Table 2 Hierarchical model equations of the SRM analytical chain for the native transition signals I and labeled transition signals I*

Protein concentration

y _p

No labeled protein introduced

p = 1, …, P

Peptide concentration before chromatography

\( {\displaystyle \begin{array}{c}{H}_i(y)=\sum \limits_{p=1}^P{g}_{ip}{d}_{ip}{y}_p\\ {}{\kappa}_i={H}_i(y)+N\left({\gamma}_{\kappa}\right)\end{array}} \)

\( {\kappa}_i^{\ast } \)

i = 1, …, S

Selected ion concentration before fragmentation

ξ _i κ _i

\( {\xi}_i{\kappa}_i^{\ast } \)

i = 1, …, S

Signal of transition at time t _n

\( {\kappa}_i{\xi}_{il}{C}_{il}^T\left({\tau}_i,{\lambda}_i\right)\left({t}_n\right) \)

\( {\kappa}_i^{\ast }{\xi}_{il}{\phi}_{il}^{\ast }{C}_{il}^T\left({\tau}_i,{\lambda}_i\right)\left({t}_n\right) \)

i = 1, …, S

l = 1, …, L

n = 1, …, N

Resulting signals of selected children of targeted peptide^a

\( {\displaystyle \begin{array}{c}{G}_l\left(\boldsymbol{\kappa}, \boldsymbol{\xi}, \boldsymbol{\tau}, \boldsymbol{\lambda} \right)=\sum \limits_{i=1}^S\sum \limits_{l=1}^L{\kappa}_i{\xi}_{il}{C}_{il}^T\left({\tau}_i,{\lambda}_i\right)\\ {}{I}_l={G}_l\left(\boldsymbol{\kappa}, \boldsymbol{\xi}, \boldsymbol{\tau} \right)+N\left({\gamma}_{nl}\right)\end{array}} \)

\( {\displaystyle \begin{array}{c}{G}_l^{\ast}\left({\boldsymbol{\kappa}}^{\ast},\boldsymbol{\xi}, {\boldsymbol{\phi}}^{\ast},\boldsymbol{\tau}, \boldsymbol{\lambda} \right)=\sum \limits_{i=1}^S\sum \limits_{l=1}^L{\kappa}_i^{\ast }{\xi}_{il}{\phi}_{il}^{\ast }{C}_{il}^T\left({\tau}_i,{\lambda}_i\right)\\ {}{I}_l^{\ast }={G}_l^{\ast}\left({\boldsymbol{\kappa}}^{\ast},\boldsymbol{\xi}, {\boldsymbol{\phi}}^{\ast},\boldsymbol{\tau} \right)+N\left({\gamma}_{nl}^{\ast}\right)\end{array}} \)

i = 1, …, S

l = 1, …, L