---
title: "Calculations"
output: 
  bookdown::html_document2:
    base_format: rmarkdown::html_vignette
    toc: true
    number_sections: true
    always_allow_html: yes
pkgdown: 
    as_is: true
bibliography: hadex2_docs.bib
vignette: >
  %\VignetteIndexEntry{Calculations}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

This article explains how HaDeX2 processes raw HDX-MS data files into aggregated datasets used for downstream analysis and visualization. It describes the processing of the data, the calculation workflow, and the results. For the visualization methods of described values, see `vignette("visualization")`.

# Initial processing of the data 

Let's start with a glimpse of the datafile - in this case, in `cluster` format from DynamX. For more information about accepted datafiles see `vignette("data_input")` 

```{r warning=FALSE}
library(HaDeX2)
head(alpha_dat, 6)
```

As we see, the data file has a very specific structure and is not informative yet. In the file, we have $m/z$ values for each $z$ value (charge) for each time point ($Exposure$) for each state of each peptide, repeated as many times as measurement was repeated (each measurement should be repeated at least three times). 

Our aim is to have one result with an uncertainty of the measurement per each peptide in each biological state in each measured time point - data in this format allows further calculations, e.q. calculating deuterium uptake values.

For a better understanding of the process of aggregating the data, see the `vignette("transformation")` article.

Within each replicate of the measurement (we recognize each replicate by the $File$ value), the $m/z$ values are provided for each possible $z$ value. The $m/z$ values are in the $Center$ column, as it is a geometrical centroid calculated from the isotopic envelope. Firstly, we have to calculate the mass value, measured experimentally:

$$expMass = z \times (Center-protonMass)\tag{1}$$
To aggregate data from different $z$ values, we have to calculate the mean mass weighted by intensity. Additional information about this step and how the weighted mean impacts the results can be found in `vignette("mass_comparison")` article. 

$$aggMass = \frac{1}{N}\sum_{k = 1}^{N}Inten_k{\cdot}pepMass_k\tag{2}$$.

Where:

* $aggMass$ - mass of the peptide [Da], average from replicates,
* $Inten_k$ - intensity of the measurement,
* $pepMass$ - mass of the peptide, calculated using Equation 1.

As we use the aggregated result from the replicates, we need to calculate an uncertainty associated with the measurement. We use the mean value as the final result, so we need to calculate error as a standard deviation of the mean, according to the Equation 3:

$$u(\vec{x}) = \sqrt{\frac{ \sum_{i=1}^n \left( x_{i} - \overline{x} \right)^2}{n(n-1)}}\tag{3}$$.

Where:

* $x_{i}$ - measured value from a replicate,
* $\overline{x}$ - mean value from all of the replicates,
* $n$ - number of replicates.

Now we have the format we want for further calculations.

# Calculation of the deuterium uptake

HaDeX package provides the calculated values in different forms. All of them are provided with associated uncertainty of the measurement. All of the uncertainties are derived from the formula - the Law of propagation of uncertainty:

$$u_{c}(y) = \sqrt{\sum_{k} \left[ \frac{\partial y}{\partial x_{k}} u(x_{k}) \right]^2}$$

## Deuterium uptake

Deuterium uptake is the increase of the mass of the peptide in time $t$. The minimal exchange control $m_{t_0}$ is mass measured directly after adding the buffer (before the start of the exchange), and $m_t$ is the mass measured in chosen time point $t$. The value is in Daltons [Da].

$$D = m_{t} - m_{t_0}$$
The uncertainty associated with deuterium uptake [Da] (based on equation 3):

$$u_c(D) = \sqrt{u(m_t)^2 + u(m_{t_0})^2}$$

## Fractional deuterium uptake

Fractional deuterium uptake is the ratio of the increase of the mass in time $t$ to the maximal exchange control. The maximal exchange control $m_{t_{100}}$ is measured after a long time (chosen by the experimenter, usually 1440 min = 24 h). It is assumed that after this long time, the exchange is finished. The minimal exchange control $m_{t_0}$ is mass measured directly after adding the buffer (before the start of the exchange), and $m_t$ is the mass measured in chosen time point $t$. This value is a percentage value [%]. 

$$D_{frac} = \frac{m_{t} - m_{t_0}}{m_{t_{100}} - m_{t_0}}$$

The uncertainty associated with fractional deuterium uptake [%] (based on equation 3):

$$u_{c}(D_{frac}) = \sqrt{ \left[ 
\frac{1}{m_{t_{100}}-m_{t_0}} u(m_{t}) \right]^2 + 
\left[ \frac{m_{t} - m_{t_{100}}}{(m_{t_{100}}-m_{t_0})^2} u(m_{t_0}) \right]^2 + 
\left[ \frac{m_{t_0} - m_{t}}{(m_{t_{100}}-m_{t_0})^2} u (m_{t_{100}}) \right]^2}$$

## Theoretical deuterium uptake

Theoretical deuterium uptake is the increase of mass in time $t$ compared with the theoretical value of the peptide mass without any exchange ($MHP$ - a mass of the singly charged monoisotopic molecular ion), and $m_t$ is the mass measured in chosen time point $t$. This value is in daltons [Da]:

$$D_{theo} = m_{t} - MHP$$

The uncertainty associated with theoretical deuterium uptake [Da] (the $MHP$ value is a constant without measurement uncertainty - based on the equation 3):

$$u(D_{theo}) = u(m_t)$$

## Theoretical fractional deuterium uptake

Theoretical fractional deuterium uptake is the ratio of the increase of mass in time $t$ compared with a theoretical value of the mass of the peptide without any exchange to the possible theoretical increase of the mass, based on the maximal potential uptake of the peptide (based on the peptide sequence). This value is a percentage value [%].

$$D_{theo, frac} = \frac{m_{t}-MHP}{MaxUptake \times protonMass}$$

The uncertainty associated with theoretical fractional deuterium uptake [%] (based on the equation 3):

$$u(D_{theo, frac}) = \left|\frac{1}{MaxUptake \times protonMass} u(D_{t}) \right|$$

# Differential values

Differential value is the way to see how the deuterium uptake differs between two biological states. It allows seeing if the possible difference is statistically important (more information below). This value is calculated as the difference between the previously described (in a chosen form) deuterium uptake of the first and second states. 

$$diff = D_{1} - D_{2}$$
The uncertainty associated with the difference of deuterium uptake (based on the equation 3):

$$u_{c}(diff) = \sqrt{u(D_{1})^2 + u(D_{2})^2}$$

The convenient way to present results calculated as described is the comparison plot and differential plot (Woods' plot).

# Deuterium uptake averaging 

Averaging of the deuterium uptake values is inspired by an already published solution [@keppel_mapping_2015]. This process can be conducted of any deuterium uptake variation - fractional or absolute, with theoretical or experimental control, single-state or differential. 

For each residue $i$ there is a subset of peptides ($n$) covering said residue. Then, the final $D_{i, agg}$ is calculated from the subset of $D_k$ (where $0<k<n$), with weights $w_k$ inverse proportional to the max uptake of peptide ($MaxUptake_k$) - the shortest the peptide the highest the possibility that the uptake took place in said residue:

$$
w_k = \frac{\frac{1}{MaxUptake_k}}{\sum_n \frac{1}{MaxUptake_k}}
$$
The weights are normalized:

$$
\sum_n w_k = 1
$$
Then, the $D_{i, agg}$ is a weighted average of set of $D_n$.

$$
D_{i, agg} = \frac{\sum_n w_k \cdot D_k}{\sum_n w_k}
$$
This process is done for each residue $i$ to cover whole protein data. 

As for previous values, also for $D_{i, agg}$ the uncertainty is calculated:

$$
u(D_{i, agg}) = \sqrt{ \sum_{n} \left[ \frac{  w_k \cdot U(D_k)}{ \sum_n w_k } \right] ^2}
$$


# Back-exchange

Back-exchange is a reverse exchange - from already exchange deuterium back to hydrogen. 

Back-exchange ($bex$) is calculated for each peptide and is defined as the difference between fractional  theoretical deuterium uptake of the experimental maximal deuterium control and 100% of exchange. 

$$
bex = 100 - D_{theo, frac}(t_{100})
$$


# AUC {#AUC}

The Area Under the Curve (AUC) represents the integral of the deuterium uptake curve. In this context, higher AUC values characterize rapidly exchanging peptides, while lower values signify minimal exchange. To constrain the results to a $[0, 1]$ range, the AUC is normalized against the maximum theoretical uptake at the last measured time point. Consequently, an AUC exceeding 1 serves as a diagnostic indicator of back-exchange.

$$
\textrm{AUC} = \sum_i^{n-1} (D_{frac, i+1} + D_{frac, i}) \cdot (t_{\textrm{norm}, i+1} - t_{\textrm{norm}, i}) \cdot \frac{1}{2}
$$