R/exchange_rates.R

In powerHaDeX: Efficient Simulation of HDX-MS Data and Tools for the Statistical Analysis

#' F constants
#'
#' @description Constants related to water, base and acid in cal/(K * mol)
#'
#' @param temp_kelvin temperature reaction in Kelvins
#' @param gas_constant gas constant (1/(dT * R) = 1.9858775)
#'
#' @return list of constants \code{Fta} for acid, \code{Ftb} for base and
#' \code{Ftw} for water.
#'
#' @keywords internal
#'
#' @export
get_F_const = function(temp_kelvin, gas_constant) {
    # Unit: cal / mol
    if(temp_kelvin == 0) stop("Temperature in Kelvin can not be 0.")
    Ea_A = 14000
    Ea_B = 17000
    Ea_W = 19000
    Q7 = (1 / temp_kelvin - 1 / 293) / gas_constant
    list(Fta = exp(-Q7 * Ea_A),
         Ftb = exp(-Q7 * Ea_B),
         Ftw = exp(-Q7 * Ea_W))
}


#' Constant related to...
#'
#' @param mol_type character, "poly" or "oligo". If "oligo" then the calculated
#' constants are multiplied by c according to the considered condition (base,
#' water or acid).
#' @param exchange type of exchange - "HD" for hydrogen to deuterium,
#' "DH" for deuterium to hydrogen (back-exchange). Default "HD".
#'
#' @return list of Ka,Kb and Kw corresponding to the chosen \code{mol_type} and
#' acid, base or water.
#'
#' @keywords internal
get_poly_const = function(mol_type, exchange = "HD") {
    match.arg(mol_type, c("poly", "oligo"))
    match.arg(exchange, c("HD", "DH"))

    if (exchange == "HD") {
        Ka_exponent = 1.62
        Kb_exponent = 10.05
        Kw_exponent = -1.5
    } else {
        Ka_exponent = 1.4
        Kb_exponent = 9.87
        Kw_exponent = -1.6
    }
    Ka_poly = (10^(Ka_exponent)) / 60
    Kb_poly = (10^(Kb_exponent)) / 60
    Kw_poly = (10^(Kw_exponent)) / 60
    if (mol_type == "poly") {
        list(Ka = Ka_poly,
             Kb = Kb_poly,
             Kw = Kw_poly)
    } else {
        Kb_oligo = Kb_poly * 1.35
        Ka_oligo = Ka_poly * 2.34
        Kw_oligo = Kw_poly * 1.585
        list(Ka = Ka_oligo,
             Kb = Kb_oligo,
             Kw = Kw_oligo)
    }
}


#' Calculating pKc values
#'
#' @description calculates supplementary constants for aspartic acid (Asp),
#' glutamic acid (Glu) and histidine (His). Values for mentioned amino acids are
#' pH and temperature dependent, in contrary to the rest amino acids with fixed
#' values.
#'
#' @inheritParams get_F_const
#' @inheritParams get_poly_const
#'
#' @details Depending on provided \code{exchange} direction tabular values of
#' exponents E_{const} are assigned. For \code{Asp}, \code{Glu} and \code{His}
#' the \code{pKc} constants are calculated based on the energies of activation
#' for given amino acid and the chosen \code{exchange} direction.
#'
#' @return  The function returns a list of \code{asp}, \code{glu} and \code{his}
#' (\code{pKc} values corresponding to amino acids).
#'
#' @keywords internal

get_pkc <- function(temp_kelvin, gas_constant, exchange = "HD") {

    # Unit: cal / mol
    if(temp_kelvin == 0) stop("Temperature in Kelvin can not be 0.")
    match.arg(exchange, c("HD", "DH"))

    if (exchange == "HD") {

        Ea_Asp <- 1000
        Asp_exponent <- -4.48
        Glu_exponent <- -4.93
        His_exponent <- -7.42
    } else {

        Ea_Asp <- 960
        Asp_exponent <- -3.87
        Glu_exponent <- -4.33
        His_exponent <- -7
    }

    Ea_Glu <- 1083
    Ea_His <- 7500
    pKc_Asp <- -log10(10^(Asp_exponent)*exp(-1*Ea_Asp*((1/temp_kelvin-1/278)/gas_constant)))
    pKc_Glu <- -log10(10^(Glu_exponent)*exp(-1*Ea_Glu*((1/temp_kelvin-1/278)/gas_constant)))
    pKc_His <- -log10(10^(His_exponent)*exp(-1*Ea_His*((1/temp_kelvin - 1/278)/gas_constant)))
    list(asp = pKc_Asp,
         glu = pKc_Glu,
         his = pKc_His)
}


#' Exchange constant for Hydrogen-Deuterium Exchange.
#'
#' @param pH reaction pH
#' @param pkc_consts constants calculated via \code{\link[powerHaDeX]{get_pkc}}
#' @param k_consts constants calculated via
#' \code{\link[powerHaDeX]{get_poly_const}}
#'
#' @return a matrix named \code{constants} of tabular and calculated constants
#' (specifically for \code{Asp}, \code{Glu}, \code{His}, \code{C−Term} and
#' \code{NHMe})
#'
#' @keywords internal

get_exchange_constants <- function(pH, pkc_consts, k_consts) {

    constants <- matrix(
        c(0, 0, 0, 0,
          -0.59, -0.32, 0.0767122542818456, 0.22,
          0, 0, 0, 0, # Asp
          -0.90, -0.12, 0.69, 0.60,
          -0.58, -0.13, 0.49, 0.32,
          -0.54, -0.46, 0.62, 0.55,
          -0.74, -0.58, 0.55, 0.46,
          -0.22, 0.218176047120386, 0.267251569286023, 0.17,
          0, 0, 0, 0, # Glu
          -0.6, -0.27, 0.24, 0.39,
          -0.47, -0.27, 0.06, 0.2,
          0, 0, 0, 0, # His
          -0.91, -0.59, -0.73, -0.23,
          -0.57, -0.13, -0.576252727721677, -0.21,
          -0.56, -0.29, -0.040, 0.12,
          -0.64, -0.28, -0.00895484265292644, 0.11,
          -0.52, -0.43, -0.235859464059171, 0.0631315866300978,
          0, -0.194773472023435, 0, -0.24,
          0, -0.854416534276379, 0, 0.6,
          -0.437992277698594, -0.388518934646472, 0.37, 0.299550285605933,
          -0.79, -0.468073125742265, -0.0662579798400606, 0.20,
          -0.40, -0.44, -0.41, -0.11,
          -0.41, -0.37, -0.27, 0.050,
          -0.739022273362575, -0.30, -0.701934483299758, -0.14,
          0, -1.32,0,1.62,
          0, 0, -1.8, 0, #C-Term
          0, 0, 0, 0, # -NHMe
          0, 0.293, 0, -0.197
        ), ncol = 4, byrow = TRUE)

    constants[3,1] <- log10(10^(-0.9 - pH) / (10^(-pkc_consts[["asp"]]) + 10^(-pH)) + 10^(0.9 - pkc_consts[["asp"]]) / (10^(-pkc_consts[["asp"]]) + 10^(-pH)))
    constants[3,2] <- log10(10^(-0.12 - pH) / (10^(-pkc_consts[["asp"]]) + 10^(-pH)) + 10^(0.58 - pkc_consts[["asp"]]) / (10^(-pkc_consts[["asp"]]) + 10^(-pH)))
    constants[3,3] <- log10(10^(0.69 - pH) / (10^(-pkc_consts[["asp"]]) + 10^(-pH)) + 10^(-0.3 - pkc_consts[["asp"]]) / (10^(-pkc_consts[["asp"]]) + 10^(-pH)))
    constants[3,4] <- log10(10^(0.6 - pH) / (10^(-pkc_consts[["asp"]]) + 10^(-pH)) + 10^(-0.18 - pkc_consts[["asp"]]) / (10^(-pkc_consts[["asp"]]) + 10^(-pH)))

    constants[9,1] <- log10(10^(-0.6 - pH) / (10^(-pkc_consts[["glu"]]) + 10^(-pH)) + 10^(-0.9 - pkc_consts[["glu"]]) / (10^(-pkc_consts[["glu"]]) + 10^(-pH)))
    constants[9,2] <- log10(10^(-0.27 - pH) / (10^(-pkc_consts[["glu"]]) + 10^(-pH)) + 10^(0.31 - pkc_consts[["glu"]]) / (10^(-pkc_consts[["glu"]]) + 10^(-pH)))
    constants[9,3] <- log10(10^(0.24 - pH) / (10^(-pkc_consts[["glu"]]) + 10^(-pH)) + 10^(-0.51 - pkc_consts[["glu"]]) / (10^(-pkc_consts[["glu"]]) + 10^(-pH)))
    constants[9,4] <- log10(10^(0.39 - pH) / (10^(-pkc_consts[["glu"]]) + 10^(-pH)) + 10^(-0.15 - pkc_consts[["glu"]]) / (10^(-pkc_consts[["glu"]]) + 10^(-pH)))

    constants[12,1] <- log10(10^(-0.8 - pH) / (10^(-pkc_consts[["his"]]) + 10^(-pH)) + 10^(-pkc_consts[["his"]]) / (10^(-pkc_consts[["his"]]) + 10^(-pH)))
    constants[12,2] <- log10(10^(-0.51 - pH) / (10^(-pkc_consts[["his"]]) + 10^(-pH)) + 10^(-pkc_consts[["his"]]) / (10^(-pkc_consts[["his"]]) + 10^(-pH)))
    constants[12,3] <- log10(10^(0.8 - pH) / (10^(-pkc_consts[["his"]]) + 10^(-pH)) + 10^(-0.1 - pkc_consts[["his"]]) / (10^(-pkc_consts[["his"]]) + 10^(-pH)))
    constants[12,4] <- log10(10^(0.83 - pH) / (10^(-pkc_consts[["his"]]) + 10^(-pH)) + 10^(0.14 - pkc_consts[["his"]]) / (10^(-pkc_consts[["his"]]) + 10^(-pH)))

    constants[26,1] <- log10(10^(0.05 - pH) / (10^(-pkc_consts[["glu"]]) + 10^(-pH)) + 10^(0.96 - pkc_consts[["glu"]]) / (10^(-pkc_consts[["glu"]]) + 10^(-pH)))

    constants[27,1] <- log10(135.5 / (k_consts[["Ka"]] * 60))
    constants[27,3] <- log10(2970000000 / (k_consts[["Kb"]] * 60))

    constants

}


#' Hydrogen-deuterium or back-exchange exchange rates
#'
#' @description Calculate exchange rates that are required to obtain exchange
#' probabilities.
#'
#' @param sequence peptide amino acid sequence as a character vector of amino
#' acids
#' @param temperature temperature of the reaction (Celsius). Default to 15.
#' @param if_corr logical. PH correction indicator. Default value \code{FALSE}.
#' The value of pH is equal to pD. If there is correction, the pD = pH + 0.4.
#' (Conelly et al 1993)
#' @inheritParams get_poly_const
#' @inheritParams get_exchange_constants
#'
#' @details   The correction of \code{pH} is taken into account for calculation
#' of \code{pD}:
#' \deqn{pD = pH + 0.4 * if_corr}
#' Next, the provided temperature is converted into K and the internal functions
#' \code{\link[powerHaDeX]{get_F_const}},
#' \code{\link[powerHaDeX]{get_poly_const}} and
#' \code{\link[powerHaDeX]{get_pkc}} are evaluated.
#'
#' Using the obtained matrix of constants and provided \code{sequence}
#' \code{F_a} and \code{F_b} are calculated for each amino acid in the sequence,
#' concerning the previous and next amino acid. For the amino acids in the
#' middle of the sequence, the following formula is used:
#'
#' \deqn{F_x = 10^{ previous_x + current_x}}
#'
#' where \code{x} is either \code{a} or \code{b}, and \code{previous_x} is the
#' acid/base factor for a previous amino acid in the sequence, and
#' \code{current_x} for the amino acid it is calculated for. If the amino acid
#' is next to the C- or N-term, the term-effect is taken into account.
#'
#' Finally, the exchange rate \code{k_c} for the amino acid is the sum of
#' catalysis constants for acid, base and water (Conelly et al, 1993). Namely:
#'
#' \deqn{k_c = k_{acid} + k_{base} + k_{water}}
#' where
#' \deqn{k_{acid} = F_a * K_a + D * F_ta,}
#' \deqn{k_{base} = F_b* K_b + OD * F_tb,}
#' \deqn{k_{water} = F_b * K_w * F_tw}
#'
#' where \code{D} and \code{OD} indicates deuterium  and deuterium oxide
#' concentration, \code{F_a} and \code{F_b} are values calculated specifically
#' for given amino acid, as described before, \code{K_a} and \code{K_b} are
#' values computed by \code{\link[powerHaDeX]{get_poly_const}} function, based
#' on the mole type, \code{F_ta}, \code{F_tb} and \code{F_tw} are values
#' computed by \code{\link[powerHaDeX]{get_F_const}} function.
#'
#' @return The obtained exchange rates are stored in vector \code{kcHD} or
#' \code{kcDH} according to the exchange direction. They are used to calculate
#' the exchange probabilities thus both \code{kcHD} and \code{kcDH} are
#' necessary as we take the possibility of back-exchange into account.
#'
#' @keywords internal
#'
#' @export

get_exchange_rates <- function(sequence,
                               exchange = "HD",
                               pH = 9,
                               temperature = 15,
                               mol_type = "poly",
                               if_corr = FALSE) {
    assert(checkLogical(if_corr))
    assert(checkChoice(mol_type, c("poly", "oligo")))
    assert(checkChoice(exchange, c("HD", "DH")))
    assert(checkFALSE(temperature == -273.15))

    if (exchange == "HD") {
        pd <- pH + 0.4 * if_corr
        D <- 10^(-pd) # [D + ]
        OD <- 10^(pd - 15.05) # [OD - ]
    } else {
        D <- 10 ^ (-pH) # [D + ]
        OD <- 10 ^ (pH - 14.17) # [OD - ]
    }

    gas_constant <- 1.9858775
    temp_kelvin <- temperature + 273.15
    F_consts <- get_F_const(temp_kelvin, gas_constant)
    poly_consts <- get_poly_const(mol_type, exchange)
    pkc_consts <- get_pkc(temp_kelvin, gas_constant, exchange)

    AAs <- strsplit('ARDdNCsGEeQHILKMFPpSTWYVncma', "")[[1]]
    constants <- get_exchange_constants(pH, pkc_consts, poly_consts)
    sequence <- c("n", sequence, "c")
    N <- length(sequence)

    if (N <= 2) stop("Length of sequence must be greater than 0")


    kcDH = rep(0, N)

    for (i in 1:N) {

        if (i == 1 || sequence[i] == 'P' || sequence[i] == 'p' || sequence[i] == 'a') {

            next()
        } else {

            if (i %in% c(1, 2, N) || sequence[i] %in% c("P", "a", "p")) {

                Fa <- 0
                Fb <- 0
            } else {

                j <- which(AAs == sequence[i])
                k <- which(AAs == sequence[i - 1])

                if (i <= 3 && i + 1 == N && sequence[i - 1] != "a" && sequence[i] != "m") {

                    Fa <- 10 ^ (constants[j, 1] + constants[k, 2] + constants[25, 2] + constants[26, 1])
                    Fb <- 10 ^ (constants[j, 3] + constants[k, 4] + constants[25, 4] + constants[26, 3])
                } else {

                    if (i <= 3 && sequence[i - 1] != "a") {

                        Fa <- 10 ^ (constants[j, 1] + constants[k, 2] + constants[25, 2])
                        Fb <- 10 ^ (constants[j, 3] + constants[k, 4] + constants[25, 4])
                    } else {

                        if (i + 1 == N && sequence[i] != "m") {

                            Fa <- 10 ^ (constants[j, 1] + constants[k, 2] + constants[26, 1])
                            Fb <- 10 ^ (constants[j, 3] + constants[k, 4] + constants[26, 3])
                        } else {

                            Fa <- 10^(constants[j, 1] + constants[k, 2])
                            Fb <- 10^(constants[j, 3] + constants[k, 4])
                        }
                    }
                }
            }
            kcDH[i] <- Fa * poly_consts[["Ka"]] * D * F_consts[["Fta"]] +
                Fb * poly_consts[["Kb"]] * OD * F_consts[["Ftb"]] +
                Fb * poly_consts[["Kw"]] * F_consts[["Ftw"]]
        }
    }
    kcDH[2:(N - 1)]
}