EstimateKinetics

Introduction

The goal of most standard, single label, NR-seq experiments is to assess the kinetics of RNA synthesis and degradation. In EZbakR, this is the task of EstimateKinetics(). This vignette will discuss the basics of the standard analysis implemented by EstimateKinetics(), and touch on two alternative analysis strategies that EstimateKinetics() also implements: non-steady-state and short-feed analyses.

library(EZbakR)

The basics

Running the standard kdeg and ksyn estimation

Below I demonstrate how to estimate synthesis and degradation rate constants using the standard analysis workflow and simulated data:

# Simulate data
simdata <- EZSimulate(nfeatures = 300, nreps = 2)

# Create an `EZbakRData` object
ezbdo <- EZbakRData(simdata$cB, simdata$metadf)

# Estimate fractions
ezbdo <- EstimateFractions(ezbdo)
#> Estimating mutation rates
#> Summarizing data for feature(s) of interest
#> Averaging out the nucleotide counts for improved efficiency
#> Estimating fractions
#> Processing output

# Estimate synthesis and degradation rate constants
ezbdo <- EstimateKinetics(ezbdo,
                          strategy = "standard")

Setting strategy = "standard" is not strictly necessary, but I do it here to highlight that this is an argument that you can alter. If you have multiple fractions tables, then you will need to specify aspects of the metadata of the table you want to use. Under the hood, EZbakR is passing these arguments to EZget(), so check out the vignette on EZget() to better understand how this works:

# Create another fractions table
ezbdo2 <- EstimateFractions(ezbdo, overwrite = FALSE)
#> Estimating mutation rates
#> Summarizing data for feature(s) of interest
#> Averaging out the nucleotide counts for improved efficiency
#> Estimating fractions
#> Processing output

# Estimate synthesis and degradation rate constants
  # Use the 1st identical fractions table you created
ezbdo2 <- EstimateKinetics(ezbdo2,
                          repeatID = 1,
                          strategy = "standard")

In this case, repeatID has to be specified because the two fractions table present are identical in all other metadata aspects. Other metadata that you can specify are:

• features: Set of features analyzed, or a subset of features unique to the table of interest.
• populations: Mutation populations analyzed. Relevant if one of the tables is from a single-label analysis, and the other is from a multi-label analysis. EstimateKinetics()’s standard analysis strategy is only compatible with single-label analyses.
• fraction_design: The fraction design table used for the analysis of interest. Rarely will be necessary to specify in this case.

What is the “standard” strategy?

The standard strategy makes the following assumptions:

1. You have conducted a pulse-label experiment. That means, cells were labeled for a period of time that I will refer to as the label time, and RNA was extracted after this labeling. This is in contrast to a pulse-chase design, where you label for a certain amount of time (the pulse time) and then chase with regular nucleotide for a certain amount of time (the chase time). Eventually, EZbakR will support kinetic parameter inference from a pulse-chase design, but I will say more about that at the end of this vignette.
2. All of the populations of RNA you analyzed were at steady-state during the course of metabolic labeling. This means that their levels were constant and not changing in response to some perturbation during the labeling.
3. RNA synthesis is a 0-th order, constant rate process, and RNA degradation is a 1st-order, single rate constant process.

If all of these assumptions are met, then the dynamics of a given RNA population are well described by the following differential equation:

$$\begin{align} \frac{\text{dR}}{\text{dt}} = k_{\text{syn}} - k_{\text{deg}}*\text{R} \end{align}$$ where $\text{R}$ represents the concentration of RNA over time and $k_{\text{syn}}$ and $k_{\text{deg}}$ are the synthesis and degradation rate constants, respectively. The general solution to this differential equation is:

$$\begin{align} \text{R(t)} = \frac{ k_{\text{syn}}}{k_{\text{deg}}} + (\text{R}_\text{o} - \frac{ k_{\text{syn}}}{k_{\text{deg}}})* e^{-\text{tl}*k_{\text{deg}}} \end{align}$$

where $\text{R}_\text{o}$ is the initial concentration of the RNA and $\text{tl}$ is the label time. Thus the concentration of new RNA ($\text{R}_\text{o} = 0$) as a function of time is:

$$\begin{align} \text{R}_{\text{new}}\text{(t)} = \frac{ k_{\text{syn}}}{k_{\text{deg}}}*(1 - e^{-\text{tl}*k_{\text{deg}}}) \end{align}$$ As we are assuming that the total RNA concentration is not changing during the labeling, and since the steady-state concentration of RNA (solve for R setting $\frac{\text{dR}}{\text{dt}} = 0$) is $\frac{ k_{\text{syn}}}{k_{\text{deg}}}$, we get that the fraction of RNA that is new (denoted $\theta$) is:

$$\begin{align} \theta = 1 - e^{-\text{tl}*k_{\text{deg}}} \end{align}$$ We can thus solve for the degradation rate constant, and use the normalized read counts to infer the synthesis rate constant.

$$\begin{align} k_{\text{deg}} &= -\frac{\text{log}(1 - \theta)}{\text{tl}} \\ k_{\text{syn}} &= (\text{normalized read count})*k_{\text{deg}} \end{align}$$

NOTE: degradation rate constants estimated in this manner will be in absolute units (1/time), whereas synthesis rate constants will be in relative units (read counts/time). Thus, degradation rate constant estimation is “internally normalized” (the fraction new is a ratio of two unnormalized quantities that would have the same normalization scale factor), whereas synthesis rate constant estimates require normalization across libraries and samples. At this stage, EZbakR uses TMM normalization (as in packages like edgeR and DESeq2) to calculate the normalized read counts. In the near future, it will also support users providing external scale factors (e.g., those derived from spike-ins).

Short feed analyses

Sometimes it is advantageous to use a fairly short label time. For example, NR-seq is commonly used as a replacement for enrichment based metabolic labeling methods (e.g., TT-seq), and a short label time is used to ensure that there is limited degradation of newly synthesized RNA during the feed. EZbakR formalizes this strategy and provides a way by which to estimate synthesis and degradation rate constants in this context.

How to run it

Everything is the same as it was in the standard case, except now we set strategy = "shortfeed":

# Estimate synthesis and degradation rate constants
ezbdo <- EstimateKinetics(ezbdo,
                          strategy = "shortfeed")

What is the “shortfeed” strategy?

In the case of an extremely short label feed time (relative to the average half-life of RNA being analyzed), we can assume that there is no degradation newly synthesized RNA during the label time. Thus, the following differential equation and solution to said ODE describes the dynamics of the new RNA:

$$\begin{align} \frac{\text{dR}}{\text{dt}} &= k_{\text{syn}} \\ \text{R}_{\text{new}}\text{(t)} &= k_{\text{syn}}*\text{t} \end{align}$$ Assuming that the concentration of new RNA is proportional to the normalized number of new reads yields the following estimate for $k_{\text{syn}}$:

$$k_{\text{syn}} = \frac{\theta*(\text{normalized read count})}{\text{tl}}$$ Assuming the RNA levels are at steady-state (as in the standard analysis), yields the following estimate for the degradation rate constant:

$$\begin{align} \text{normalized read count} &= \frac{k_{\text{syn}}}{k_{\text{deg}}} \\ k_{\text{deg}} &= \frac{k_{\text{syn}}}{\text{normalized read count}} \\ k_{\text{deg}} &= \frac{\theta}{\text{tl}} \end{align}$$

The calculus enthusiasts among you may recognize that this estimate for $k_{\text{deg}}$ is equivalent to a 1st order Taylor series approximation of the “standard” estimate strategy. This is no coincidence, as the assumption of a short feed is equivalent to assuming that $\theta$ is close to 0 (most of the RNA is old). It may also seem odd that this estimate is bounded between 0 and 1/tl, but this just reflects the assumption made by this strategy that RNA half lives are much shorter than the label time.

Non-steady-state (NSS) analyses

NOTE: The strategy described here is a slight modification of that discussed in Narain et al., 2021. If you use EZbakR’s “NSS” strategy, please cite that paper as well as EZbakR.

One of the key assumptions made by the “standard” analysis strategy is the assumption that RNA levels are at steady-state during the labeling. This assumption can be violated if a perturbation was applied to cells shortly before the start of, or during, labeling. It can also be violated if you are studying a process (e.g., development) where the state of the cells is changing over the course of the experiment. EstimateKinetic() can adjust its kinetic parameter estimation strategy accordingly, though it requires making some assumptions about how the experiment was performed. See below for details

Requirements of the “NSS” strategy

In this case, it is important to discuss some key aspects about how this strategy works before showcasing it. Namely:

1. Instead of assuming the fraction new in a given sample is related to the degradation rate constant, EZbakR will compare the old read levels in the labeled sample and a set of samples that provide information about RNA abundances at the start of the labeling.
2. EZbakR assumes that your metadf contains two additional columns. One should specify which samples belong to the same biological condition/analysis time point and thus should assay similar RNA populations. The other should specify which group of samples identified by the first column provide information about the RNA population that existed at the start of labeling.

For example, suppose you have a time course with the following samples:

1. “sampleA”: -s4U data collected at the start of the time course
2. “sampleB”: +s4U data collected after a 2 hour labeling period that started at the start of the time course.
3. “sampleC”: +s4U data collected after a 2 hour labeling period that started at a time point equivalent to the end of the labeling in “sampleB”.

In this case, “sampleA” provides information about the initial RNA concentrations for “sampleB”, and “sampleB” provides information about the initial RNA concentrations for “sampleC”. Therefore, your metadf could look like:

metadf <- data.frame(
  sample = c('sampleA',
             'sampleB',
             'sampleC'),
  tl = c(0, 2, 2),
  group = c("a", 
            "b", 
            "c"),
  reference = c("a",
                "a",
                "c")
)

One thing to note is that the value you enter for “sampleA”’s reference is technically arbitrary, as it is a -s4U sample that you will not be estimating degradation rate constants from. Thus, you could enter any string for its value in the column named “reference” in this example.

How to run it

The only difference in this case is that you have to tell EZbakR which metadf columns correspond to the two additional columns explained above. The relevant parameters are:

1. grouping_factor: Which column defines groups of samples which assay the same RNA population?
2. reference_factor: Which column defines the groups of samples which assay the RNA population that existed at the start of labeling?

# new metadf
metadf <- data.frame(sample = paste0('sample', 1:6),
                 tl = c(2, 2, 2, 2, 0, 0),
                 group = c('b', 'b',
                           'c', 'c',
                           'a', 'b'),
                 ref = c('a', 'a',
                         'b', 'b',
                         'a', 'a'))

# Create an `EZbakRData` object
ezbdo <- EZbakRData(simdata$cB, metadf)

# Estimate fractions
ezbdo <- EstimateFractions(ezbdo)
#> Estimating mutation rates
#> Summarizing data for feature(s) of interest
#> Averaging out the nucleotide counts for improved efficiency
#> Estimating fractions
#> Processing output

# Estimate synthesis and degradation rate constants
ezbdo <- EstimateKinetics(ezbdo,
                          strategy = "NSS",
                          grouping_factor = "group",
                          reference_factor = "ref")

In this case, the only experimental details column of the metadf is called “treatment”, so this is specified in grouping_factors.

What is the “NSS” strategy?

You can see Narain et al. for an alternative discussion of this strategy.

We would like to relax the assumption that the total RNA levels are unchanging over the course of the labeling. We can do this by considering the dynamics of the old RNA during the labeling. If we assume that a single, unchanging degradation rate constant describes their turnover, then the following ODE and solution describe its dynamics:

$$\begin{align} \frac{\text{dR}}{\text{dt}} &= -k_{\text{deg}}*\text{R} \\ \text{R}_{\text{old}}\text{(t)} &= \text{R}_{\text{o}}*e^{-k_{\text{deg}}*\text{t}} \end{align}$$ In the steady-state case, $\text{R}_{\text{o}}$ would be equivalent to the steady-state RNA level ($\frac{ k_{\text{syn}}}{k_{\text{deg}}}$). In the away from steady-state case though, this assumption is no longer valid. Despite this, assume for a second that we could estimate $\text{R}_{\text{o}}$. Under this assumption, we could then estimate $k_{\text{deg}}$ as follows:

$$k_{\text{deg}} = -\frac{\text{log}(\text{R}_{\text{old}}\text{(t)}/\text{R}_{\text{o}})}{\text{tl}}$$

We can then return to the general solution of the differential equation described in the ‘standard’ analysis strategy section:

$$\begin{align} \text{R(t)} &= \frac{ k_{\text{syn}}}{k_{\text{deg}}} + (\text{R}_\text{o} - \frac{ k_{\text{syn}}}{k_{\text{deg}}})* e^{-\text{tl}*k_{\text{deg}}} \\ \text{R}_{\text{new}}\text{(t)} &= \frac{ k_{\text{syn}}}{k_{\text{deg}}}*(1 - e^{-\text{tl}*k_{\text{deg}}}) \end{align}$$ where in the second line we set $\text{R}_\text{o} = 0$ as there was no new RNA at the start of labeling, and derive an estimator for $k_{\text{syn}}$:

$$k_{\text{syn}} = \frac{\theta*(\text{normalized read count)}*k_{\text{deg}}}{1 - e^{-k_{\text{deg}}*\text{tl}}}$$ This yields the same estimator as in the steady-state case when $\theta = 1 - e^{-k_{\text{deg}}*\text{tl}}$. To make this strategy work, we just need to estimate the starting levels of old RNA. Thus, you need some sort of RNA-seq data from a timepoint equivalent to the start of labeling.

One thing you may be wondering is just how much of non-steady-state analysis this really is if we are still assuming constant rate constants? What if the rates of synthesis and degradation are being actively modulated by the cell during the label time? I won’t present a detailed proof here, but it turns out that the strategy laid out above is the best that you can do. In the case where rate constants are changing during the label time, you can think of the estimates from this strategy as time averages of the true rate constants, averaged over the labeling period.

Pulse-chase analyses

The infrastructure is in place for EZbakR to support pulse-chase analyses of kinetic parmaeters (e.g., metadf can accept specification of a pulse and chase time). Thus, in the future, EZbakR will provide such support. That being said, I would like to take some time to argue against almost ever doing a pulse-chase NR-seq experiment. Pulse-chase experimental designs suffer from a number of shortcomings and are almost never necessary in NR-seq experiments.

There are many classic examples of pulse-chase experimental designs being used to assess the kinetics of RNA metabolism. In these cases though, the point of the pulse was to create a species of RNA whose dynamics are completely driven by the degradation kinetics of the RNA. For example, the pioneering studies from the Parker lab elucidating the mechanisms of RNA degradation involved pulsing with a particular nutrient to stimulate transcription of a construct. The chase then washed out that nutrient to shut off transcription from the construct, meaning that RNA produced from it will exclusively degrade post-chase. When you pulse with a metabolic label like s⁴U though, you have already created a species of RNA whose dynamics are completely degradation driven: the unlabeled RNA. Following the metabolic label pulse with a nucleotide chase is thus redundant, switching the degradation driven population from the unlabeled RNA to the labeled RNA.

In addition, pulse-chase experiments often necessitate extensive exposure of your cells to the metabolic label, which increases the possibility of cytotoxic effects of metabolic label. Finally, analysis of pulse-chase experiments is more complicated than a pulse-label design. To estimate the kinetics of degradation, you need to know what fraction of the RNA for each feature of interest was labeled after the pulse. This means that estimating the synthesis and degradation kinetics requires two separate samples, RNA extracted after the pulse and RNA extracted after the chase. With the pulse alone, you can get all of the information that you can get with both though!! You want multiple time points? Just do multiple pulses. Pulse-labels are easier to perform and easier to analyze than pulse-chases.