An out‐of‐equilibrium definition of protein turnover

Protein turnover (PT) has been formally defined only in equilibrium conditions, which is ill‐suited to quantify PT during dynamic processes that occur during embryogenesis or (extra) cellular signaling. In this Hypothesis, we propose a definition of PT in an out‐of‐equilibrium regime that allows the quantification of PT in virtually any biological context. We propose a simple mathematical and conceptual framework applicable to a broad range of available data, such as RNA sequencing coupled with pulsed‐SILAC datasets. We apply our framework to a published dataset and show that stimulation of mouse dendritic cells with LPS leads to a proteome‐wide change in PT. This is the first quantification of PT out‐of‐equilibrium, paving the way for the analysis of biological systems in other contexts.


INTRODUCTION
Proteins are the main workhorses of cells, thereby defining their function. 1,2 Even at steady-state, proteins are constantly produced and lost through degradation and/or dilution ( Figure 1A). 3 The level of a given protein is determined by the interplay between these two antagonistic processes. [4][5][6] The so-called protein turnover (PT) depends on synthesis and degradation rates, along with dilution occurring during cell division. Protein synthesis rates depend on mRNA translation rates and mRNA copy numbers, which are determined by their synthesis and degradation rates (mRNA turnover). Pioneering studies using radioactive and isotope labeling of newly synthesized proteins have revealed quantitative differences between global protein synthesis rates in different organs. 7 Protein synthesis and degradation rates also differ markedly in undifferentiated cell types -such as stem cells -as compared to more differentiated cells. 8 Murine hematopoietic stem cells need to tightly control their PT to maintain their function. 9 It has also been shown that hematopoietic, epidermal, and colonic epithelium

k(t)
Protein decay

s(t)
Protein synthesis P(t) Protein level Protein level dynamics Protein turnover dynamics (B) (A)

(I) (H) (K) (J)
F I G U R E 1 RETHINKING protein turnover as fluxes. (A) Protein level is the result of synthesis and degradation/decay dynamics processes described by time-dependent rates s -synthesis rate -and k -decay rate. Protein-level dynamics can be described using ordinary differential equations (ODEs). (B) Protein turnover (τ PT ) dynamics focuses on flows of protein in the system more than on the protein level. Protein turnover depends on two new time-dependent variables, the equilibrium protein turnover (PT) τ PT,eq , and the imbalance θ. (C) Sensitivity analysis of the protein turnover, defined here as the inverse characteristic time τ PT with respect to the production rate s and the decay rate k. The protein turnover is plotted as the fold-increase (FI) in protein turnover with respect to a reference point, represented by the crossed circle. The protein be applied to existing data measuring proteome dynamics, and that this new conceptual framework highlights that bone-marrow-derived mouse dendritic cells exhibit proteome-wide protein turnover changes upon LPS stimulation. 15

EXPERIMENTAL AND THEORETICAL STATE OF THE ART
The first quantifications of the so-called "protein turnover" in mammalian cells were performed over 80 years ago. 16,17 While since then methodological developments exploded, general assumptions to quantify protein turnover remained similar. Namely, steady-state is generally assumed, symmetrizing the problem of measuring protein synthesis and degradation rates. 6,18 Only a few studies acknowledge or try to address the problem of protein turnover in non-steady-state conditions. 6,19,20 Thus, protein turnover was mainly captured through the computation of the protein degradation rate using pulse-and-chase methodologies. 6 First studies relied on radioactive isotope-labeled amino acids, incorporated into newly synthesized proteins, these later being detected using mass spectrometry, 16,17 scintillation counting, 21 autoradiography of 1D or 2D gels. 22,23 Due to the numerous limitations of such methods, 24 biochemical tools were developed. Chemical inhibition of protein synthesis (cycloheximide, puromycin, etc.) or degradation (MG-132, bortezomib, lactacystin, etc.) offered new insights into the protein turnover. 3,25,26 Non-radioactive isotope labeling in combination with mass spectrometry was then developed in the early 2000s 27-29 before further developments leading eventually to stable isotope labeling of amino acid in cell culture (SILAC). [30][31][32] Dynamic SILAC was successfully used in the last decade to measure protein turnover of hundreds to thousands of proteins. 4,[33][34][35][36][37] To reduce measurement variability 36,37 and to quantify separately protein synthesis and degradation, three channels SILAC 8,15,36 and SILAC-TMT 37-40 were later developed. This last method, along with careful normalization procedure 41 and data-independent acquisition based-protocol, 42 is today pushed further to quantify changes in protein turnover in single cells. 43,44 These methods were used to decipher which steps of gene expression are the main determinants of protein turnover. 4,15,33,45 Moreover, it has been shown that protein turnover is influenced by subcel-lular protein localization, [45][46][47][48] protein function, 45 cell-cell contact and signaling, 49 belonging to a protein complex, 45,[50][51][52][53] cell identity   and differentiation, 8,15,19,54-57 such as immunologic state, 8,15,56 but   also amino acid/codon composition, 58 cell microenvironment, 49,59 and cell plasticity. 60 It also varies for orthologous proteins in different species. [61][62][63][64][65] Another important regulator of PT is post-translational modifications (PTMs). The addition of chemical groups through methylation, phosphorylation, glycosylation, acetylation, hydroxylation, SUMOylation, and paradigmatically (K48 poly-) ubiquitination, are key in activating or inactivating protein degrons, that is, protein domains that regulate protein stability. 66 Importantly, a new methodological development called site-resolved protein turnover (SPOT) highlights that peptides with different PTMs (peptidoforms) can have diverging turnovers, suggesting that PTMs with diverging turnover may distinguish states of differential protein stability, structure, localization, enzymatic activity, or protein-protein interactions. 67 Methodological developments to investigate the effects of PTMs on protein turnover are also ongoing efforts in the field. 40,66,[68][69][70][71][72] Importantly, the exponential decay expected from the steady-state assumption when performing a pulse-and-chase experiment is not always observed. 5,18,[73][74][75][76] Indeed, for certain subsets of proteins, the probability that any given protein molecule is degraded can change in an age-dependent manner. The decay of these proteins follows then a non-exponential decay. 18,[73][74][75][76] In these cases, more complex models depicting protein turnover are necessary. Similarly, more complex kinetic models have been suggested to analyze pulse-and-chase isotope labeling experiments, using so-called two-or three-compartment models. 65,77 As already elegantly discussed in, 6 when the system is not at a steady-state, that is, during dynamic processes such as differentiation, new mathematical frameworks are needed to quantify protein turnover from pulse-and-chase-based experiments. So far, the dynamics of these rates have been approximated as linear, 15,78 while the true dynamics can be more complex. Kinetic models of gene expression assuming more complex dynamics of the rates are also used to infer the protein, 19 or RNA, 79 turnover dynamics but remain studyspecific. There is nowadays no methodological consensus to quantify rates dynamics. Development of new methodological and analytical frameworks 19,38,43,79,80 and data modeling tools for rate-varying systems in 81,82 are ongoing efforts in the field.

Protein turnover at steady-state
Most of the current literature describes protein turnover, explicitly or implicitly, through its "steady-state definition". 6 The underlying assumption implies that the dynamics of protein level P can be described by a simple ordinary differential equation (ODE - Figure 1A): in which s and k are the production and the decay rate of the protein, respectively. This equation is readily solvable for s and k constants.
The dynamics of P follow canonical exponential relaxation dynamics (assuming P(t = 0) = 0 without loss of generality): where the level of protein at equilibrium P eq is given by: Thus, the characteristic response time is given by 1/k. The protein turnover (characteristic time) τ PT is usually defined as the time to renew half of protein levels at equilibrium P eq 6,7 and reads: Beyond the equilibrium definition of protein turnover: Turnover as competing fluxes The common definition of the protein half-life or protein turnover characteristic time τ PT relies on the hypothesis that s and k are constants, that is, time-independent ( Figure 1A), and that a steady-state can be reached. This assumption is unsustainable in most biologically relevant conditions, 14,83 since protein synthesis and decay rates often vary over time, for example, during the cell cycle 78 or after cell stimulation. 15 Therefore, protein levels at equilibrium P eq ; and thus, the characteristic time of protein turnover τ PT cannot be defined. To overcome these difficulties, new quantities are required that account for out-of-equilibrium properties of biological systems. 6 To derive such quantities, we first stress the time dependency of s and k in the governing ODE previously introduced ( Figure 1A): Then, we decided to focus on fluxes that dictate protein input and output in the cell ( Figure 1B). We split the total flux into influx, J in,t , and outflux, J out,t , which correspond to protein synthesis and decay: Protein synthesis and decay rates cannot be directly compared because their dimensions are different. Synthesis is often assumed to be a zero-order reaction while protein degradation is best approximated as a first-order reaction 73 ( Figure 1A). In contrast, comparing fluxes is meaningful and physically correct, independently of the order of these reactions because they are both expressed in (arbitrary unit of) molecules per hour. Thus, we defined the average flux J tot,t : Given a system with an average flux J tot,t and an instantaneous protein level P(t), we next defined the (time-dependent) protein turnover characteristic time τ PT ( Figure 1B and 1C) as: Physically, this represents the time for a system experiencing a constant flux J tot to "metabolize" (synthesize and/or degrade) an amount P(t) of proteins. Finally, using the previous equations, τ PT can be written as: where PT,eq . = 1∕k eq is the equilibrium protein turnover ( Figure 1D) and θ is the imbalance between influx and outflux: with τ p,t = P(t)/s(t). Importantly, θ quantifies to which extent the system is out-of-equilibrium ( Figure 1E). The value of θ also indicates which flux dominates in the system and it is important to understand which component of the system is mostly affecting the protein turnover: As expected, when θ = 1, we have τ PT (t eq ) = τ PT,eq = 1/k eq ( Figure 1F and 1G ).
While we used a deterministic framework along the previous mathematical development, this reasoning can be easily extended to stochastic systems (Box 1). Under linearity assumptions, the deterministic case is the average of a population of cells subject to stochastic dynamics.

Sensitivity analysis of τ PT
To characterize τ PT , we first conducted a sensitivity analysis. Fixing the protein content P in the gene expression system, we varied the protein synthesis rate s and the degradation rate k. We computed the characteristic time of protein turnover τ PT ( Figure 1C) and its components: the equilibrium characteristic time τ PT,eq ( Figure 1D) and the imbalance θ ( Figure 1E). As expected, the characteristic time of protein turnover τ PT

Protein turnover characteristic time at the limits
Studying the limits of τ PT will give some intuitions about how the protein turnover characteristic time behaves according to changes in the protein synthesis rate s, the protein degradation rate k, and the protein level P.
When the synthesis and/or the degradation rate are infinite, that is an infinite number of proteins can be degraded or synthesized per unit of time, the characteristic time of protein turnover τ PT vanishes, meaning that the renewal of the system is instantaneous, as expected.
In these previous cases, when the synthesis or the degradation rate is zero, the characteristic time of protein turnover τ PT is defined by the remaining occurring process, that is, degradation or synthesis, respectively. This is an expected observation, which is at the basis of the analysis of pulse and chase data (exponential decay).
This last limit indicates that in absence of protein synthesis or degradation, the time to renew the protein system is infinite. In other words, the protein system will remain stable indefinitely, as expected.

Elements for a first-principled derivation
Here we give some hints to derive our redefinition of the protein turnover (characteristic time) from first (physical) principles. This derivation gives some intuitions on the physical meaning of our redefinition of the protein turnover.
Discrete system. First, let us define our system as discrete. Its dynamics can be modeled by means of markovian processes. Assume the following birth and death process, with n t ∈ {0, 1, . . . } the number of proteins at time t: Master equation. The probabilistic time evolution of the previous system can be described by a chemical master equation. 84,85 Let P(n|k t ,s t ,t) the probability of the system to be in state n at time t, knowing the rates of the expression system at time t are s t and k t . The master equation reads:

s t , t)
It is worth noting that the system dynamics is fully described taking into account the metadynamic of the system P(k t ,s t ) such that Total flux: Instantaneous exit probability flux. The probability of the system to leave its state for (k t ,s t ) fixed, is given by:

s t , t)
We recognize the total flux J tot : Integrating the equation we obtain, knowing that P(n|k t ,s t ,t = 0) = 1: The factor 2 can be interpreted by the fact that we can exit the system's state in two ways.
Escape rate per particle and characteristic time of protein turnover. One can now assume that the exit probability P(n|k t ,s t ,t) can be factorized as: Continued in which we have, ∀i ∈ (1, 2, . . . , n): where we recognize τ PT : In other words, we found that the exit rate per particle depends only on τ PT,eq and θ, so on the steady-state exit rate per particle "corrected" by how far the system is from steady-state: = Π t,eq × Π t,non−eq decreases, and thus the protein turnover (1∕ PT ) increases, when both s and k increase ( Figure 1C). This change in protein turnover is mediated by a change in the equilibrium protein turnover ( Figure 1D) due to the change in the degradation rate k ( Figure 1B, D) and in the imbalance θ ( Figure 1E). The same conclusions can be drawn from the time evolution of τ PT after acute perturbation of an equilibrium system by a step change in the protein synthesis rate and the degradation rate ( Figure 1F-G -note that the relationship between protein levels and changes in k and s were already described in 6 which we show here again for the sake of the clarity of our argumentation). As expected, when the system reaches a steady-state ( Figure 1F), τ PT converges towards τ PT,eq ( Figure 1G). In summary, a flux-based definition of PT allows tracking the dynamics of transient changes in protein turnover, while the equilibrium definition would not detect and quantify the protein turnover changes 6 ( Figure 1G).

τ PT and protein age
To gain insights into the interpretability of protein turnover, we wondered how the characteristic time τ PT is linked to protein age. In the previous definition of the protein turnover characteristic time, the latter corresponds to the protein half-life and gives thus the protein population average age at steady-state (that is independent of the synthesis rate). When the system is out of steady-state this simple relationship breaks down. We thus investigated if our redefinition of protein turnover can predict changes in the average age of the protein population. We performed stochastic simulations of the simple chemical system presented before ( Figure 1H). 85 At the time 100 h, we acutely changed the synthesis rate s from 100 proteins/hour to a value between 0 and 500 proteins/hour -putting the system out-of-equilibrium -and followed τ PT and the average protein age over time ( Figure 1I-J). While after a long time (t ≫ τ PT ) the average protein age converges towards its steady-state and initial value, instantaneous protein age depends on the actual value of the synthesis rate and the intensity of the perturbation. As expected, increasing the protein turnover characteristic time τ PT ( Figure 1I) (thus reducing protein turnover) leads to an increase in the average protein age ( Figure 1J).
While the exact relation between the relaxation dynamics of τ PT and the average protein age is non-trivial, we observe overlapping while delayed relaxation dynamics of the two observables ( Figure 1K).

Error sensitivity of τ PT
To measure the protein level P, the protein synthesis rate s and protein degradation rate k usually involves complex and often noisy time course proteomics and transcriptomics experiments. 6 pairs (s, k) tested (Figure 2A). In particular, the error is increasing when we are far from the equilibrium such that P ≪ P eq . Mathematically, this is well demonstrated by looking at the case P ≫ P eq : When P ≪ P eq the previous simplification does not appear anymore and the error on τ PT increases as the error on P. Positive errors on s and k lead to an underestimation of the protein turnover for P ≪ P eq and P ≫ P eq respectively. For negative errors ( Figure 2B which the error on τ PT is lower than 5% of the true value ( Figure 2E).
The dependency on the system state, namely if P ≫ P eq or P ≪ P eq , is also confirmed ( Figure 2E).

Guidelines to quantify protein turnover using a Bayesian approach
To make our definition of protein turnover easy to use, hereafter we propose guidelines to design and analyze experiments for robust quantification of protein turnover. These aim to determine if protein turnover is different across experimental conditions or if it is changing over time in a single condition. This framework considers experimental and modeling noise/errors and their propagation to the protein turnover characteristic time using a Bayesian framework. First, it is essential to design an adequate Control condition ( Figure 2F), which can be non-trivial. For example, to quantify changes of protein turnover in a Test condition, a Control condition in which protein turnover of protein A is known to be constant over time is optimal. As illustrated by our case study, this is not trivial as protein turnover can change over time even in seemingly constant culture conditions. In this case, the question is whether protein turnover of protein A changes more in the Test condition than in the Control condition. Second, we recommend to maximize the number of experimental replicates for proteomics and transcriptomics in both Test and Control conditions to quantify technical and biological variability in our system. This will also increase the robustness of further modeling, incorporating the contribution of data variability in the modeling itself. Third, we suggest merging all replicates per condition, to have a fully Bayesian modeling framework. The error model of the data, or the biological variability prior, should be defined directly in the Bayesian inference algorithm used (in the likelihood function chosen for instance). We suggest using a Bayesian inference algorithm such as the one presented in. 81,82 Similarly, modeling different peptides from the same protein separately permits to assess and incorporate in the data analysis potential bias and uncertainties coming from mass-spectrometry quantification. 19 Fourth, after modeling, we will have access to the posterior distribu- Before applying our analytical framework to the data from Jovanovic et al., 15 we assessed whether the analysis of RNA and protein levels along with gene expression rates allows to determine specific and shared genome-wide responses. In addition, we investigated if there were systematic biases in the data, for example, a systematic fold-change of the synthesis rate upon LPS stimulation with respect to the initial time point. We first analyzed the

LPS condition MOCK condition (H) (G) (F) (E) (D) (C) (B) (A)
F I G U R E 3 DESCRIPTIVE overview of gene expression remodeling in DCs after LPS stimulation. 2D density and scatterplots of fold-change in protein level with respect to fold-change in RNA level for LPS (A -r 2 = 0.19) or MOCK condition (C -r 2 = 0.0006). FI stands for fold-increase. The fold-change was computed for one condition between the levels measured at 12 h versus those measured at 0 h. 2D density and scatterplots of the protein degradation rate k with respect to the protein synthesis rate s measured at time 12 h in LPS (B -r 2 = 0.03) and MOCK condition (D -r 2 = 0.03). Distributions of the rates (protein synthesis rate s, RNA translation k tr. , protein degradation k) values, and levels (protein and RNA) inferred and measured at time 0 h (blue) and 12 h (red) in the LPS (E) and MOCK condition (F). Kernel density estimates are shown. p-values are computed using a Kolmogorov-Smirnov statistical test. Protein-specific rates after LPS (G) or MOCK (H) stimulation (12 h) with respect to protein-specific rates before it (0 h).
correlation between gene expression rates and their output in terms of fold-increases in RNA and protein levels. We found that changes in protein and RNA levels were poorly correlated in both MOCKand LPS-stimulated conditions ( Figure 3A and 3C). Similarly, we found no clear correlations between protein-specific protein synthesis and degradation rates ( Figure 3B and 3D). This hints at variable strategies to regulate gene expression, protein levels, and dynamics in response to perturbations. Moreover, these results also suggest that there is no trivial bias in the inferred and measured values of protein, RNA levels and s and k introduced by experimental or computational procedures.
To describe how global gene expression is altered upon stimulation with LPS, we first studied the changes in gene expression rates in both MOCK and LPS stimulation conditions. We computed fold-changes in protein synthesis rate s, RNA translation rate k tr , protein degradation rate k, and in RNA and protein levels ( Figure 3E-F). LPS stimulation of DCs leads to statistically significant alteration of RNA levels and proteome-wide protein synthesis rate s, RNA translation rate k tr , and protein degradation rate k. Interestingly, protein levels do not exhibit such statistically significant alteration. This suggests that even though proteome composition is altered 15 (Figure 3A), total protein content, proportions, and protein expression regimes remain relatively F I G U R E 4 LPS stimulation of DCs leads to a global increase in protein turnover. 2D density and scatterplots of the equilibrium protein turnover characteristic time τ PT,eq at timepoint 12 h with respect to timepoint 0 h in LPS (A) or MOCK (F) conditions. 2D density and scatterplots of the imbalance θ at timepoint 12 h with respect to timepoint 0 h in LPS (B) or MOCK (G) conditions. 2D density and scatterplot of the protein turnover characteristic time τ PT at timepoint 12 h with respect to timepoint 0 h in LPS (C) or MOCK (H) conditions. The gray dashed lines represent the average of the distributions. 88%, respectively 91%, of measured proteins exhibit an increased protein turnover after 12 h of LPS (D), or respectively MOCK (I), stimulation of the DCs, with respect to time point 0 h. The dashed lines represent the average of the distributions. GO (biological processes) terms enrichment analysis was performed on proteins with the highest protein turnover increase (red bars, n = 200) or with the highest protein turnover decrease (blue bars, n = 200) in LPS (E) and MOCK (J) conditions. Only the 10 GO (biological processes) terms with the lowest adjusted p values (with all p value < 0.05) are shown.
stable. However, it is important to note that because MS uses the same amount of total protein input for each sample, we cannot exclude that differences in total protein content per cell between conditions could limit the robustness of this conclusion. The same observations can be made in the MOCK condition, even though the change in RNA distribution appears weaker than in LPS condition ( Figure 3F).
Surprisingly, protein synthesis and degradation rates vary substantially in both MOCK and LPS stimulation conditions between 0 and 12 h (Figure 3E-H). This indicates that cultured mouse bone marrowderived DCs are not at steady-state. This is likely caused by the change in environment that these cells encounter when seeded into cell culture conditions after isolation.

Changes in protein turnover of mouse DCs upon LPS stimulation
To decipher how protein turnover changes during LPS stimulation of DCs, we took advantage of our redefinition of protein turnover and its mathematical framework. We observed that 12 h of LPS or MOCK LPS stimulation, and even more so after 12 h of stimulation. 88%, respectively 91%, of measured proteins exhibit an increased protein turnover after LPS, respectively MOCK, stimulation of DCs for 12 h, with respect to the absence of stimulation ( Figure 4D and 4I).
Interestingly, proteins involved in immune response triggered by LPS are over-represented in proteins with the highest protein turnover increase in the LPS condition ( Figure 4E) but not in the MOCK condition ( Figure 4J).

Quantification of errors and statistical significance of protein turnover analysis in Jovanovic et al. data
We next assessed how technical errors and/or biological variability would affect protein turnover analyses. We analyzed separately and compared the two replicates done in each condition in Jovanovic et al.
We find that the protein degradation rate, and to a lesser extent the protein synthesis rate, exhibit the higher relative errors of the variables used to compute the protein turnover ( Figure 5A). While P and RNA are directly measured, k and s have to be inferred, thus suggesting that most of the data variability comes from gene expression rate inference. When computing the fold-increase in protein turnover separately for the two replicates, we found that only 5% of measured proteins exhibit a relative error lower than 10% for both time points 0 h and 12 h ( Figure 5B -red points -and Figure 5C). Importantly, the distribution of protein turnover fold-increase of these low-error genes is similar to the whole proteome ( Figure 5D). This suggests that the analysis of protein turnover changes caused by LPS stimulation is robust to inter-replicate variability, that is, technical errors and/or biological variability, even though having only two replicates limits the robustness of statistical analysis for individual proteins ( Figure 5E).
We then performed differential protein turnover variation analysis between LPS and MOCK conditions. We first merged the replicates, assuming that technical errors would not affect the conclusions of the subsequent analysis. We selected the proteins for which there is at least a two-fold change in protein turnover between LPS and MOCK conditions between 0 and 12 h ( Figure 5F -red points and red dotted-lines). We filtered these proteins into two groups according to their decrease in protein turnover (or τ PT increase, orange), or increase in protein turnover (or τ PT decrease, cyan) after LPS stimulation ( Figure 5F). When then performed Gene Ontology (GO)term enrichment analysis on these two groups, which recovered the conclusions made by analyzing LPS stimulation alone ( Figure 5G vs. Figure 5E), with an over-representation of proteins involved in immune response triggered by LPS with the highest protein turnover increase ( Figure 5G). In contrast, we observe no statistically significant GO term enrichment in the set of proteins with the highest protein turnover decrease ( Figure 5G).

CONCLUSIONS AND OUTLOOK
Here we show how the redefinition of protein turnover through flux analysis can be applied to existing out-of-equilibrium data and how it can lead to new biological insights. We showed, using differential protein turnover variation analysis, that LPS stimulation of bone-marrow-derived DCs leads to an increased turnover of proteins involved in the acute immune response. Specific regulatory strategies may have evolved for given subsets of the proteome to balance the velocity at which cells respond to stimuli, energy consumption related to the cost of translation and protein degradation, and the maintenance of a high-quality proteome. 83,86 Importantly, we cannot exclude that changes in the turnover of some proteins might be due to changes in their allocation to protein complexes altering their stability, 45,[50][51][52][53] or changes in PTM profiles mediated by LPS stimulation. 40,66,[68][69][70][71][72] Our proposed redefinition of protein turnover is operative and could be applied to a broad range of data. Quantifying protein turnover using our approach needs only protein synthesis rate s, degradation rate k, and protein level P ( Figure 1B). All these quantities can be obtained through available techniques such as pulsed SILAC, 15 BONCAT, 87 or live-cell imaging. 78 New techniques such as singlecell proteomics 43,44 and ARTseq-FISH 88 may also benefit from our redefinition and generalization of protein turnover in the near future.
In summary, our redefinition of protein turnover allows to quantify protein turnover dynamics in out-of-equilibrium conditions such as cell proliferation and differentiation. It should thus be instrumental in improving our understanding of cell identity dynamics, fate bifurcation, and maintenance.

AUTHOR CONTRIBUTIONS
Benjamin Martin conceptualized the project and analyzed the data.
Benjamin Martin and David M. Suter interpreted the data and wrote the manuscript.

ACKNOWLEDGMENTS
This work is supported by EPFL dotation to UPSUTER. We thank Nikolai Slavov, Maike Hansen, Sun Shoujie, Joanna Dembska, Andrea Salati and Félix Naef for discussions and for carefully reading the manuscript.

CONFLICT OF INTEREST STATEMENT
Authors declare no competing interests.

DATA AVAILABILITY STATEMENT
The codes that support the findings of this study are available from the corresponding authors upon reasonable request.