are similarly mixed and shown in Supplementary Materials Figure 10.

Figure 5.

The experimental noise changes the dependency of the accuracy of the rate constant estimations on I₀(t) and ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ time courses. Shown are the relative absolute errors of rate constant estimations from I₀(t) (the x-axis) and ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ (the y-axis) for peptides with I₀(0) > I₁(0). Shown are the results for more than 13000 peptides. The red line is the identity line.

The obtained results are important for protein turnover estimations in high-throughput experiments. For high mass peptides, the first heavy mass isotopomer is often the most abundant. In this case, it may be preferable (because of a better signal-to-noise ratio) to use I₁(t) or I₂(t) rather than I₀(t) for time course modeling. However, direct use of the heavy mass isotopomers will result in wrong estimations, as was illustrated in examples and explained based on the closed-form equation. The formulas for the transformed ${\tilde{\mathrm{I}}}_1\left(\mathrm{t}\right)$ and ${\tilde{\mathrm{I}}}_2\left(\mathrm{t}\right)$, which reproduce the true rate constants, are provided. It should be noted that when the magnitude of change in RAs of heavy mass isotopomers is small, it falls below levels of reliable quantification, thus limits the utility of these signals for the determination of rate constants. However, when the changes are detectable, the provided analyses show that the transformed RAs have increased depletion (during label incorporation) compared to the original RAs of M₁ and M₂.

Conclusion.

Heavy water metabolic labeling, followed by LC-MS, has been used for studying in vivo protein turnover in high-throughput. Exponential decay models have been developed to extract the degradation rate constants from the time course of RAs of mass isotopomers. Recently derived closed-form equations describe the dynamics of RAs of the heavy mass isotopomers. Mechanistic insights from the equations show that the time course dynamics of the heavy mass isotopomers are not governed by exponential depletion. The forms are rather complex, but each one of them does contain an exponential decay term. Appropriate transformations convert the original RAs to exponential decays. The final extraction of the degradation rate constants requires an additional quantity, peptide enrichment in deuterium. The closed-form equations allow the estimation of this quantity at each labeling time point.

Abbreviations:

BWE

body water enrichment in deuterium

CPSM

Carbamoyl-phosphate synthase mitochondrial

Eq

equation

FFT

fast-Fourier transform

LC-MS

liquid chromatography and mass spectrometry

NEH

number of exchangeable hydrogens

RA

relative abundance