NPARC 1.0.1

This vignette shows how to reproduce the analysis described by Childs, Bach, Franken et al. (2019): Non-parametric analysis of thermal proteome profiles reveals novel drug-binding proteins using the `NPARC`

package.

Load necessary packages.

```
library(dplyr)
library(magrittr)
library(ggplot2)
library(broom)
library(knitr)
library(NPARC)
```

First, we load data from a staurosporine TPP experiment (Savitski et al. 2014). The necessary ETL (extract, transform, load) steps have already been conducted, including download from the supplements of the respective publication and conversion into tidy format.

`data("stauro_TPP_data_tidy")`

Before applying any further transformations and filters we create a copy of the imported data.

`df <- stauro_TPP_data_tidy`

Letâ€™s perform a first check of the imported data:

```
df %>%
mutate(compoundConcentration = factor(compoundConcentration),
replicate = factor(replicate),
dataset = factor(dataset)) %>%
summary()
```

```
## dataset uniqueID relAbundance temperature
## Staurosporine:307080 Length:307080 Min. : 0.00 Min. :40.0
## Class :character 1st Qu.: 0.17 1st Qu.:46.0
## Mode :character Median : 0.57 Median :53.5
## Mean : 0.58 Mean :53.5
## 3rd Qu.: 0.95 3rd Qu.:61.0
## Max. :394.98 Max. :67.0
## NA's :70990
## compoundConcentration replicate uniquePeptideMatches
## 0 :153540 1:153540 Min. : 0.00
## 20:153540 2:153540 1st Qu.: 2.00
## Median : 6.00
## Mean : 10.36
## 3rd Qu.: 13.00
## Max. :351.00
## NA's :63400
```

The displayed data contains the following columns:

`dataset`

: The dataset containing the measurements of several TMT-10 experiments. In each experiment, cells were treated with a vehicle or with the compound in one or two concentrations, and measured at ten different temperatures.`uniqueID`

: The unique identifier for each protein. In the given dataset, it contains the gene symbol concatenated by IPI id. Example: 15 KDA PROTEIN._IPI00879051, 16 KDA PROTEIN._IPI00903282, 17 KDA PROTEIN._IPI00878120, 17 KDA PROTEIN._IPI00736161, 176 KDA PROTEIN._IPI00894060, 18 KDA PROTEIN._IPI00644570 .`relAbundance`

: The relative signal intensity of the protein in each experiment, scaled to the intensity at the lowest temperature.`temperature`

: The temperatures corresponding to each of the ten measurements in a TMT experiment.`compoundConcentration`

The concentration of the administered compound in \(\mu M\).`replicate`

: The replicate number in each experimental group. Each pair of vehicle and treatment experiments was conducted in two replicates.`uniquePeptideMatches`

: The number of unique peptides with which a protein was identified.

The imported data contains **307080** rows with entries for **7677** proteins.

First, we remove all abundances that were not found with at least one unique peptide, or for which a missing value was recorded.

```
df %<>% filter(uniquePeptideMatches >= 1)
df %<>% filter(!is.na(relAbundance))
```

Next, we ensure that the dataset only contains proteins reproducibly observed with full melting curves in both replicates and treatment groups per dataset. A full melting curve is defined by the presence of measurements at all 10 temperatures for the given experimental group.

```
# Count full curves per protein
df %<>%
group_by(dataset, uniqueID) %>%
mutate(n = n()) %>%
group_by(dataset) %>%
mutate(max_n = max(n)) %>%
ungroup
table(distinct(df, uniqueID, n)$n)
```

```
##
## 10 20 30 40
## 992 809 993 4505
```

We see that the majority of proteins contain 40 measurements. This corresponds to two full replicate curves per experimental group. We will focus on these in the current analysis.

```
# Filter for full curves per protein:
df %<>%
filter(n == max_n) %>%
dplyr::select(-n, -max_n)
```

The final data contains **180200** rows with entries for **4505** proteins.
This number coincides with the value reported in Table 1 of the corresponding publication (Childs et al. 2019).

We first illustrate the principles of nonparametric analysis of response curves (NPARC) on an example protein (STK4) from the staurosporine dataset. The same protein is shown in Figures 1 and 2 of the paper.

We first select all entries belonging to the desired protein and dataset:

`stk4 <- filter(df, uniqueID == "STK4_IPI00011488")`

The table `stk4`

has 40 rows with measurements of four experimental groups. They consist of two treatment groups (vehicle: \(0~\mu M\) staurosporine, treatment: \(20~\mu M\) staurosporine) with two replicates each. Let us look at the treatment group of replicate 1 for an example:

```
stk4 %>% filter(compoundConcentration == 20, replicate == 1) %>%
dplyr::select(-dataset) %>% kable(digits = 2)
```

uniqueID | relAbundance | temperature | compoundConcentration | replicate | uniquePeptideMatches |
---|---|---|---|---|---|

STK4_IPI00011488 | 1.00 | 40 | 20 | 1 | 8 |

STK4_IPI00011488 | 1.03 | 43 | 20 | 1 | 8 |

STK4_IPI00011488 | 1.06 | 46 | 20 | 1 | 8 |

STK4_IPI00011488 | 1.03 | 49 | 20 | 1 | 8 |

STK4_IPI00011488 | 0.92 | 52 | 20 | 1 | 8 |

STK4_IPI00011488 | 0.93 | 55 | 20 | 1 | 8 |

STK4_IPI00011488 | 0.78 | 58 | 20 | 1 | 8 |

STK4_IPI00011488 | 0.44 | 61 | 20 | 1 | 8 |

STK4_IPI00011488 | 0.21 | 64 | 20 | 1 | 8 |

STK4_IPI00011488 | 0.12 | 67 | 20 | 1 | 8 |

To obtain a first impression of the measurements in each experimental group, we generate a plot of the measurements:

```
stk4_plot_orig <- ggplot(stk4, aes(x = temperature, y = relAbundance)) +
geom_point(aes(shape = factor(replicate), color = factor(compoundConcentration)), size = 2) +
theme_bw() +
ggtitle("STK4") +
scale_color_manual("staurosporine (mu M)", values = c("#808080", "#da7f2d")) +
scale_shape_manual("replicate", values = c(19, 17))
print(stk4_plot_orig)
```

We will show how to add the fitted curves to this plot in the following steps.

To assess whether there is a significant difference between both treatment groups, we will fit a null model and an alternative models to the data. The null model fits a sigmoid melting curve through all data points irrespective of experimental condition. The alternative model fits separate melting curves per experimental group .

We use the `NPARC`

package function `fitSingleSigmoid`

to fit the null model:

`nullFit <- NPARC:::fitSingleSigmoid(x = stk4$temperature, y = stk4$relAbundance)`

The function returns an object of class `nls`

:

`summary(nullFit)`

```
##
## Formula: y ~ (1 - Pl)/(1 + exp((b - a/x))) + Pl
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Pl 0.0000 0.1795 0.000 1.00000
## a 692.6739 226.9106 3.053 0.00419 **
## b 12.5048 4.4989 2.780 0.00851 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1814 on 37 degrees of freedom
##
## Algorithm "port", convergence message: relative convergence (4)
```

The function `augment`

from the `broom`

package provides a convenient way to obtain the predictions and residuals at each temperature in tabular format. By appending the returned predictions and residuals to our measurements, we ensure that relevant data is collected in the same table and can be added to the plot for visualization. The residuals will be needed later for construction of the test statistic:

`nullPredictions <- broom::augment(nullFit)`

Let us look at the values returned by `augment`

at two consecutive temperatures. Note that, while the predictions will be the same for each experiment at a given temperature, the residuals will differ because they were computed by comparing the predictions to the actual measurements:

`nullPredictions %>% filter(x %in% c(46, 49)) %>% kable()`

x | y | .fitted | .resid |
---|---|---|---|

46 | 1.0591140 | 0.9278000 | 0.1313139 |

49 | 1.0333794 | 0.8363683 | 0.1970111 |

46 | 0.9449568 | 0.9278000 | 0.0171568 |

49 | 0.9187253 | 0.8363683 | 0.0823571 |

46 | 0.8661451 | 0.9278000 | -0.0616550 |

49 | 0.7139894 | 0.8363683 | -0.1223788 |

46 | 0.8717407 | 0.9278000 | -0.0560594 |

49 | 0.7068211 | 0.8363683 | -0.1295471 |

Now we can append these values to our data frame and show the predicted curve in the plot:

```
stk4$nullPrediction <- nullPredictions$.fitted
stk4$nullResiduals <- nullPredictions$.resid
stk4_plot <- stk4_plot_orig + geom_line(data = stk4, aes(y = nullPrediction))
print(stk4_plot)
```

Next we fit the alternative model. Again, we compute the predicted values and the corresponding residuals by the `broom::augment()`

function. To take the compound concentration as a factor into account, we iterate over both concentrations and fit separate models to each subset. We implement this by first grouping the data using the function `dplyr::group_by()`

, and starting the model fitting by `dplyr::do()`

.

```
alternativePredictions <- stk4 %>%
# Fit separate curves per treatment group:
group_by(compoundConcentration) %>%
do({
fit = NPARC:::fitSingleSigmoid(x = .$temperature, y = .$relAbundance, start=c(Pl = 0, a = 550, b = 10))
broom::augment(fit)
}) %>%
ungroup %>%
# Rename columns for merge to data frame:
dplyr::rename(alternativePrediction = .fitted,
alternativeResiduals = .resid,
temperature = x,
relAbundance = y)
```

Add the predicted values and corresponding residuals to our data frame:

```
stk4 <- stk4 %>%
left_join(alternativePredictions,
by = c("relAbundance", "temperature",
"compoundConcentration")) %>%
distinct()
```

Add the curves predicted by the alternative model to the plot. Conceptually, it corresponds to the plot shown in Figures 2 (A)/(B) of the paper.

```
stk4_plot <- stk4_plot +
geom_line(data = distinct(stk4, temperature, compoundConcentration, alternativePrediction),
aes(y = alternativePrediction, color = factor(compoundConcentration)))
print(stk4_plot)
```

This plot summarizes Figures 2(A) and 2(B) in the corresponding publication (Childs et al. 2019).

In order to quantify the improvement in goodness-of-fit of the alternative model relative to the null model, we compute the sum of squared residuals (RSS):

```
rssPerModel <- stk4 %>%
summarise(rssNull = sum(nullResiduals^2),
rssAlternative = sum(alternativeResiduals^2))
kable(rssPerModel, digits = 4)
```

rssNull | rssAlternative |
---|---|

1.2181 | 0.0831 |

These values will be used to construct the \(F\)-statistic according to

\[\begin{equation} \label{eq:f_stat} {F} = \frac{{d}_{2}}{{d}_{1}} \cdot \frac{{RSS}^{0} - {RSS}^{1}}{{RSS}^{1}}. \end{equation}\]To compute this statistic and to derive a p-value, we need the degrees of freedom \({d}_{1}\) and \({d}_{2}\). As described in the paper, they cannot be analytically derived due to the correlated nature of the measurements. The paper describes how to estimate these values from the RSS-values of all proteins in the dataset. In the following Section, we illustrate how to repeat the model fitting for all proteins of a dataset and how to perform hypothesis testing on these models.

This section describes the different steps of the NPARC workflow for model fitting and hyothesis testing. Note that the package also provides a function `runNPARC()`

that performs all of the following steps with one single function call.

In order to analyze all datasets as described in the paper, we fit null and alternative models to all proteins using the package function `NPARCfit`

:

`BPPARAM <- BiocParallel::SerialParam(progressbar = FALSE)`

```
fits <- NPARCfit(x = df$temperature,
y = df$relAbundance,
id = df$uniqueID,
groupsNull = NULL,
groupsAlt = df$compoundConcentration,
BPPARAM = BPPARAM,
returnModels = FALSE)
```

`## Starting model fitting...`

`## ... complete`

`## Elapsed time: 38.41 secs`

`## Flagging successful model fits...`

`## ... complete.`

`## Evaluating model fits...`

`## Evaluating models ...`

`## ... complete.`

`## Computing model predictions and residuals ...`

`## ... complete.`

`## Starting model fitting...`

`## ... complete`

`## Elapsed time: 1.54 mins`

`## Flagging successful model fits...`

`## ... complete.`

`## Evaluating model fits...`

`## Evaluating models ...`

`## ... complete.`

`## Computing model predictions and residuals ...`

`## ... complete.`

`str(fits, 1)`

```
## List of 2
## $ predictions: tibble [360,152 Ã— 7] (S3: tbl_df/tbl/data.frame)
## $ metrics : tibble [13,515 Ã— 15] (S3: tbl_df/tbl/data.frame)
```

The returned object `fits`

contains two tables. The table `metrics`

contains the fitted parameters and goodness-of-fit measures for the null and alternative models per protein and group. The table `predictions`

contains the corresponding predicted values and residuals per model.

```
fits$metrics %>%
mutate(modelType = factor(modelType), nCoeffs = factor(nCoeffs), nFitted = factor(nFitted), group = factor((group))) %>%
summary
```

```
## modelType id tm a
## alternative:9010 Length:13515 Min. : 43.56 Min. : 0.0
## null :4505 Class :character 1st Qu.: 50.21 1st Qu.: 837.2
## Mode :character Median : 52.85 Median : 1103.1
## Mean : 54.09 Mean : 1434.2
## 3rd Qu.: 56.29 3rd Qu.: 1489.3
## Max. :298.46 Max. :15000.0
## NA's :787 NA's :12
## b pl aumc resid_sd
## Min. : 0.00001 Min. :0.00000 Min. : 4.87 Min. :0.007522
## 1st Qu.: 16.12000 1st Qu.:0.05282 1st Qu.:11.56 1st Qu.:0.035043
## Median : 21.44813 Median :0.07976 Median :14.46 Median :0.049799
## Mean : 27.04499 Mean :0.14086 Mean :15.03 Mean :0.064155
## 3rd Qu.: 28.40228 3rd Qu.:0.14449 3rd Qu.:18.06 3rd Qu.:0.074672
## Max. :250.00000 Max. :1.50000 Max. :37.42 Max. :1.896479
## NA's :12 NA's :12 NA's :12 NA's :12
## rss loglik tm_sd nCoeffs
## Min. : 0.00113 Min. :-70.47 Min. :0.000e+00 3 :13503
## 1st Qu.: 0.02922 1st Qu.: 27.31 1st Qu.:0.000e+00 NA's: 12
## Median : 0.06361 Median : 37.21 Median :0.000e+00
## Mean : 0.19888 Mean : 40.00 Mean :9.812e+11
## 3rd Qu.: 0.15090 3rd Qu.: 49.53 3rd Qu.:1.000e+00
## Max. :79.39804 Max. :123.31 Max. :2.622e+15
## NA's :12 NA's :12 NA's :787
## nFitted conv group
## 20 :8999 Mode :logical 0 :4505
## 40 :4504 FALSE:12 20 :4505
## NA's: 12 TRUE :13503 NA's:4505
##
##
##
##
```

```
fits$predictions %>%
mutate(modelType = factor(modelType), group = factor((group))) %>%
summary
```

```
## modelType id x y
## alternative:179991 Length:360152 Min. :40.0 Min. :0.0000
## null :180161 Class :character 1st Qu.:46.0 1st Qu.:0.1563
## Mode :character Median :53.5 Median :0.5784
## Mean :53.5 Mean :0.5644
## 3rd Qu.:61.0 3rd Qu.:0.9571
## Max. :67.0 Max. :8.2379
## NA's :12 NA's :12
## .fitted .resid group
## Min. :0.01099 Min. :-1.156805 0 : 89986
## 1st Qu.:0.15962 1st Qu.:-0.025273 20 : 90005
## Median :0.58448 Median : 0.000311 NA's:180161
## Mean :0.56001 Mean : 0.004385
## 3rd Qu.:0.96267 3rd Qu.: 0.028124
## Max. :1.50000 Max. : 7.240276
## NA's :12 NA's :12
```

The results of the STK4 example from earlier can be selected from this object as follows.

First, we check the RSS values of the null and alterantive models:

```
stk4Metrics <- filter(fits$metrics, id == "STK4_IPI00011488")
rssNull <- filter(stk4Metrics, modelType == "null")$rss
rssAlt <- sum(filter(stk4Metrics, modelType == "alternative")$rss) # Summarize over both experimental groups
rssNull
```

`## [1] 1.218132`

`rssAlt`

`## [1] 0.08314745`

Next, we plot the predicted curves per model and experimental group:

```
stk4Predictions <- filter(fits$predictions, modelType == "alternative", id == "STK4_IPI00011488")
stk4_plot_orig +
geom_line(data = filter(stk4Predictions, modelType == "alternative"),
aes(x = x, y = .fitted, color = factor(group))) +
geom_line(data = filter(stk4Predictions, modelType == "null"),
aes(x = x, y = .fitted))
```

In order to compute \(F\)-statistics per protein and dataset according to Equation (), we need to know the degrees of freedom of the corresponding null distribution. If we could assume independent and identically distributed (iid) residuals, we could compute them from the number of fitted values and model parameters. In the following, we will show why this simple equation is not appropriate for the curve data we are working with.

First, we compute the test statistics and p-values with theoretical degrees of freedom. These would be true for the case of iid residuals:

```
modelMetrics <- fits$metrics
fStats <- NPARCtest(modelMetrics, dfType = "theoretical")
```

Let us take a look at the computed degrees of freedom:

```
fStats %>%
filter(!is.na(pAdj)) %>%
distinct(nFittedNull, nFittedAlt, nCoeffsNull, nCoeffsAlt, df1, df2) %>%
kable()
```

nFittedNull | nFittedAlt | nCoeffsNull | nCoeffsAlt | df1 | df2 |
---|---|---|---|---|---|

40 | 40 | 3 | 6 | 3 | 34 |

We plot the \(F\)-statistics against the theoretical \(F\)-distribution to check how well the null distribution is approximated now:

```
ggplot(filter(fStats, !is.na(pAdj))) +
geom_density(aes(x = fStat), fill = "steelblue", alpha = 0.5) +
geom_line(aes(x = fStat, y = df(fStat, df1 = df1, df2 = df2)), color = "darkred", size = 1.5) +
theme_bw() +
# Zoom in to small values to increase resolution for the proteins under H0:
xlim(c(0, 10))
```

`## Warning: Removed 177 rows containing non-finite values (stat_density).`

`## Warning: Removed 177 row(s) containing missing values (geom_path).`